Addition - Knowledge

6356: 5637: 6351:{\displaystyle {\begin{aligned}\mathbf {A} +\mathbf {B} &={\begin{bmatrix}a_{11}&a_{12}&\cdots &a_{1n}\\a_{21}&a_{22}&\cdots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\cdots &a_{mn}\\\end{bmatrix}}+{\begin{bmatrix}b_{11}&b_{12}&\cdots &b_{1n}\\b_{21}&b_{22}&\cdots &b_{2n}\\\vdots &\vdots &\ddots &\vdots \\b_{m1}&b_{m2}&\cdots &b_{mn}\\\end{bmatrix}}\\&={\begin{bmatrix}a_{11}+b_{11}&a_{12}+b_{12}&\cdots &a_{1n}+b_{1n}\\a_{21}+b_{21}&a_{22}+b_{22}&\cdots &a_{2n}+b_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}+b_{m1}&a_{m2}+b_{m2}&\cdots &a_{mn}+b_{mn}\\\end{bmatrix}}\\\end{aligned}}} 3323: 1757: 2222:, they must be expressed with common units . For example, adding 50 milliliters to 150 milliliters gives 200 milliliters. However, if a measure of 5 feet is extended by 2 inches, the sum is 62 inches, since 60 inches is synonymous with 5 feet. On the other hand, it is usually meaningless to try to add 3 meters and 4 square meters, since those units are incomparable; this sort of consideration is fundamental in 5084: 1765: 1280: 1619: 5234: 1721: 1882: 3247: 2059: 1937: 386: 3376: 47: 1127: 7016: 6616: 739: 5225:, also in 1872, although his formalism was slightly different. One must prove that this operation is well-defined, dealing with co-Cauchy sequences. Once that task is done, all the properties of real addition follow immediately from the properties of rational numbers. Furthermore, the other arithmetic operations, including multiplication, have straightforward, analogous definitions. 2321:" (usually ticking off fingers), and arriving at five. This strategy seems almost universal; children can easily pick it up from peers or teachers. Most discover it independently. With additional experience, children learn to add more quickly by exploiting the commutativity of addition by counting up from the larger number, in this case, starting with three and counting "four, 6824: 2317:. When given a problem that requires that two items and three items be combined, young children model the situation with physical objects, often fingers or a drawing, and then count the total. As they gain experience, they learn or discover the strategy of "counting-on": asked to find two plus three, children count three past two, saying "three, four, 272: 3184:. When the result of an addition exceeds the value of a digit, the procedure is to "carry" the excess amount divided by the radix (that is, 10/10) to the left, adding it to the next positional value. This is correct since the next position has a weight that is higher by a factor equal to the radix. Carrying works the same way in binary: 616: 6367: 661: 501: 889: 804: 4547:

A straightforward computation shows that the equivalence class of the result depends only on the equivalences classes of the summands, and thus that this defines an addition of equivalence classes, that is integers. Another straightforward computation shows that this addition is the same as the above

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One possible fix is to consider collections of objects that can be easily divided, such as pies or, still better, segmented rods. Rather than solely combining collections of segments, rods can be joined end-to-end, which illustrates another conception of addition: adding not the rods but the lengths

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in 1872. The commutativity and associativity of real addition are immediate; defining the real number 0 to be the set of negative rationals, it is easily seen to be the additive identity. Probably the trickiest part of this construction pertaining to addition is the definition of additive inverses.

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can be added by a simple modification of the above process. One aligns two decimal fractions above each other, with the decimal point in the same location. If necessary, one can add trailing zeros to a shorter decimal to make it the same length as the longer decimal. Finally, one performs the same

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Given a set with an addition operation, one cannot always define a corresponding subtraction operation on that set; the set of natural numbers is a simple example. On the other hand, a subtraction operation uniquely determines an addition operation, an additive inverse operation, and an additive

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Although this definition can be useful for concrete problems, the number of cases to consider complicates proofs unnecessarily. So the following method is commonly used for defining integers. It is based on the remark that every integer is the difference of two natural integers and that two such

6636:, the set of available numbers is restricted to a finite subset of the integers, and addition "wraps around" when reaching a certain value, called the modulus. For example, the set of integers modulo 12 has twelve elements; it inherits an addition operation from the integers that is central to 3168:

Adding two "1" digits produces a digit "0", while 1 must be added to the next column. This is similar to what happens in decimal when certain single-digit numbers are added together; if the result equals or exceeds the value of the radix (10), the digit to the left is incremented:

7979:(p. 73) compares adding measuring rods to adding sets of cats: "For example, inches can be subdivided into parts, which are hard to tell from the wholes, except that they are shorter; whereas it is painful to cats to divide them into parts, and it seriously changes their nature." 4810: 6931:. In some contexts, such as the integers, distributivity over addition and the existence of a multiplicative identity is enough to uniquely determine the multiplication operation. The distributive property also provides information about addition; by expanding the product 381:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,} 5095:

Unfortunately, dealing with multiplication of Dedekind cuts is a time-consuming case-by-case process similar to the addition of signed integers. Another approach is the metric completion of the rational numbers. A real number is essentially defined to be the limit of a

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7 + 9 = 16, and the digit 1 is the carry. An alternate strategy starts adding from the most significant digit on the left; this route makes carrying a little clumsier, but it is faster at getting a rough estimate of the sum. There are many alternative methods.

3135: 6611:{\displaystyle {\begin{bmatrix}1&3\\1&0\\1&2\end{bmatrix}}+{\begin{bmatrix}0&0\\7&5\\2&1\end{bmatrix}}={\begin{bmatrix}1+0&3+0\\1+7&0+5\\1+2&2+1\end{bmatrix}}={\begin{bmatrix}1&3\\8&5\\3&3\end{bmatrix}}} 734:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,} 429: 2884:

Since the end of the 20th century, some US programs, including TERC, decided to remove the traditional transfer method from their curriculum. This decision was criticized, which is why some states and counties did not support this experiment.

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circuits which in turn may be combined into more complex logical operations. In modern digital computers, integer addition is typically the fastest arithmetic instruction, yet it has the largest impact on performance, since it underlies all

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Different nations introduce whole numbers and arithmetic at different ages, with many countries teaching addition in pre-school. However, throughout the world, addition is taught by the end of the first year of elementary school.

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work directly with physical quantities, so their addition mechanisms depend on the form of the addends. A mechanical adder might represent two addends as the positions of sliding blocks, in which case they can be added with an

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Maximization is commutative and associative, like addition. Furthermore, since addition preserves the ordering of real numbers, addition distributes over "max" in the same way that multiplication distributes over addition:

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one replaces multiplication with addition and addition with maximization. In this context, addition is called "tropical multiplication", maximization is called "tropical addition", and the tropical "additive identity" is

1056: 4952: 4563:. Here, the semigroup is formed by the natural numbers and the group is the additive group of integers. The rational numbers are constructed similarly, by taking as semigroup the nonzero integers with multiplication. 970: 4118:

This recursive formulation of addition was developed by Dedekind as early as 1854, and he would expand upon it in the following decades. He proved the associative and commutative properties, among others, through

1502: 5216: 2329:"), either through experience or rote memorization. Once some facts are committed to memory, children begin to derive unknown facts from known ones. For example, a child asked to add six and seven may know that 7551:. Its usual definition combines integration, subtraction, and multiplication. In general, convolution is useful as a kind of domain-side addition; by contrast, vector addition is a kind of range-side addition. 4679: 4687: 2852:

As students grow older, they commit more facts to memory, and learn to derive other facts rapidly and fluently. Many students never commit all the facts to memory, but can still find any basic fact quickly.

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Some authors think that "carry" may be inappropriate for education; Van de Walle (p. 211) calls it "obsolete and conceptually misleading", preferring the word "trade". However, "carry" remains the standard

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bitwise logical operations in conjunction with bitshift operations as shown in the pseudocode below. Both XOR and AND gates are straightforward to realize in digital logic allowing the realization of

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Addition in other bases is very similar to decimal addition. As an example, one can consider addition in binary. Adding two single-digit binary numbers is relatively simple, using a form of carrying:

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The standard algorithm for adding multidigit numbers is to align the addends vertically and add the columns, starting from the ones column on the right. If a column exceeds nine, the extra digit is "

3338:, also called a counting frame, is a calculating tool that was in use centuries before the adoption of the written modern numeral system and is still widely used by merchants, traders and clerks in 7299: 2785:: Since zero is the additive identity, adding zero is trivial. Nonetheless, in the teaching of arithmetic, some students are introduced to addition as a process that always increases the addends; 1356: 611:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,} 5642: 526: 7898:"introduces the novelty of writing the sum above the addends"; it is unclear whether Karpinski is claiming this as an original invention or simply the introduction of the practice to Europe. 3834:. Such overflow bugs may be hard to discover and diagnose because they may manifest themselves only for very large input data sets, which are less likely to be used in validation tests. The 1081: 5070: 1736:

When two or more disjoint collections are combined into a single collection, the number of objects in the single collection is the sum of the numbers of objects in the original collections.

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Schubert, E. Thomas, Phillip J. Windley, and James Alves-Foss. "Higher Order Logic Theorem Proving and Its Applications: Proceedings of the 8th International Workshop, volume 971 of."

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is the exponential part. Addition requires two numbers in scientific notation to be represented using the same exponential part, so that the two significands can simply be added.

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Baez (p. 37) explains the historical development, in "stark contrast" with the set theory presentation: "Apparently, half an apple is easier to understand than a negative apple!"

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is one more, or 13. Such derived facts can be found very quickly and most elementary school students eventually rely on a mixture of memorized and derived facts to add fluently.

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of those values combined. The example in the adjacent image shows two columns of three apples and two apples each, totaling at five apples. This observation is equivalent to the

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pseudocarry. Many implementations are, in fact, hybrids of these last three designs. Unlike addition on paper, addition on a computer often changes the addends. On the ancient

2958: 496:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,} 3360:. It made use of a gravity-assisted carry mechanism. It was the only operational mechanical calculator in the 17th century and the earliest automatic, digital computer. 6712:. Unlike most addition operations, addition of ordinal numbers is not commutative. Addition of cardinal numbers, however, is a commutative operation closely related to the 3372:

followed Pascal, building the second functional mechanical calculator in 1709, a calculating clock made of wood that, once setup, could multiply two numbers automatically.

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A number-line visualization of the algebraic addition 2 + 4 = 6. A "jump" that has a distance of 2 followed by another that is long as 4, is the same as a translation by 6.

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Textbook constructions are usually not so cavalier with the "lim" symbol; see Burrill (p. 138) for a more careful, drawn-out development of addition with Cauchy sequences.

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Proceedings of the International Congress of Mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume II: Invited lectures. Tropical Geometry and its Applications

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One set has 3 shapes while the other set has 2. The total amount of shapes are 5, which is a consequence of the addition of the objects from the two sets (3 + 2 = 5).

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Akian, Marianne; Bapat, Ravindra; Gaubert, Stephane (2005). "Min-plus methods in eigenvalue perturbation theory and generalised Lidskii-Vishik-Ljusternik theorem".

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The commutativity and associativity of rational addition is an easy consequence of the laws of integer arithmetic. For a more rigorous and general discussion, see

4265:; because −6 and 4 have different signs, their absolute values are subtracted, and since the absolute value of the negative term is larger, the answer is negative. 884:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{base}}^{\text{exponent}}\\\scriptstyle {\text{base}}^{\text{power}}\end{matrix}}\right\}\,=\,} 8532: 799:{\displaystyle \scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.} 1740:

This interpretation is easy to visualize, with little danger of ambiguity. It is also useful in higher mathematics (for the rigorous definition it inspires, see

2978: 7065:, then their sum is approximately equal to their maximum. This approximation is extremely useful in the applications of mathematics, for example in truncating 7467:

describes the addition of arbitrarily many numbers, usually more than just two. It includes the idea of the sum of a single number, which is itself, and the

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Beckmann, S. (2014). The twenty-third ICMI study: primary mathematics study on whole numbers. International Journal of STEM Education, 1(1), 1-8. Chicago

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Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers

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The intuitive approach, inverting every element of a cut and taking its complement, works only for irrational numbers; see Enderton p. 117 for details.

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The verifications are carried out in Enderton p. 104 and sketched for a general field of fractions over a commutative ring in Dummit and Foote p. 263.

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Department of the Army (1961) Army Technical Manual TM 11-684: Principles and Applications of Mathematics for Communications-Electronics . Section 5.1

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and adding board, both addends are destroyed, leaving only the sum. The influence of the abacus on mathematical thinking was strong enough that early

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is either 1 or 3. This finding has since been affirmed by a variety of laboratories using different methodologies. Another 1992 experiment with older

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Dummit and Foote p. 224. For this argument to work, one still must assume that addition is a group operation and that multiplication has an identity.

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wrote, "In the addition of cipher, or subtraction of it, the quantity, positive or negative, remains the same", corresponding to the unary statement

1013: 10068: 10027: 9581: 4883: 4099:. On the other hand, some sources prefer to use a restricted recursion theorem that applies only to the set of natural numbers. One then considers 2349:

Children are often presented with the addition table of pairs of numbers from 0 to 9 to memorize. Knowing this, children can perform any addition.

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Bronstein, Ilja Nikolaevič; Semendjajew, Konstantin Adolfovič (1987) . "2.4.1.1.". In Grosche, Günter; Ziegler, Viktor; Ziegler, Dorothea (eds.).

4153:). The integer zero is a special third case, being neither positive nor negative. The corresponding definition of addition must proceed by cases: 10244: 9124: 8161: 1108: 244: 196:

Performing addition is one of the simplest numerical tasks to do. Addition of very small numbers is accessible to toddlers; the most basic task,

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Addition and Subtraction: A Cognitive Perspective. Interpretations of Number Operations and Symbolic Representations of Addition and Subtraction

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Enderton calls this statement the "Absorption Law of Cardinal Arithmetic"; it depends on the comparability of cardinals and therefore on the

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A common construction of the set of real numbers is the Dedekind completion of the set of rational numbers. A real number is defined to be a

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number. Linear combinations are especially useful in contexts where straightforward addition would violate some normalization rule, such as

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was limited by its carry mechanism, which forced its wheels to only turn one way so it could add. To subtract, the operator had to use the

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When an original length is extended by a given amount, the final length is the sum of the original length and the length of the extension.

4805:{\displaystyle {\frac {3}{4}}+{\frac {1}{8}}={\frac {3\times 8+4\times 1}{4\times 8}}={\frac {24+4}{32}}={\frac {28}{32}}={\frac {7}{8}}} 4088:

Again, there are minor variations upon this definition in the literature. Taken literally, the above definition is an application of the

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texts often claimed that in the process of adding "a number to a number", both numbers vanish. In modern times, the ADD instruction of a

3251: 7121:, their cardinal sum is exactly equal to the greater of the two. Accordingly, there is no subtraction operation for infinite cardinals. 4609: 3896:

of finite sets, (the cardinality of a set is the number of elements in the set), then it is appropriate to define their sum as follows:

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To prove the usual properties of addition, one must first define addition for the context in question. Addition is first defined on the

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European Congress of Mathematics: Barcelona, July 10–14, 2000, Volume I. Dequantization of Real Algebraic Geometry on Logarithmic Paper

3420:. The simplest architecture is the ripple carry adder, which follows the standard multi-digit algorithm. One slight improvement is the 3216:. The 1 is carried to the left, and the 0 is written at the bottom of the rightmost column. The second column from the right is added: 8594:

According to a survey of the nations with highest TIMSS mathematics test scores; see Schmidt, W., Houang, R., & Cogan, L. (2002).

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The fact that addition is commutative is known as the "commutative law of addition" or "commutative property of addition". Some other

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does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication.

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was, more specifically, the result of this construction applied to the equivalences classes under isomorphisms of the objects of an

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below). However, it is not obvious how one should extend this version of addition to include fractional numbers or negative numbers.

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There are many binary operations that can be viewed as generalizations of the addition operation on the real numbers. The field of

179:, meaning that when one adds more than two numbers, the order in which addition is performed does not matter. Repeated addition of 9275: 8719: 7304:

which becomes more accurate as the base of the logarithm increases. The approximation can be made exact by extracting a constant

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in both ways, one concludes that addition is forced to be commutative. For this reason, ring addition is commutative in general.

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operation, and general coproducts are perhaps the most abstract of all the generalizations of addition. Some coproducts, such as

1641:. This is appropriate not only because the sum of two positive numbers is greater than either, but because it was common for the 8491:

Algorithms and Architectures for Parallel Processing: 10th International Conference, ICA3PP 2010, Busan, Korea, May 21–23, 2010

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system, starting with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient

10169: 10146: 10085: 10049: 10008: 9987: 9951: 9904: 9759: 9738: 9715: 9691: 9672: 9653: 9622: 9598: 9562: 9532: 9513: 9490: 9469: 9450: 9427: 9350: 9322: 9301: 9265: 9234: 9200: 9172: 9149: 9102: 8869: 8823: 8795: 8426: 8213: 8100: 8038: 7813: 7678: 8234:"First-grade basic facts: An investigation into teaching and learning of an accelerated, high-demand memorization standard" 7230: 6785: 2786: 2897:

addition process as above, except the decimal point is placed in the answer, exactly where it was placed in the summands.

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balls from a box; the youngest responded well for small numbers, while older subjects were able to compute sums up to 5.

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Addition, along with subtraction, multiplication and division, is considered one of the basic operations and is used in

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A number-line visualization of the unary addition 2 + 4 = 6. A translation by 4 is equivalent to four translations by 1.

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to add upward, contrary to the modern practice of adding downward, so that a sum was literally higher than the addends.

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and for scaling vectors. A familiar vector space is the set of all ordered pairs of real numbers; the ordered pair (

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There are even more generalizations of multiplication than addition. In general, multiplication operations always

8459: 8025:(in German). Vol. 1. Translated by Ziegler, Viktor. Weiß, Jürgen (23 ed.). Thun and Frankfurt am Main: 7570: 5247: 3489: 895: 7131: 3870:, positive fractions are added before negative numbers are even considered; this is also the historical route.) 3130:{\displaystyle 2.34\times 10^{-5}+5.67\times 10^{-6}=2.34\times 10^{-5}+0.567\times 10^{-5}=2.907\times 10^{-5}} 976: 10586: 10581: 10230: 3457: 1893:, meaning that one can change the order of the terms in a sum, but still get the same result. Symbolically, if 1094: 392: 230: 9922:

Viro, Oleg (2001). Cascuberta, Carles; Miró-Roig, Rosa Maria; Verdera, Joan; Xambó-Descamps, Sebastià (eds.).

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Tying these observations together, tropical addition is approximately related to regular addition through the

7217:. Some authors prefer to replace addition with minimization; then the additive identity is positive infinity. 4551:

This way of defining integers as equivalence classes of pairs of natural numbers, can be used to embed into a

2188:. For instance, 3 is the successor of 2 and 7 is the successor of 6. Because of this succession, the value of 10540: 10384: 9886: 8062:

Hempel, C.G. (2001). The philosophy of Carl G. Hempel: studies in science, explanation, and rationality. p. 7

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Hosch, W.L. (Ed.). (2010). The Britannica Guide to Numbers and Measurement. The Rosen Publishing Group. p. 38

6916:, where it relates multiplication of infinitesimal group elements with addition of vectors in the associated 8846:

Lipschutz, S., & Lipson, M. (2001). Schaum's outline of theory and problems of linear algebra. Erlangga.

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See Viro 2001 for an example of the sophistication involved in adding with sets of "fractional cardinality".

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is often taken to be their sum as real numbers modulo 2π. This amounts to an addition operation on the

5482: 4464: 10555: 10550: 2270:, between 18 and 35 months, exploited their development of motor control by allowing them to retrieve 8647:

Enderton (p. 79) observes, "But we want one binary operation +, not all these little one-place functions."

6640:. The set of integers modulo 2 has just two elements; the addition operation it inherits is known in 2912: 8902:

The set still must be nonempty. Dummit and Foote (p. 48) discuss this criterion written multiplicatively.

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identity; for this reason, an additive group can be described as a set that is closed under subtraction.

200:, can be performed by infants as young as five months, and even some members of other animal species. In 3830:

occurs, resulting in an incorrect answer. Unanticipated arithmetic overflow is a fairly common cause of

2799:. Doubles facts form a backbone for many related facts, and students find them relatively easy to grasp. 2293:

monkeys performed similarly to human infants. More dramatically, after being taught the meanings of the

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he may mean a great variety of things, but not so great a variety as he will mean by 'multiplication'."

8635: 8153: 7894:, saying it was about as common as adding downwards. On the other hand, Karpinski (p. 103) writes that 3282: 10303: 4819:

are the same; in this case, one can simply add the numerators while leaving the denominator the same:

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is 8, because 8 is the successor of 7, which is the successor of 6, making 8 the 2nd successor of 6.

1991:)? Given that addition is associative, the choice of definition is irrelevant. For any three numbers 1928:

are commutative, such as multiplication, but many others, such as subtraction and division, are not.

1524:; this terminology carries over to the summation of multiple terms. This is to be distinguished from 62: 31: 10377: 7548: 7505:

combine multiplication and summation; they are sums in which each term has a multiplier, usually a

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Complex numbers are added by adding the real and imaginary parts of the summands. That is to say:

9112: 7946: 7496: 1666: 69: 7656: 3951: 2775:: Adding 1 or 2 is a basic task, and it can be accomplished through counting on or, ultimately, 1230: 10509: 10473: 10344: 10339: 10119: 10113: 7518: 7492: 6946: 4571: 4120: 3854:, addition is then extended to progressively larger sets that include the natural numbers: the 3361: 2239:

Studies on mathematical development starting around the 1980s have exploited the phenomenon of

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The numbers or the objects to be added in general addition are collectively referred to as the

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Histoire des Instruments et Machines à Calculer, Trois Siècles de Mécanique Pensante 1642–1942

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in set theory. These give two different generalizations of addition of natural numbers to the

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was a series of bugs where overflow errors occurred due to use of a 2-digit format for years.

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system is the fluent recall or derivation of the 100 single-digit "addition facts". One could

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is negative, positive, or zero itself, and he used words rather than algebraic symbols. Later

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A second interpretation of addition comes from extending an initial length by a given length:

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Columnar addition – the numbers in the column are to be added, with the sum written below the

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to be 2, and they are comparatively surprised when a physical situation seems to imply that

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wrote, "zero becomes the same as what is added to it", corresponding to the unary statement

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The Development of Arithmetic Concepts and Skills. Two perspectives on addition development

9939: 9793: 9257: 8815: 8026: 7784: 7660: 7476: 7102: 6832: 5476:) in the plane. The sum of two vectors is obtained by adding their individual coordinates: 4602:, but a conceptually simpler definition involves only integer addition and multiplication: 4317: 4089: 4021:, a mechanism that allows common elements to be separated out and therefore counted twice. 3424:

design, again following human intuition; one does not perform all the carries in computing

3294: 2758:, but pattern-based strategies are more enlightening and, for most people, more efficient: 2223: 2219: 1541: 1371: 1152: 1135: 168: 153: 38: 9961: 9914: 9136:

The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English

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Addition of two complex numbers can be done geometrically by constructing a parallelogram.

2841:: An advanced strategy uses 10 as an intermediate for sums involving 8 or 9; for example, 2042:

becomes important. In the standard order of operations, addition is a lower priority than

216:, where research on the most efficient implementations of addition continues to this day. 148:, another area of mathematics, addition can also be performed on abstract objects such as 8: 10324: 9389: 7062: 6681: 5566: 4552: 3827: 3440: 3436: 3413: 3380: 2918: 2039: 1949: 1359: 1291:

There are also situations where addition is "understood", even though no symbol appears:

9943: 9797: 8530:"Extra, Extra – Read All About It: Nearly All Binary Searches and Mergesorts are Broken" 8191: 6831:

In the real and complex numbers, addition and multiplication can be interchanged by the

1864:, because each unary addition operation has an inverse unary subtraction operation, and 10535: 10414: 10158: 9929: 9890: 9859: 9813: 9783: 9502: 9439: 9251: 9220: 9161: 9134: 9091: 8255: 7918: 7778: 7565: 7502: 7454: 7070: 6928: 6901: 6709: 6685: 6637: 6633: 6627: 4959: 4567: 4142: 3998: 3465: 3453: 3421: 3316: 3301: 2963: 2868: 2862: 2181: 1540:, many authors did not consider the first addend an "addend" at all. Today, due to the 185: 9873: 9279: 8716: 1708:

Addition is used to model many physical processes. Even for the simple case of adding

10576: 10209: 10192: 10165: 10142: 10123: 10081: 10045: 10004: 9998: 9983: 9947: 9923: 9900: 9823: 9755: 9749: 9734: 9727: 9711: 9687: 9668: 9649: 9628: 9618: 9594: 9558: 9528: 9509: 9486: 9465: 9446: 9423: 9365: 9346: 9318: 9312: 9297: 9261: 9230: 9196: 9168: 9145: 9098: 8865: 8858: 8819: 8791: 8422: 8209: 8193: 8096: 8034: 7809: 7788: 7674: 7575: 7534: 7313: 7209: 5457: 4309: 3835: 3417: 3327: 2893: 2301:

was able to compute the sum of two numerals without further training. More recently,

2298: 2290: 2075: 1729: 201: 9552: 7728:

Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers

10454: 10064: 10023: 9957: 9910: 9869: 9577: 9289: 9120: 8582: 8245: 7694: 7666: 7560: 6817: 6771: 6759: 5595:

Matrix addition is defined for two matrices of the same dimensions. The sum of two

5418: 5076: 4986: 4579: 4150: 2827:

are usually memorized early and can be used for deriving other facts. For example,

2071: 1925: 1789: 1674: 1560: 1544:

of addition, "augend" is rarely used, and both terms are generally called addends.

149: 9343:

Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction

7053:)" is a binary operation similar to addition. In fact, if two nonnegative numbers 5468:) is interpreted as a vector from the origin in the Euclidean plane to the point ( 3346:, and elsewhere; it dates back to at least 2700–2300 BC, when it was used in 2115:

in 628 AD, although he wrote it as three separate laws, depending on whether

10519: 10035: 8974: 8723: 8536: 8416: 7895: 7544: 7510: 7317: 7309: 7118: 7097: 6813: 6720: 6705: 5590: 5443: 5426: 5421:

is centrally concerned with such generalized operations, and they also appear in

5097: 4595: 4146: 3893: 3859: 3369: 3263: 3258: 3146: 2294: 1829: 157: 121: 9093:

Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics

7935: 7751: 7670: 6912:. The formula is still a good first-order approximation in the broad context of 6700:

A far-reaching generalization of addition of natural numbers is the addition of

6672:

The general theory of abstract algebra allows an "addition" operation to be any

4570:

to the case of any commutative semigroup. Without the cancellation property the

3289:. The most common situation for a general-purpose analog computer is to add two 1712:, there are many possible interpretations and even more visual representations. 1700:, meaning "and". It appears in mathematical works dating back to at least 1489. 10429: 10424: 10400: 10319: 10314: 10137:

Davison, David M.; Landau, Marsha S.; McCracken, Leah; Thompson, Linda (1999).

9928:. Progress in Mathematics. Vol. 201. Basel: Birkhäuser. pp. 135–146. 9777: 9645: 7887: 7514: 6924: 6809: 6781: 6713: 6701: 5449: 4138: 4018: 3879: 3847: 3485: 3449: 3357: 2796: 2302: 2286: 2043: 1709: 1670: 1669:; Boethius also used several other terms for the addition operation. The later 1642: 1529: 1363: 1139: 811: 532: 137: 89: 81: 10180: 7833: 7808:(reprint of 1st ed.). Malabar, FL: Robert E. Krieger Publishing Company. 2139: 1051:{\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,} 10570: 10349: 10204:

Weaver, J. Fred (1982). "Addition and Subtraction: A Cognitive Perspective".

10196: 9810:

Mathematics Unlimited – 2001 and Beyond. From Finite Sets to Feynman Diagrams

9632: 9413: 8955:

For an example of left and right distributivity, see Loday, especially p. 15.

8002: 7530: 7066: 6689: 6641: 5358: 4982: 4947:{\displaystyle {\frac {1}{4}}+{\frac {2}{4}}={\frac {1+2}{4}}={\frac {3}{4}}} 3514:

with the sum this must be explicitly requested, typically with the statement

3353: 2755: 1945: 176: 7595:"Addend" is not a Latin word; in Latin it must be further conjugated, as in 4261:|, with the sign of the term whose absolute value is larger. As an example, 3209:). The top row shows the carry bits used. Starting in the rightmost column, 1552: 9805: 8550: 8511:

The identity of the augend and addend varies with architecture. For ADD in

8418:

The Universal History of Computing: From the Abacus to the Quantum Computer

8197: 7891: 7645:= 3. Sets of fingers are handy; sets of apples are preferred by textbooks." 7506: 7019: 6645: 5574: 5453: 5222: 4978: 4313: 3831: 3461: 3432: 3223:

again; the 1 is carried, and 0 is written at the bottom. The third column:

2255:

dolls manipulated behind a screen demonstrated that five-month-old infants

2252: 2050:, multiplication and division, but is given equal priority to subtraction. 1646: 9612: 8281:

Schmidt, W., Houang, R., & Cogan, L. (2002). "A coherent curriculum".

3322: 10504: 10468: 10298: 10293: 9701: 9226: 7540: 7522: 6917: 6755: 6677: 6673: 4816: 3863: 3356:

invented the mechanical calculator in 1642; it was the first operational

2326: 2240: 2106: 1890: 1861: 1537: 1143: 965:{\displaystyle \scriptstyle {\sqrt{\scriptstyle {\text{radicand}}}}\,=\,} 417: 141: 133: 77: 10222: 9895: 9837: 9818: 9788: 9705: 8259: 4574:

from the semigroup into the group may be non-injective. Originally, the

3416:

execute integer addition in electronic digital computers, usually using

1756: 10483: 10328: 8933:, properly speaking, a mathematician may mean practically anything. By 7472: 7458: 7109:

The approximation becomes exact in a kind of infinite limit; if either

6909: 6728: 6660:, which in turn generalizes to addition operations on many-dimensional 5422: 4583: 3851: 3473: 2805:: Sums such as 6 + 7 = 13 can be quickly derived from the doubles fact 2277:

Even some nonhuman animals show a limited ability to add, particularly

2248: 2247:

look longer at situations that are unexpected. A seminal experiment by

2161: 2070:

to any number, does not change the number; this means that zero is the

2067: 190: 180: 73: 6900:

This identity allows multiplication to be carried out by consulting a

6762:. Subtraction is itself a sort of inverse to addition, in that adding 3826:

On a computer, if the result of an addition is too large to store, an

3488:

often replaces the augend with the sum but preserves the addend. In a

1948:, which means that when three or more numbers are added together, the 1497:{\displaystyle \sum _{k=1}^{5}k^{2}=1^{2}+2^{2}+3^{2}+4^{2}+5^{2}=55.} 120:

items, addition can also be defined and executed without referring to

10514: 10439: 10434: 9934: 9864: 9608: 8250: 8233: 7757: 7499:. Integration over a zero-dimensional manifold reduces to summation. 7468: 7464: 7221: 6927:

over addition; this requirement is formalized in the definition of a

6913: 6905: 6732: 6724: 5211:{\displaystyle \lim _{n}a_{n}+\lim _{n}b_{n}=\lim _{n}(a_{n}+b_{n}).} 5083: 5075:

This definition was first published, in a slightly modified form, by

4556: 2271: 1800:, in an algebraic sense, or it can be interpreted as the addition of 1728:

Possibly the most basic interpretation of addition lies in combining

1681: 1662: 1602: 1379: 1284: 1001: 94: 7917:

Cajori, Florian (1928). "Origin and meanings of the signs + and -".

7752:

Shmerko, V.P.; Yanushkevich , Svetlana N. ; Lyshevski, S.E. (2009).

6908:

and computing addition by hand; it also enables multiplication on a

6758:

can be thought of as a kind of addition—that is, the addition of an

3884:

There are two popular ways to define the sum of two natural numbers

1860:

plays a passive role. The unary view is also useful when discussing

1764: 10478: 10444: 8342: 8092: 7488: 7482: 6812:, the product may still make sense; for example, multiplication by 6649: 5570: 5378: 3312: 3298: 3278: 2751: 2314: 2282: 2047: 1618: 1279: 920: 743: 213: 161: 117: 10369: 9314:

Elementary and Middle School Mathematics: Teaching developmentally

9293: 5233: 4674:{\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+bc}{bd}}.} 1556: 9419:

Beyond Infinity: An Expedition to the Outer Limits of Mathematics

9394:

Interactive Mathematics Miscellany and Puzzles (cut-the-knot.org)

9244:

California State Board of Education mathematics content standards

4132: 3855: 3290: 2795:: Adding a number to itself is related to counting by two and to 2747: 2278: 2267: 1881: 1720: 1685: 1626:, one of the first English arithmetic texts, in the 15th century. 1358:

This notation can cause confusion, since in most other contexts,

205: 172: 145: 129: 9836:

Litvinov, Grigory; Maslov, Victor; Sobolevskii, Andreii (1999).

8085:

Moebs, William; et al. (2022). "1.4 Dimensional Analysis".

4137:

The simplest conception of an integer is that it consists of an

3246: 2325:." Eventually children begin to recall certain addition facts (" 7431:{\displaystyle \max(a,b)=\lim _{h\to 0}h\log(e^{a/h}+e^{b/h}).} 6949:

is an arithmetic operation remotely related to addition. Since

6657: 3477: 3343: 3335: 3286: 2281:. In a 1995 experiment imitating Wynn's 1992 result (but using 2244: 2058: 670: 209: 125: 10156:

Bunt, Lucas N.H.; Jones, Phillip S.; Bedient, Jack D. (1976).

10136: 10099:(in French). Presses universitaires de France. pp. 20–28. 4873:{\displaystyle {\frac {a}{c}}+{\frac {b}{c}}={\frac {a+b}{c}}} 4566:

This construction has been also generalized under the name of

3319:

mechanism, is an important limitation to overall performance.

1936: 6661: 6653: 5578: 3533: 3481: 3375: 3347: 3266: 3193:

In this example, two numerals are being added together: 01101

1689: 1605: 1548: 51: 46: 9554:

An Invitation to General Algebra and Universal Constructions

9249: 8343:"Reviews of TERC: Investigations in Number, Data, and Space" 6999:. However, division is not left distributive over addition; 3428:, but one bypasses the group of 9s and skips to the answer. 1126: 7475:. An infinite summation is a delicate procedure known as a 7015: 5353:, interpreted as points of the complex plane, is the point 3339: 3270: 1631: 1615:"to increase", one gets "augend", "thing to be increased". 1564: 829: 792: 667: 550: 435: 278: 9462:

From Sticks and Stones: Personal Adventures in Mathematics

2305:

have demonstrated an ability to perform basic arithmetic.

2038:

When addition is used together with other operations, the

9274: 8516: 8512: 8379:

Dale R. Patrick, Stephen W. Fardo, Vigyan Chandra (2008)

5456:

is an algebraic structure that allows for adding any two

3190:

0 1 1 0 1 + 1 0 1 1 1 ————————————— 1 0 0 1 0 0 = 36

9614:

Mathematical Methods for Optical Physics and Engineering

7665:. Palgrave, London: The MacMillan Press Ltd. p. 1. 7578:(also known as cryptarithms), puzzles involving addition 6648:" function. A similar "wrap around" operation arises in 3464:. To increase speed, modern designs calculate digits in 3431:

In practice, computational addition may be achieved via

1611:

results in "addend", "thing to be added". Likewise from

9684:

Introduction to Languages and the Theory of Computation

9360:

Wynn, Karen (1998). "Numerical competence in infants".

8585:

chapters 4 and 5, for example, follow this development.

8231: 8204:

Children's mathematics: Cognitively guided instruction

7886:

Schwartzman (p. 212) attributes adding upwards to the

7485:

a finite set is equivalent to summing 1 over the set.

6823: 6565: 6478: 6427: 6376: 6039: 5851: 5670: 3315:, where the efficiency of addition, in particular the 2769:

reduces the number of "addition facts" from 100 to 55.

1808:. Under the latter interpretation, the parts of a sum 1066: 1017: 980: 942: 939: 899: 853: 835: 832: 827: 781: 770: 759: 756: 750: 707: 700: 697: 683: 676: 673: 665: 627: 577: 556: 553: 548: 511: 462: 441: 438: 433: 396: 347: 326: 305: 284: 281: 276: 7333: 7294:{\displaystyle \log(a+b)\approx \max(\log a,\log b),} 7233: 7134: 6844: 6735:, are named to evoke their connection with addition. 6723:, disjoint union is seen as a particular case of the 6370: 5640: 5485: 5250: 5119: 5007: 4886: 4825: 4690: 4612: 4467: 3954: 3019: 2986: 2966: 2927: 2900:

As an example, 45.1 + 4.34 can be solved as follows:

2871:" into the next column. For example, in the addition 1391: 1351:{\displaystyle 3{\frac {1}{2}}=3+{\frac {1}{2}}=3.5.} 1309: 1233: 1189: 1155: 1065: 1016: 979: 938: 898: 826: 749: 664: 626: 547: 510: 432: 395: 275: 10058: 9996: 9977: 8018: 7864: 7862: 4111: +", and pastes these unary operations for all 3176:

7 + 9 → 6, carry 1 (since 7 + 9 = 16 = 6 + (1 × 10))

3173:

5 + 5 → 0, carry 1 (since 5 + 5 = 10 = 0 + (1 × 10))

2206:, making addition iterated succession. For example, 1586: 9881:Mikhalkin, Grigory (2006). Sanz-Solé, Marta (ed.). 9588: 10157: 10111: 9726: 9504:The Glass Wall: Why Mathematics Can Seem Difficult 9501: 9438: 9160: 9133: 9090: 8857: 8201: 7934: 7430: 7293: 7197: 6889: 6610: 6350: 5631:matrix computed by adding corresponding elements: 5554: 5334: 5210: 5064: 4946: 4872: 4804: 4673: 4536: 4458:Addition of ordered pairs is done component-wise: 4308:. So, one can define formally the integers as the 3975: 3468:; these schemes go by such names as carry select, 3164:1 + 1 → 0, carry 1 (since 1 + 1 = 2 = 0 + (1 × 2)) 3129: 2999: 2972: 2952: 1496: 1350: 1263: 1213: 1173: 1075: 1050: 989: 964: 908: 883: 798: 733: 636: 610: 520: 495: 405: 380: 9804: 9190: 8634:For a version that applies to any poset with the 7859: 7069:. However, it presents a perpetual difficulty in 6695: 4009:. An alternate version of this definition allows 3311:Addition is also fundamental to the operation of 2903:4 5 . 1 0 + 0 4 . 3 4 ———————————— 4 9 . 4 4 167:Addition has several important properties. It is 10568: 10155: 9281:Adding It Up: Helping Children Learn Mathematics 8860:Mathematical methods for physics and engineering 8855: 8616:Begle p. 49, Johnson p. 120, Devine et al. p. 75 7356: 7334: 7258: 7162: 7141: 7073:, essentially since "max" is not invertible. If 6968:, division is right distributive over addition: 5167: 5144: 5121: 4985:of rationals that is closed downward and has no 4103:to be temporarily "fixed", applies recursion on 3269:. If the addends are the rotation speeds of two 521:{\displaystyle \scriptstyle {\text{difference}}} 8856:Riley, K.F.; Hobson, M.P.; Bence, S.J. (2010). 8303: 8301: 8299: 8297: 8295: 8293: 8291: 4815:Addition of fractions is much simpler when the 4589: 3368:, which required as many steps as an addition. 1673:terms "adden" and "adding" were popularized by 1076:{\displaystyle \scriptstyle {\text{logarithm}}} 10160:The Historical roots of Elementary Mathematics 10097:Le Calcul Mécanique. Que Sais-Je ? n° 367 9246:Adopted December 1997, accessed December 2005. 8539:. Official Google Research Blog, June 2, 2006. 7933: 5065:{\displaystyle a+b=\{q+r\mid q\in a,r\in b\}.} 140:. Addition belongs to arithmetic, a branch of 10385: 10238: 8493:. Proceedings. Vol. 1. Springer, 2010. p. 194 8405:Truitt and Rogers pp. 1;44–49 and pp. 2;77–78 8238:Journal for Research in Mathematics Education 8232:Henry, Valerie J.; Brown, Richard S. (2008). 7769: 7767: 4444:. This allows identifying the natural number 3383:" logic circuit that adds two binary digits, 2789:may help rationalize the "exception" of zero. 2033:(1 + 2) + 3 = 3 + 3 = 6 = 1 + 5 = 1 + (2 + 3) 1940:2 + (1 + 3) = (2 + 1) + 3 with segmented rods 1102: 637:{\displaystyle \scriptstyle {\text{product}}} 238: 10139:Mathematics: Explorations & Applications 10112:Baroody, Arthur; Tiilikainen, Sirpa (2003). 10034: 9838:Idempotent mathematics and interval analysis 9341:Fosnot, Catherine T.; Dolk, Maarten (2001). 9310: 8685:Adding and Subtracting Fractions, Grades 5–8 8318:"Vertical addition and subtraction strategy" 8288: 7629:From Enderton (p. 138): "...select two sets 5056: 5020: 4115:together to form the full binary operation. 2218:To numerically add physical quantities with 2160:Within the context of integers, addition of 1374:of related numbers can be expressed through 204:, students are taught to add numbers in the 9131: 7920:A History of Mathematical Notations, Vol. 1 5335:{\displaystyle (a+bi)+(c+di)=(a+c)+(b+d)i.} 4024:The other popular definition is recursive: 3892:. If one defines natural numbers to be the 3330:including the addition and carry mechanisms 3297:); this can be accomplished roughly with a 2913:Scientific notation § Basic operations 2164:also plays a special role: for any integer 1637:"the highest, the top" and associated verb 1121: 909:{\displaystyle \scriptstyle {\text{power}}} 10392: 10378: 10245: 10231: 9478: 9340: 8683:Schyrlet Cameron, and Carolyn Craig (2013) 8073:Mathematics for Elementary School Teachers 8007:Mathematics curriculum in school education 7764: 7730:. Princeton University Press, 2014. p. 161 7495:, or more precisely and generally, over a 7441:In this sense, the maximum operation is a 7198:{\displaystyle a+\max(b,c)=\max(a+b,a+c).} 2123:refined the concept; around the year 830, 1661:, if not to earlier Roman writers such as 1630:"Sum" and "summand" derive from the Latin 1547:All of the above terminology derives from 1109: 1095: 990:{\displaystyle \scriptstyle {\text{root}}} 245: 231: 37:"Add" redirects here. For other uses, see 10252: 9933: 9894: 9880: 9863: 9850:Loday, Jean-Louis (2002). "Arithmetree". 9817: 9787: 9387: 9250:Devine, D.; Olson, J.; Olson, M. (1991). 9111: 9088: 8515:see Horowitz and Hill p. 679; for ADD in 8249: 8151: 8084: 7877:Karpinski pp. 56–57, reproduced on p. 104 7747: 7745: 4017:to possibly overlap and then takes their 1818:play asymmetric roles, and the operation 1532:. Some authors call the first addend the 1295:A whole number followed immediately by a 1046: 1042: 960: 956: 879: 875: 729: 725: 606: 602: 587: 583: 566: 562: 491: 487: 472: 468: 451: 447: 406:{\displaystyle \scriptstyle {\text{sum}}} 376: 372: 357: 353: 336: 332: 315: 311: 294: 290: 10041:Quantum Computing: A Gentle Introduction 9639: 9222:The Mathematics of the Elementary School 9158: 8436: 8421:. New York: John Wiley & Sons, Inc. 8227: 8225: 7754:Computer arithmetics for nanoelectronics 7100:, perhaps even returning zero. See also 7081:, then a straightforward calculation of 7014: 6822: 6684:with such an addition operation include 5232: 5082: 4058:. Define the general sum recursively by 3374: 3321: 3245: 2057: 1935: 1880: 1848:addends, it is more appropriate to call 1763: 1755: 1719: 1617: 1278: 1125: 45: 10017: 9747: 9571: 9550: 9459: 8548: 5221:This definition was first published by 2074:for addition, and is also known as the 1696:) is an abbreviation of the Latin word 1299:indicates the sum of the two, called a 1272: 14: 10569: 10203: 10178: 9681: 9436: 9362:The Development of Mathematical Skills 8785: 8665:K. Smith p. 234, Sparks and Rees p. 66 8381:Electronic Digital System Fundamentals 7916: 7800: 7773: 7742: 6796:times, then the sum is the product of 5565:This addition operation is central to 5555:{\displaystyle (a,b)+(c,d)=(a+c,b+d).} 4537:{\displaystyle (a,b)+(c,d)=(a+c,b+d).} 3841: 2906: 2229: 10373: 10226: 10038:; Polak, Wolfgang H. (4 March 2011). 9849: 9724: 9700: 9522: 9499: 9412: 9218: 8809: 8551:"The Risks Digest Volume 4: Issue 45" 8549:Neumann, Peter G. (2 February 1987). 8414: 8222: 7990:Using number lines with 5–8 year olds 7923:. The Open Court Company, Publishers. 7831: 7654: 6738: 6621: 4181:is zero, treat it as an identity. If 2765:: Mentioned above, using the pattern 2308: 1955:As an example, should the expression 1751: 10208:. Taylor & Francis. p. 60. 9921: 9607: 9359: 7955:participating institution membership 7827: 7825: 7448: 5109:. Addition is defined term by term: 5091:using Cauchy sequences of rationals. 4414:is a natural number, one can denote 3285:to balance forces on an assembly of 2888: 2746:The prerequisite to addition in the 1741: 10399: 9980:Advanced Computer Arithmetic Design 9729:Principles of Mathematical Analysis 9662: 9400:from the original on April 26, 2006 9253:Elementary Mathematics for Teachers 8812:Foundations of Discrete Mathematics 8788:Functions of One Complex Variable I 8714:Ferreirós p. 135; see section 6 of 8196:; Franke, Megan Loef; Levi, Linda; 8154:"Elephants have a head for figures" 8152:Randerson, James (21 August 2008). 7927: 6890:{\displaystyle e^{a+b}=e^{a}e^{b}.} 5432: 2053: 1138:"+" between the terms; that is, in 24: 10105: 9557:(2.3 ed.). General Printing. 7453:Incrementation, also known as the 5412: 5228: 4141:(which is a natural number) and a 3873: 3304:, but a better design exploits an 2921:, numbers are written in the form 2176:is the least integer greater than 1703: 1222: 1142:. The result is expressed with an 25: 10598: 9163:A History of Computing Technology 8929:Linderholm (p. 49) observes, "By 8747:Lecture Notes in Computer Science 8717:Stetigkeit und irrationale Zahlen 8598:. American educator, 26(2), p. 4. 8368:Decimals and Fractions: It's Easy 8164:from the original on 2 April 2015 7822: 7662:First-Year Technician Mathematics 6667: 2741: 2313:Typically, children first master 2234: 2105:This law was first identified in 1715: 9665:Introduction to Smooth Manifolds 9525:The Nature of Modern Mathematics 8031:B.G. Teubner Verlagsgesellschaft 7783:(1st ed.). Binghamton, NY: 7491:is a kind of "summation" over a 5654: 5646: 5581:are all represented by vectors. 4973:Construction of the real numbers 3536:allow this to be abbreviated as 3366:Pascal's calculator's complement 3277:. A hydraulic adder can add the 2953:{\displaystyle x=a\times 10^{b}} 1931: 1876: 171:, meaning that the order of the 10059:Truitt, T.; Rogers, A. (1960). 9997:Horowitz, P.; Hill, W. (2001). 9978:Flynn, M.; Oberman, S. (2001). 9751:Calculus: Early Transcendentals 9464:. Science Research Associates. 9061: 9052: 9043: 9034: 9025: 9016: 9007: 8998: 8989: 8980: 8967: 8958: 8949: 8940: 8923: 8914: 8905: 8896: 8887: 8878: 8849: 8840: 8831: 8803: 8779: 8770: 8761: 8752: 8739: 8730: 8708: 8699: 8690: 8677: 8668: 8659: 8650: 8641: 8628: 8619: 8610: 8601: 8588: 8576: 8565:from the original on 2014-12-28 8542: 8522: 8505: 8496: 8483: 8474: 8465: 8452: 8408: 8399: 8386: 8383:The Fairmont Press, Inc. p. 155 8373: 8360: 8335: 8310: 8275: 8266: 8185: 8176: 8145: 8136: 8127: 8118: 8109: 8078: 8065: 8056: 8047: 8012: 7995: 7982: 7970: 7961: 7910: 7901: 7880: 7871: 7850: 7602: 7589: 7571:Parallel addition (mathematics) 7543:is used to add two independent 7096:can accumulate an unacceptable 5603:(pronounced "m by n") matrices 4997:is defined element by element: 4966: 4036:, that is the number following 3490:high-level programming language 3448:as well as such basic tasks as 92:results in the total amount or 54:, a popular choice in textbooks 10141:(TE ed.). Prentice Hall. 9617:. Cambridge University Press. 9589:Dummit, D.; Foote, R. (1999). 8864:. Cambridge University Press. 8462:in Pascal's calculator article 8366:Rebecca Wingard-Nelson (2014) 8033:, Leipzig). pp. 115–120. 7733: 7720: 7711: 7687: 7648: 7623: 7422: 7380: 7363: 7349: 7337: 7285: 7261: 7252: 7240: 7189: 7165: 7156: 7144: 6696:Set theory and category theory 5546: 5522: 5516: 5504: 5498: 5486: 5323: 5311: 5305: 5293: 5287: 5272: 5266: 5251: 5202: 5176: 4528: 4504: 4498: 4486: 4480: 4468: 4253:to be the difference between | 3970: 3958: 3904:) be the cardinality of a set 3281:in two chambers by exploiting 3140: 1134:Addition is written using the 1039: 1031: 13: 1: 10541:Conway chained arrow notation 9887:European Mathematical Society 9874:10.1016/S0021-8693(02)00510-0 9388:Bogomolny, Alexander (1996). 9193:A Survey of Basic Mathematics 9074: 8208:. Portsmouth, NH: Heinemann. 6750: 4316:of natural numbers under the 4243:have different signs, define 4165:| be its absolute value. Let 3692:// left bitshift carry by one 3250:Addition with an op-amp. See 3188:1 1 1 1 1 (carried digits) 1871: 1563:words derived from the Latin 1181:("one plus two equals three") 183:is the same as counting (see 10094: 10003:(2 ed.). Cambridge UP. 9191:Sparks, F.; Rees C. (1979). 9132:Schwartzman, Steven (1994). 8964:Compare Viro Figure 1 (p. 2) 8920:Lee p. 526, Proposition 20.9 8489:Yeo, Sang-Soo, et al., eds. 8446: 8394:The common school arithmetic 7616: 7308:, named by analogy with the 7045:The maximum operation "max ( 5361:three of whose vertices are 4590:Rational numbers (fractions) 3510:; if the goal is to replace 3241: 2809:by adding one more, or from 2357: 2155: 2138:. In the 12th century, 1952:does not change the result. 124:, using abstractions called 7: 9754:(4 ed.). Brooks/Cole. 9733:(3 ed.). McGraw-Hill. 9686:(3 ed.). McGraw-Hill. 9574:Foundations of Real Numbers 9527:(3rd ed.). Wadsworth. 9311:Van de Walle, John (2004). 8088:University Physics Volume 1 8075:. Cengage Learning. Sec 2.3 7671:10.1007/978-1-349-02405-6_1 7554: 7010: 5584: 5377:is the point such that the 4448:with the equivalence class 4126: 4040:in the natural numbers, so 3391:, along with a carry input 3273:, they can be added with a 2062:5 + 0 = 5 with bags of dots 175:does not matter, and it is 68:) is one of the four basic 10: 10603: 10493:Inverse for right argument 10061:Basics of Analog Computers 9707:Category Theory in Context 9640:Enderton, Herbert (1977). 9508:. Teachers College Press. 9482:Mathematics Made Difficult 9159:Williams, Michael (1985). 8636:descending chain condition 8471:Flynn and Overman pp. 2, 8 8392:P.E. Bates Bothman (1837) 8022:Taschenbuch der Mathematik 6680:operation on a set. Basic 6625: 5588: 5441: 5437: 4989:. The sum of real numbers 4970: 4598:can be computed using the 4216:are both negative, define 4189:are both positive, define 4130: 3976:{\displaystyle N(A\cup B)} 3877: 3326:Part of Charles Babbage's 3144: 2910: 2860: 1901:are any two numbers, then 1840:. Instead of calling both 1828:is viewed as applying the 1622:Redrawn illustration from 1587: 1378:, which compactly denotes 1264:{\displaystyle 3+3+3+3=12} 61:(usually signified by the 36: 29: 10551:Knuth's up-arrow notation 10528: 10492: 10453: 10407: 10260: 10075: 9479:Linderholm, Carl (1971). 9441:The Mathematical Universe 9276:National Research Council 9117:The History of Arithmetic 8480:Flynn and Overman pp. 1–9 8442: 7942:Oxford English Dictionary 7599:"the number to be added". 4436:the equivalence class of 4421:the equivalence class of 4364:The equivalence class of 4290:are equal if and only if 4079:1 + 1 = 1 + 0 = (1 + 0) = 3908:. Take two disjoint sets 3528:. Some languages such as 3446:floating-point operations 3398:, producing the sum bit, 2704: 2669: 2634: 2599: 2564: 2529: 2494: 2459: 2424: 2389: 1885:4 + 2 = 2 + 4 with blocks 1008: 1000: 930: 919: 818: 810: 656: 648: 539: 531: 424: 416: 267: 259: 32:Addition (disambiguation) 10556:Steinhaus–Moser notation 10018:Jackson, Albert (1960). 9572:Burrill, Claude (1967). 9551:Bergman, George (2005). 9437:Dunham, William (1994). 9364:. Taylor & Francis. 9317:(5e ed.). Pearson. 9089:Ferreirós, José (1999). 9058:Rieffel and Polak, p. 16 8786:Conway, John B. (1986), 8445:, p. 48 (1994); Quoting 7582: 4600:least common denominator 4586:as semigroup operation. 3548: 2856: 2344: 2213: 2198:can also be seen as the 2078:. In symbols, for every 1788:can be interpreted as a 1584:Proto-Indo-European root 1507: 1214:{\displaystyle 5+4+2=11} 1122:Notation and terminology 76:, the other three being 9748:Stewart, James (1999). 9380:Mathematical exposition 8810:Joshi, Kapil D (1989), 8415:Ifrah, Georges (2001). 8370:Enslow Publishers, Inc. 7947:Oxford University Press 7497:differentiable manifold 6652:, where the sum of two 5357:obtained by building a 5100:of rationals, lim 4684:As an example, the sum 4173:be integers. If either 3502:does not change either 2980:is the significand and 2819:: Sums of the form 5 + 100:mathematical expression 10179:Poonen, Bjorn (2010). 10080:(in French). Hermann. 10076:Marguin, Jean (1994). 10000:The Art of Electronics 9725:Rudin, Walter (1976). 9642:Elements of Set Theory 9460:Johnson, Paul (1975). 9286:National Academy Press 9219:Begle, Edward (1975). 9183:Elementary mathematics 7992:. Nelson Thornes. p. 8 7549:distribution functions 7432: 7295: 7208:For these reasons, in 7199: 7042: 6891: 6828: 6612: 6352: 5556: 5336: 5238: 5212: 5092: 5066: 4948: 4874: 4806: 4675: 4572:semigroup homomorphism 4538: 4121:mathematical induction 4107:to define a function " 3977: 3710:// Recursive algorithm 3551:// Iterative algorithm 3410: 3402:, and a carry output, 3331: 3255: 3131: 3001: 3000:{\displaystyle 10^{b}} 2974: 2954: 2877:¹ 27 + 59 ———— 86 2063: 1941: 1886: 1769: 1761: 1742:§ Natural numbers 1725: 1657:date back at least to 1627: 1536:. In fact, during the 1498: 1412: 1376:capital sigma notation 1352: 1288: 1271:(see "multiplication" 1265: 1215: 1175: 1131: 1077: 1052: 991: 966: 910: 885: 800: 735: 638: 612: 522: 497: 407: 382: 88:. The addition of two 55: 10587:Mathematical notation 10582:Elementary arithmetic 10546:Grzegorczyk hierarchy 10254:Elementary arithmetic 10185:Girls' Angle Bulletin 10118:. Routledge. p. 9770:Mathematical research 9682:Martin, John (2003). 9593:(2 ed.). Wiley. 9500:Smith, Frank (2002). 8816:John Wiley & Sons 8638:, see Bergman p. 100. 8596:A coherent curriculum 8502:Karpinski pp. 102–103 8396:. Henry Benton. p. 31 8307:Fosnot and Dolk p. 99 7907:Karpinski pp. 150–153 7838:mathworld.wolfram.com 7785:John Wiley & Sons 7457:, is the addition of 7445:version of addition. 7433: 7296: 7200: 7077:is much greater than 7018: 6892: 6827:A circular slide rule 6826: 6784:can be thought of as 6745:elementary arithmetic 6613: 6353: 5557: 5337: 5236: 5213: 5086: 5067: 4971:Further information: 4949: 4875: 4807: 4676: 4561:cancellation property 4539: 4131:Further information: 4094:partially ordered set 3978: 3878:Further information: 3868:mathematics education 3378: 3325: 3306:operational amplifier 3249: 3132: 3002: 2975: 2955: 2333:and then reason that 2121:Indian mathematicians 2112:Brahmasphutasiddhanta 2061: 1939: 1884: 1767: 1759: 1723: 1621: 1570:, which is in turn a 1499: 1392: 1353: 1282: 1266: 1221:(see "associativity" 1216: 1176: 1174:{\displaystyle 1+2=3} 1129: 1078: 1053: 992: 967: 911: 886: 801: 736: 639: 613: 523: 498: 408: 383: 224:Arithmetic operations 49: 10095:Taton, René (1963). 9889:. pp. 827–852. 9808:; Dolan, J. (2001). 9543:Advanced mathematics 9523:Smith, Karl (1980). 9022:Litvinov et al. p. 3 8027:Verlag Harri Deutsch 7717:Devine et al. p. 263 7655:Lewis, Rhys (1974). 7331: 7231: 7132: 7103:Loss of significance 6842: 6833:exponential function 6682:algebraic structures 6368: 5638: 5483: 5248: 5117: 5005: 4884: 4823: 4688: 4610: 4465: 4318:equivalence relation 4032:be the successor of 3952: 3456:access and fetching 3017: 2984: 2964: 2925: 2831:can be derived from 2813:but subtracting one. 2763:Commutative property 2224:dimensional analysis 2180:, also known as the 1967:be defined to mean ( 1856:in this case, since 1582:"to give", from the 1542:commutative property 1389: 1307: 1231: 1187: 1153: 1063: 1014: 977: 936: 896: 824: 747: 662: 624: 545: 508: 430: 393: 273: 39:ADD (disambiguation) 30:For other uses, see 27:Arithmetic operation 10520:Super-logarithm (4) 10479:Root extraction (3) 10036:Rieffel, Eleanor G. 9944:2000math......5163V 9798:2004math......2090A 8322:primarylearning.org 8192:Carpenter, Thomas; 7945:(Online ed.). 7832:Weisstein, Eric W. 7806:Decimal Computation 7780:Decimal Computation 7503:Linear combinations 7455:successor operation 7063:orders of magnitude 7003:is not the same as 6788:. If a single term 6686:commutative monoids 5567:classical mechanics 4310:equivalence classes 3842:Addition of numbers 3828:arithmetic overflow 3362:Pascal's calculator 3283:Newton's second law 2919:scientific notation 2907:Scientific notation 2843:8 + 6 = 8 + 2 + 4 = 2835:by adding one more. 2285:instead of dolls), 2230:Performing addition 2040:order of operations 1950:order of operations 1624:The Art of Nombryng 1593:"to give"; thus to 10536:Ackermann function 10430:Exponentiation (3) 10425:Multiplication (2) 10266: 10020:Analog Computation 9852:Journal of Algebra 9843:Reliable Computing 9663:Lee, John (2003). 8884:Cheng, pp. 124–132 8722:2005-10-31 at the 8535:2016-04-01 at the 8194:Fennema, Elizabeth 8009:. Springer. p. 204 7988:Mosley, F (2001). 7699:www.mathsisfun.com 7428: 7370: 7316:, and taking the " 7291: 7195: 7071:numerical analysis 7043: 6887: 6829: 6739:Related operations 6638:musical set theory 6634:modular arithmetic 6628:Modular arithmetic 6622:Modular arithmetic 6608: 6602: 6551: 6464: 6413: 6348: 6346: 6338: 6018: 5837: 5552: 5332: 5239: 5208: 5175: 5152: 5129: 5093: 5062: 4960:field of fractions 4944: 4870: 4802: 4671: 4576:Grothendieck group 4568:Grothendieck group 4534: 4145:(generally either 3973: 3452:generation during 3411: 3332: 3256: 3127: 2997: 2970: 2950: 2863:Carry (arithmetic) 2309:Childhood learning 2251:in 1992 involving 2064: 2003:, it is true that 1942: 1887: 1770: 1762: 1752:Extending a length 1726: 1628: 1494: 1348: 1289: 1261: 1211: 1171: 1132: 1073: 1072: 1048: 1047: 987: 986: 962: 961: 948: 906: 905: 881: 880: 869: 866: 848: 796: 795: 790: 787: 776: 765: 731: 730: 719: 716: 713: 706: 692: 689: 682: 634: 633: 608: 607: 596: 593: 572: 518: 517: 493: 492: 481: 478: 457: 403: 402: 378: 377: 366: 363: 342: 321: 300: 186:Successor function 56: 10564: 10563: 10457:for left argument 10367: 10366: 10362: 10361: 10171:978-0-13-389015-0 10164:. Prentice-Hall. 10148:978-0-13-435817-8 10087:978-2-7056-6166-3 10063:. John F. Rider. 10051:978-0-262-01506-6 10010:978-0-521-37095-0 9989:978-0-471-41209-0 9953:978-3-7643-6417-5 9906:978-3-03719-022-7 9761:978-0-534-36298-0 9740:978-0-07-054235-8 9717:978-0-486-80903-8 9693:978-0-07-232200-2 9674:978-0-387-95448-6 9655:978-0-12-238440-0 9624:978-0-511-91510-9 9600:978-0-471-36857-1 9564:978-0-9655211-4-7 9534:978-0-8185-0352-8 9515:978-0-8077-4242-6 9492:978-0-7234-0415-6 9471:978-0-574-19115-1 9452:978-0-471-53656-7 9429:978-1-541-64413-7 9352:978-0-325-00353-5 9333:Cognitive science 9324:978-0-205-38689-5 9303:978-0-309-06995-3 9267:978-0-471-85947-5 9236:978-0-07-004325-1 9202:978-0-07-059902-4 9174:978-0-13-389917-7 9167:. Prentice-Hall. 9151:978-0-88385-511-9 9104:978-0-8176-5749-9 9004:Akian et al. p. 4 8871:978-0-521-86153-3 8825:978-0-470-21152-6 8797:978-0-387-90328-6 8460:Competing designs 8428:978-0-471-39671-0 8283:American Educator 8215:978-0-325-00137-1 8102:978-1-947172-20-3 8071:R. Fierro (2012) 8040:978-3-87144-492-0 7953:(Subscription or 7868:Schwartzman p. 19 7815:978-0-89874-318-0 7680:978-1-349-02405-6 7576:Verbal arithmetic 7566:Mental arithmetic 7535:quantum mechanics 7449:Other ways to add 7355: 7314:quantum mechanics 7215:negative infinity 7210:tropical geometry 7061:are of different 6792:appears in a sum 6786:repeated addition 6772:inverse functions 5166: 5143: 5120: 4942: 4929: 4908: 4895: 4868: 4847: 4834: 4800: 4787: 4774: 4753: 4712: 4699: 4666: 4634: 4621: 4548:case definition. 4090:recursion theorem 3836:Year 2000 problem 3418:binary arithmetic 3328:Difference Engine 3313:digital computers 3252:Summing amplifier 3180:This is known as 2973:{\displaystyle a} 2894:Decimal fractions 2889:Decimal fractions 2754:all the facts by 2739: 2738: 2297:0 through 4, one 2291:cottontop tamarin 2076:additive identity 1926:binary operations 1340: 1321: 1119: 1118: 1086: 1085: 1070: 1037: 1025: 984: 954: 952: 946: 903: 863: 858: 845: 840: 785: 774: 763: 714: 711: 704: 690: 687: 680: 631: 591: 581: 570: 560: 515: 476: 466: 455: 445: 400: 361: 351: 340: 330: 319: 309: 298: 288: 202:primary education 128:instead, such as 16:(Redirected from 10594: 10529:Related articles 10394: 10387: 10380: 10371: 10370: 10342: 10317: 10296: 10275: 10263: 10262: 10247: 10240: 10233: 10224: 10223: 10219: 10200: 10175: 10163: 10152: 10133: 10100: 10091: 10072: 10055: 10031: 10014: 9993: 9965: 9937: 9918: 9898: 9877: 9867: 9833: 9821: 9801: 9791: 9765: 9744: 9732: 9721: 9697: 9678: 9659: 9636: 9604: 9591:Abstract Algebra 9585: 9568: 9538: 9519: 9507: 9496: 9475: 9456: 9444: 9433: 9409: 9407: 9405: 9375: 9356: 9328: 9307: 9271: 9240: 9206: 9178: 9166: 9155: 9139: 9128: 9119:. Rand McNally. 9113:Karpinski, Louis 9108: 9096: 9068: 9065: 9059: 9056: 9050: 9047: 9041: 9038: 9032: 9029: 9023: 9020: 9014: 9011: 9005: 9002: 8996: 8993: 8987: 8984: 8978: 8971: 8965: 8962: 8956: 8953: 8947: 8944: 8938: 8927: 8921: 8918: 8912: 8909: 8903: 8900: 8894: 8891: 8885: 8882: 8876: 8875: 8863: 8853: 8847: 8844: 8838: 8835: 8829: 8828: 8807: 8801: 8800: 8783: 8777: 8774: 8768: 8767:Ferreirós p. 128 8765: 8759: 8756: 8750: 8743: 8737: 8734: 8728: 8712: 8706: 8703: 8697: 8694: 8688: 8687:Mark Twain, Inc. 8681: 8675: 8672: 8666: 8663: 8657: 8656:Ferreirós p. 223 8654: 8648: 8645: 8639: 8632: 8626: 8623: 8617: 8614: 8608: 8605: 8599: 8592: 8586: 8580: 8574: 8573: 8571: 8570: 8555:The Risks Digest 8546: 8540: 8526: 8520: 8509: 8503: 8500: 8494: 8487: 8481: 8478: 8472: 8469: 8463: 8456: 8450: 8440: 8434: 8432: 8412: 8406: 8403: 8397: 8390: 8384: 8377: 8371: 8364: 8358: 8357: 8355: 8353: 8339: 8333: 8332: 8330: 8328: 8314: 8308: 8305: 8286: 8279: 8273: 8270: 8264: 8263: 8253: 8251:10.2307/30034895 8229: 8220: 8219: 8207: 8189: 8183: 8180: 8174: 8173: 8171: 8169: 8149: 8143: 8140: 8134: 8131: 8125: 8122: 8116: 8113: 8107: 8106: 8082: 8076: 8069: 8063: 8060: 8054: 8053:Kaplan pp. 69–71 8051: 8045: 8044: 8016: 8010: 7999: 7993: 7986: 7980: 7974: 7968: 7965: 7959: 7958: 7950: 7938: 7931: 7925: 7924: 7914: 7908: 7905: 7899: 7884: 7878: 7875: 7869: 7866: 7857: 7854: 7848: 7847: 7845: 7844: 7829: 7820: 7819: 7798: 7771: 7762: 7761: 7749: 7740: 7737: 7731: 7724: 7718: 7715: 7709: 7708: 7706: 7705: 7691: 7685: 7684: 7652: 7646: 7627: 7610: 7606: 7600: 7597:numerus addendus 7593: 7561:Lunar arithmetic 7545:random variables 7437: 7435: 7434: 7429: 7421: 7420: 7416: 7400: 7399: 7395: 7369: 7300: 7298: 7297: 7292: 7204: 7202: 7201: 7196: 7095: 7040: 7036: 7034: 7028: 7026: 7006: 7002: 6998: 6967: 6942: 6896: 6894: 6893: 6888: 6883: 6882: 6873: 6872: 6860: 6859: 6818:additive inverse 6803: 6791: 6769: 6766:and subtracting 6765: 6760:additive inverse 6706:cardinal numbers 6617: 6615: 6614: 6609: 6607: 6606: 6556: 6555: 6469: 6468: 6418: 6417: 6357: 6355: 6354: 6349: 6347: 6343: 6342: 6335: 6334: 6319: 6318: 6299: 6298: 6283: 6282: 6268: 6267: 6252: 6251: 6213: 6212: 6197: 6196: 6177: 6176: 6164: 6163: 6152: 6151: 6139: 6138: 6125: 6124: 6109: 6108: 6089: 6088: 6076: 6075: 6064: 6063: 6051: 6050: 6027: 6023: 6022: 6015: 6014: 5995: 5994: 5980: 5979: 5941: 5940: 5921: 5920: 5909: 5908: 5895: 5894: 5875: 5874: 5863: 5862: 5842: 5841: 5834: 5833: 5814: 5813: 5799: 5798: 5760: 5759: 5740: 5739: 5728: 5727: 5714: 5713: 5694: 5693: 5682: 5681: 5657: 5649: 5630: 5620: 5561: 5559: 5558: 5553: 5433:Abstract algebra 5419:abstract algebra 5373:. Equivalently, 5341: 5339: 5338: 5333: 5217: 5215: 5214: 5209: 5201: 5200: 5188: 5187: 5174: 5162: 5161: 5151: 5139: 5138: 5128: 5077:Richard Dedekind 5071: 5069: 5068: 5063: 4987:greatest element 4981:of rationals: a 4953: 4951: 4950: 4945: 4943: 4935: 4930: 4925: 4914: 4909: 4901: 4896: 4888: 4879: 4877: 4876: 4871: 4869: 4864: 4853: 4848: 4840: 4835: 4827: 4811: 4809: 4808: 4803: 4801: 4793: 4788: 4780: 4775: 4770: 4759: 4754: 4752: 4741: 4718: 4713: 4705: 4700: 4692: 4680: 4678: 4677: 4672: 4667: 4665: 4657: 4640: 4635: 4627: 4622: 4614: 4596:rational numbers 4580:abelian category 4555:any commutative 4543: 4541: 4540: 4535: 4454: 4447: 4443: 4435: 4428: 4420: 4413: 4409: 4397: 4387: 4376:contains either 4375: 4359: 4341: 4307: 4289: 4279: 4264: 4252: 4234: 4207: 4083: 4080: 4076: 4057: 4047: 4043: 3996: 3982: 3980: 3979: 3974: 3947: 3937: 3926: 3860:rational numbers 3822: 3819: 3816: 3813: 3810: 3807: 3804: 3801: 3798: 3795: 3792: 3789: 3786: 3783: 3780: 3777: 3774: 3771: 3768: 3765: 3762: 3759: 3756: 3753: 3750: 3747: 3744: 3741: 3738: 3735: 3732: 3729: 3726: 3723: 3720: 3717: 3714: 3711: 3708: 3705: 3702: 3699: 3696: 3693: 3690: 3687: 3684: 3681: 3678: 3675: 3672: 3669: 3666: 3663: 3660: 3657: 3654: 3651: 3648: 3645: 3642: 3639: 3636: 3633: 3630: 3627: 3624: 3621: 3618: 3615: 3612: 3609: 3606: 3603: 3600: 3597: 3594: 3591: 3588: 3585: 3582: 3579: 3576: 3573: 3570: 3567: 3564: 3561: 3558: 3555: 3552: 3545: 3527: 3501: 3427: 3259:Analog computers 3229: 3222: 3215: 3189: 3136: 3134: 3133: 3128: 3126: 3125: 3104: 3103: 3082: 3081: 3060: 3059: 3038: 3037: 3006: 3004: 3003: 2998: 2996: 2995: 2979: 2977: 2976: 2971: 2959: 2957: 2956: 2951: 2949: 2948: 2874: 2847: 2844: 2834: 2830: 2826: 2822: 2812: 2808: 2352: 2351: 2336: 2332: 2265: 2261: 2209: 2202:th successor of 2197: 2175: 2151: 2137: 2100: 2083: 2072:identity element 2054:Identity element 2034: 2030: 1827: 1817: 1790:binary operation 1695: 1592: 1591: 1503: 1501: 1500: 1495: 1487: 1486: 1474: 1473: 1461: 1460: 1448: 1447: 1435: 1434: 1422: 1421: 1411: 1406: 1357: 1355: 1354: 1349: 1341: 1333: 1322: 1314: 1270: 1268: 1267: 1262: 1220: 1218: 1217: 1212: 1180: 1178: 1177: 1172: 1111: 1104: 1097: 1082: 1080: 1079: 1074: 1071: 1068: 1057: 1055: 1054: 1049: 1038: 1035: 1027: 1026: 1023: 996: 994: 993: 988: 985: 982: 971: 969: 968: 963: 955: 953: 950: 947: 944: 941: 915: 913: 912: 907: 904: 901: 890: 888: 887: 882: 874: 870: 865: 864: 861: 859: 856: 847: 846: 843: 841: 838: 805: 803: 802: 797: 794: 791: 786: 783: 775: 772: 764: 761: 740: 738: 737: 732: 724: 720: 715: 712: 709: 705: 702: 699: 691: 688: 685: 681: 678: 675: 643: 641: 640: 635: 632: 629: 617: 615: 614: 609: 601: 597: 592: 589: 582: 579: 571: 568: 561: 558: 527: 525: 524: 519: 516: 513: 502: 500: 499: 494: 486: 482: 477: 474: 467: 464: 456: 453: 446: 443: 412: 410: 409: 404: 401: 398: 387: 385: 384: 379: 371: 367: 362: 359: 352: 349: 341: 338: 331: 328: 320: 317: 310: 307: 299: 296: 289: 286: 257: 256: 247: 240: 233: 226: 219: 218: 199: 122:concrete objects 104: 67: 21: 10602: 10601: 10597: 10596: 10595: 10593: 10592: 10591: 10567: 10566: 10565: 10560: 10524: 10505:Subtraction (1) 10500:Predecessor (0) 10488: 10469:Subtraction (1) 10464:Predecessor (0) 10449: 10403: 10401:Hyperoperations 10398: 10368: 10363: 10358: 10347: 10343: 10338: 10333: 10322: 10318: 10313: 10308: 10301: 10297: 10292: 10287: 10280: 10276: 10271: 10256: 10251: 10216: 10172: 10149: 10130: 10108: 10106:Further reading 10103: 10088: 10052: 10022:. McGraw-Hill. 10011: 9990: 9954: 9907: 9896:math.AG/0601041 9830: 9819:math.QA/0004133 9789:math.SP/0402090 9762: 9741: 9718: 9694: 9675: 9656: 9625: 9601: 9576:. McGraw-Hill. 9565: 9535: 9516: 9493: 9472: 9453: 9430: 9422:. Basic Books. 9403: 9401: 9372: 9353: 9325: 9304: 9268: 9256:(2e ed.). 9237: 9203: 9195:. McGraw-Hill. 9175: 9152: 9105: 9077: 9072: 9071: 9066: 9062: 9057: 9053: 9048: 9044: 9039: 9035: 9030: 9026: 9021: 9017: 9012: 9008: 9003: 8999: 8994: 8990: 8986:Enderton p. 164 8985: 8981: 8975:Axiom of Choice 8972: 8968: 8963: 8959: 8954: 8950: 8945: 8941: 8928: 8924: 8919: 8915: 8910: 8906: 8901: 8897: 8892: 8888: 8883: 8879: 8872: 8854: 8850: 8845: 8841: 8836: 8832: 8826: 8808: 8804: 8798: 8784: 8780: 8775: 8771: 8766: 8762: 8757: 8753: 8744: 8740: 8735: 8731: 8724:Wayback Machine 8713: 8709: 8705:Enderton p. 114 8704: 8700: 8695: 8691: 8682: 8678: 8673: 8669: 8664: 8660: 8655: 8651: 8646: 8642: 8633: 8629: 8624: 8620: 8615: 8611: 8606: 8602: 8593: 8589: 8581: 8577: 8568: 8566: 8547: 8543: 8537:Wayback Machine 8527: 8523: 8510: 8506: 8501: 8497: 8488: 8484: 8479: 8475: 8470: 8466: 8457: 8453: 8441: 8437: 8429: 8413: 8409: 8404: 8400: 8391: 8387: 8378: 8374: 8365: 8361: 8351: 8349: 8341: 8340: 8336: 8326: 8324: 8316: 8315: 8311: 8306: 8289: 8280: 8276: 8271: 8267: 8230: 8223: 8216: 8190: 8186: 8182:F. Smith p. 130 8181: 8177: 8167: 8165: 8150: 8146: 8141: 8137: 8132: 8128: 8123: 8119: 8114: 8110: 8103: 8083: 8079: 8070: 8066: 8061: 8057: 8052: 8048: 8041: 8017: 8013: 8000: 7996: 7987: 7983: 7975: 7971: 7966: 7962: 7952: 7932: 7928: 7915: 7911: 7906: 7902: 7896:Leonard of Pisa 7885: 7881: 7876: 7872: 7867: 7860: 7855: 7851: 7842: 7840: 7830: 7823: 7816: 7802:Schmid, Hermann 7795: 7775:Schmid, Hermann 7772: 7765: 7750: 7743: 7738: 7734: 7726:Mazur, Joseph. 7725: 7721: 7716: 7712: 7703: 7701: 7693: 7692: 7688: 7681: 7653: 7649: 7628: 7624: 7619: 7614: 7613: 7607: 7603: 7594: 7590: 7585: 7557: 7451: 7412: 7408: 7404: 7391: 7387: 7383: 7359: 7332: 7329: 7328: 7324:tends to zero: 7318:classical limit 7310:Planck constant 7232: 7229: 7228: 7133: 7130: 7129: 7119:cardinal number 7117:is an infinite 7098:round-off error 7082: 7041:= 0.001 to 1000 7038: 7032: 7030: 7024: 7023: 7013: 7004: 7000: 6969: 6950: 6932: 6878: 6874: 6868: 6864: 6849: 6845: 6843: 6840: 6839: 6801: 6789: 6767: 6763: 6753: 6741: 6721:category theory 6702:ordinal numbers 6698: 6670: 6630: 6624: 6601: 6600: 6595: 6589: 6588: 6583: 6577: 6576: 6571: 6561: 6560: 6550: 6549: 6538: 6526: 6525: 6514: 6502: 6501: 6490: 6474: 6473: 6463: 6462: 6457: 6451: 6450: 6445: 6439: 6438: 6433: 6423: 6422: 6412: 6411: 6406: 6400: 6399: 6394: 6388: 6387: 6382: 6372: 6371: 6369: 6366: 6365: 6345: 6344: 6337: 6336: 6327: 6323: 6311: 6307: 6305: 6300: 6291: 6287: 6275: 6271: 6269: 6260: 6256: 6244: 6240: 6237: 6236: 6231: 6226: 6221: 6215: 6214: 6205: 6201: 6189: 6185: 6183: 6178: 6172: 6168: 6159: 6155: 6153: 6147: 6143: 6134: 6130: 6127: 6126: 6117: 6113: 6101: 6097: 6095: 6090: 6084: 6080: 6071: 6067: 6065: 6059: 6055: 6046: 6042: 6035: 6034: 6025: 6024: 6017: 6016: 6007: 6003: 6001: 5996: 5987: 5983: 5981: 5972: 5968: 5965: 5964: 5959: 5954: 5949: 5943: 5942: 5933: 5929: 5927: 5922: 5916: 5912: 5910: 5904: 5900: 5897: 5896: 5887: 5883: 5881: 5876: 5870: 5866: 5864: 5858: 5854: 5847: 5846: 5836: 5835: 5826: 5822: 5820: 5815: 5806: 5802: 5800: 5791: 5787: 5784: 5783: 5778: 5773: 5768: 5762: 5761: 5752: 5748: 5746: 5741: 5735: 5731: 5729: 5723: 5719: 5716: 5715: 5706: 5702: 5700: 5695: 5689: 5685: 5683: 5677: 5673: 5666: 5665: 5658: 5653: 5645: 5641: 5639: 5636: 5635: 5622: 5612: 5593: 5591:Matrix addition 5587: 5484: 5481: 5480: 5446: 5444:Vector addition 5440: 5435: 5427:category theory 5415: 5413:Generalizations 5249: 5246: 5245: 5231: 5229:Complex numbers 5196: 5192: 5183: 5179: 5170: 5157: 5153: 5147: 5134: 5130: 5124: 5118: 5115: 5114: 5108: 5098:Cauchy sequence 5087:Adding π/6 and 5006: 5003: 5002: 4975: 4969: 4934: 4915: 4913: 4900: 4887: 4885: 4882: 4881: 4854: 4852: 4839: 4826: 4824: 4821: 4820: 4792: 4779: 4760: 4758: 4742: 4719: 4717: 4704: 4691: 4689: 4686: 4685: 4658: 4641: 4639: 4626: 4613: 4611: 4608: 4607: 4592: 4466: 4463: 4462: 4449: 4445: 4437: 4430: 4422: 4415: 4411: 4399: 4389: 4377: 4365: 4343: 4342:if and only if 4323: 4291: 4281: 4271: 4262: 4244: 4217: 4190: 4157:For an integer 4135: 4129: 4081: 4078: 4059: 4049: 4045: 4041: 3988: 3953: 3950: 3949: 3939: 3928: 3917: 3882: 3876: 3874:Natural numbers 3848:natural numbers 3844: 3824: 3823: 3820: 3817: 3814: 3811: 3808: 3805: 3802: 3799: 3796: 3793: 3790: 3787: 3784: 3781: 3778: 3775: 3772: 3769: 3766: 3763: 3760: 3757: 3754: 3751: 3748: 3745: 3742: 3739: 3736: 3733: 3730: 3727: 3724: 3721: 3718: 3715: 3712: 3709: 3706: 3703: 3700: 3697: 3694: 3691: 3688: 3685: 3682: 3679: 3676: 3673: 3670: 3667: 3664: 3661: 3658: 3655: 3652: 3649: 3646: 3643: 3640: 3637: 3634: 3631: 3628: 3625: 3622: 3619: 3616: 3613: 3610: 3607: 3604: 3601: 3598: 3595: 3592: 3589: 3586: 3583: 3580: 3577: 3574: 3571: 3568: 3565: 3562: 3559: 3556: 3553: 3550: 3537: 3515: 3493: 3470:carry lookahead 3425: 3407: 3396: 3370:Giovanni Poleni 3293:(referenced to 3244: 3237: 3233: 3228: 3224: 3221: 3217: 3214: 3210: 3208: 3204: 3200: 3196: 3191: 3187: 3149: 3147:Binary addition 3143: 3118: 3114: 3096: 3092: 3074: 3070: 3052: 3048: 3030: 3026: 3018: 3015: 3014: 2991: 2987: 2985: 2982: 2981: 2965: 2962: 2961: 2944: 2940: 2926: 2923: 2922: 2915: 2909: 2904: 2891: 2878: 2872: 2865: 2859: 2845: 2842: 2832: 2828: 2824: 2820: 2810: 2806: 2773:One or two more 2744: 2347: 2334: 2330: 2311: 2303:Asian elephants 2295:Arabic numerals 2263: 2259: 2237: 2232: 2216: 2207: 2189: 2169: 2158: 2143: 2128: 2088: 2079: 2056: 2032: 2031:. For example, 2004: 1934: 1879: 1874: 1830:unary operation 1819: 1809: 1754: 1718: 1710:natural numbers 1706: 1704:Interpretations 1693: 1510: 1482: 1478: 1469: 1465: 1456: 1452: 1443: 1439: 1430: 1426: 1417: 1413: 1407: 1396: 1390: 1387: 1386: 1382:. For example, 1332: 1313: 1308: 1305: 1304: 1232: 1229: 1228: 1188: 1185: 1184: 1154: 1151: 1150: 1146:. For example, 1124: 1115: 1067: 1064: 1061: 1060: 1034: 1022: 1018: 1015: 1012: 1011: 981: 978: 975: 974: 949: 943: 940: 937: 934: 933: 900: 897: 894: 893: 868: 867: 860: 855: 854: 850: 849: 842: 837: 836: 831: 828: 825: 822: 821: 789: 788: 782: 778: 777: 771: 767: 766: 760: 755: 751: 748: 745: 744: 718: 717: 708: 701: 698: 694: 693: 684: 677: 674: 669: 666: 663: 660: 659: 628: 625: 622: 621: 595: 594: 588: 578: 574: 573: 567: 557: 552: 549: 546: 543: 542: 512: 509: 506: 505: 480: 479: 473: 463: 459: 458: 452: 442: 437: 434: 431: 428: 427: 397: 394: 391: 390: 365: 364: 358: 348: 344: 343: 337: 327: 323: 322: 316: 306: 302: 301: 295: 285: 280: 277: 274: 271: 270: 251: 222: 197: 189:). Addition of 138:complex numbers 102: 65: 50:3 + 2 = 5 with 42: 35: 28: 23: 22: 15: 12: 11: 5: 10600: 10590: 10589: 10584: 10579: 10562: 10561: 10559: 10558: 10553: 10548: 10543: 10538: 10532: 10530: 10526: 10525: 10523: 10522: 10517: 10512: 10507: 10502: 10496: 10494: 10490: 10489: 10487: 10486: 10484:Super-root (4) 10481: 10476: 10471: 10466: 10460: 10458: 10451: 10450: 10448: 10447: 10442: 10437: 10432: 10427: 10422: 10417: 10411: 10409: 10405: 10404: 10397: 10396: 10389: 10382: 10374: 10365: 10364: 10360: 10359: 10336: 10334: 10320:Multiplication 10311: 10309: 10290: 10288: 10269: 10267: 10261: 10258: 10257: 10250: 10249: 10242: 10235: 10227: 10221: 10220: 10214: 10201: 10176: 10170: 10153: 10147: 10134: 10128: 10107: 10104: 10102: 10101: 10092: 10086: 10073: 10069:QA76.4 T7 10056: 10050: 10032: 10028:QA76.4 J3 10015: 10009: 9994: 9988: 9974: 9973: 9971: 9967: 9966: 9952: 9919: 9905: 9878: 9847: 9834: 9828: 9812:. p. 29. 9802: 9774: 9773: 9771: 9767: 9766: 9760: 9745: 9739: 9722: 9716: 9698: 9692: 9679: 9673: 9660: 9654: 9646:Academic Press 9637: 9623: 9605: 9599: 9586: 9569: 9563: 9547: 9546: 9544: 9540: 9539: 9533: 9520: 9514: 9497: 9491: 9476: 9470: 9457: 9451: 9434: 9428: 9414:Cheng, Eugenia 9410: 9384: 9383: 9381: 9377: 9376: 9370: 9357: 9351: 9337: 9336: 9334: 9330: 9329: 9323: 9308: 9302: 9272: 9266: 9247: 9241: 9235: 9215: 9214: 9212: 9208: 9207: 9201: 9187: 9186: 9184: 9180: 9179: 9173: 9156: 9150: 9129: 9109: 9103: 9097:. Birkhäuser. 9085: 9084: 9082: 9078: 9076: 9073: 9070: 9069: 9060: 9051: 9042: 9033: 9024: 9015: 9013:Mikhalkin p. 2 9006: 8997: 8995:Mikhalkin p. 1 8988: 8979: 8966: 8957: 8948: 8939: 8931:multiplication 8922: 8913: 8904: 8895: 8886: 8877: 8870: 8848: 8839: 8830: 8824: 8802: 8796: 8778: 8776:Burrill p. 140 8769: 8760: 8751: 8738: 8729: 8707: 8698: 8689: 8676: 8674:Enderton p. 92 8667: 8658: 8649: 8640: 8627: 8625:Enderton p. 79 8618: 8609: 8600: 8587: 8575: 8541: 8528:Joshua Bloch, 8521: 8504: 8495: 8482: 8473: 8464: 8451: 8435: 8427: 8407: 8398: 8385: 8372: 8359: 8334: 8309: 8287: 8285:, 26(2), 1–18. 8274: 8265: 8244:(2): 153–183. 8221: 8214: 8184: 8175: 8144: 8135: 8126: 8117: 8108: 8101: 8077: 8064: 8055: 8046: 8039: 8011: 8001:Li, Y., & 7994: 7981: 7969: 7960: 7926: 7909: 7900: 7879: 7870: 7858: 7849: 7821: 7814: 7793: 7763: 7741: 7732: 7719: 7710: 7686: 7679: 7647: 7621: 7620: 7618: 7615: 7612: 7611: 7601: 7587: 7586: 7584: 7581: 7580: 7579: 7573: 7568: 7563: 7556: 7553: 7450: 7447: 7439: 7438: 7427: 7424: 7419: 7415: 7411: 7407: 7403: 7398: 7394: 7390: 7386: 7382: 7379: 7376: 7373: 7368: 7365: 7362: 7358: 7354: 7351: 7348: 7345: 7342: 7339: 7336: 7302: 7301: 7290: 7287: 7284: 7281: 7278: 7275: 7272: 7269: 7266: 7263: 7260: 7257: 7254: 7251: 7248: 7245: 7242: 7239: 7236: 7206: 7205: 7194: 7191: 7188: 7185: 7182: 7179: 7176: 7173: 7170: 7167: 7164: 7161: 7158: 7155: 7152: 7149: 7146: 7143: 7140: 7137: 7012: 7009: 6898: 6897: 6886: 6881: 6877: 6871: 6867: 6863: 6858: 6855: 6852: 6848: 6810:natural number 6782:Multiplication 6752: 6749: 6740: 6737: 6714:disjoint union 6697: 6694: 6690:abelian groups 6669: 6668:General theory 6666: 6654:angle measures 6626:Main article: 6623: 6620: 6619: 6618: 6605: 6599: 6596: 6594: 6591: 6590: 6587: 6584: 6582: 6579: 6578: 6575: 6572: 6570: 6567: 6566: 6564: 6559: 6554: 6548: 6545: 6542: 6539: 6537: 6534: 6531: 6528: 6527: 6524: 6521: 6518: 6515: 6513: 6510: 6507: 6504: 6503: 6500: 6497: 6494: 6491: 6489: 6486: 6483: 6480: 6479: 6477: 6472: 6467: 6461: 6458: 6456: 6453: 6452: 6449: 6446: 6444: 6441: 6440: 6437: 6434: 6432: 6429: 6428: 6426: 6421: 6416: 6410: 6407: 6405: 6402: 6401: 6398: 6395: 6393: 6390: 6389: 6386: 6383: 6381: 6378: 6377: 6375: 6359: 6358: 6341: 6333: 6330: 6326: 6322: 6317: 6314: 6310: 6306: 6304: 6301: 6297: 6294: 6290: 6286: 6281: 6278: 6274: 6270: 6266: 6263: 6259: 6255: 6250: 6247: 6243: 6239: 6238: 6235: 6232: 6230: 6227: 6225: 6222: 6220: 6217: 6216: 6211: 6208: 6204: 6200: 6195: 6192: 6188: 6184: 6182: 6179: 6175: 6171: 6167: 6162: 6158: 6154: 6150: 6146: 6142: 6137: 6133: 6129: 6128: 6123: 6120: 6116: 6112: 6107: 6104: 6100: 6096: 6094: 6091: 6087: 6083: 6079: 6074: 6070: 6066: 6062: 6058: 6054: 6049: 6045: 6041: 6040: 6038: 6033: 6030: 6028: 6026: 6021: 6013: 6010: 6006: 6002: 6000: 5997: 5993: 5990: 5986: 5982: 5978: 5975: 5971: 5967: 5966: 5963: 5960: 5958: 5955: 5953: 5950: 5948: 5945: 5944: 5939: 5936: 5932: 5928: 5926: 5923: 5919: 5915: 5911: 5907: 5903: 5899: 5898: 5893: 5890: 5886: 5882: 5880: 5877: 5873: 5869: 5865: 5861: 5857: 5853: 5852: 5850: 5845: 5840: 5832: 5829: 5825: 5821: 5819: 5816: 5812: 5809: 5805: 5801: 5797: 5794: 5790: 5786: 5785: 5782: 5779: 5777: 5774: 5772: 5769: 5767: 5764: 5763: 5758: 5755: 5751: 5747: 5745: 5742: 5738: 5734: 5730: 5726: 5722: 5718: 5717: 5712: 5709: 5705: 5701: 5699: 5696: 5692: 5688: 5684: 5680: 5676: 5672: 5671: 5669: 5664: 5661: 5659: 5656: 5652: 5648: 5644: 5643: 5621:, is again an 5589:Main article: 5586: 5583: 5563: 5562: 5551: 5548: 5545: 5542: 5539: 5536: 5533: 5530: 5527: 5524: 5521: 5518: 5515: 5512: 5509: 5506: 5503: 5500: 5497: 5494: 5491: 5488: 5450:linear algebra 5442:Main article: 5439: 5436: 5434: 5431: 5414: 5411: 5381:with vertices 5343: 5342: 5331: 5328: 5325: 5322: 5319: 5316: 5313: 5310: 5307: 5304: 5301: 5298: 5295: 5292: 5289: 5286: 5283: 5280: 5277: 5274: 5271: 5268: 5265: 5262: 5259: 5256: 5253: 5230: 5227: 5219: 5218: 5207: 5204: 5199: 5195: 5191: 5186: 5182: 5178: 5173: 5169: 5165: 5160: 5156: 5150: 5146: 5142: 5137: 5133: 5127: 5123: 5104: 5073: 5072: 5061: 5058: 5055: 5052: 5049: 5046: 5043: 5040: 5037: 5034: 5031: 5028: 5025: 5022: 5019: 5016: 5013: 5010: 4968: 4965: 4941: 4938: 4933: 4928: 4924: 4921: 4918: 4912: 4907: 4904: 4899: 4894: 4891: 4867: 4863: 4860: 4857: 4851: 4846: 4843: 4838: 4833: 4830: 4799: 4796: 4791: 4786: 4783: 4778: 4773: 4769: 4766: 4763: 4757: 4751: 4748: 4745: 4740: 4737: 4734: 4731: 4728: 4725: 4722: 4716: 4711: 4708: 4703: 4698: 4695: 4682: 4681: 4670: 4664: 4661: 4656: 4653: 4650: 4647: 4644: 4638: 4633: 4630: 4625: 4620: 4617: 4591: 4588: 4545: 4544: 4533: 4530: 4527: 4524: 4521: 4518: 4515: 4512: 4509: 4506: 4503: 4500: 4497: 4494: 4491: 4488: 4485: 4482: 4479: 4476: 4473: 4470: 4410:otherwise. If 4362: 4361: 4267: 4266: 4139:absolute value 4128: 4125: 4086: 4085: 4019:disjoint union 3985: 3984: 3972: 3969: 3966: 3963: 3960: 3957: 3948:is defined as 3880:Natural number 3875: 3872: 3843: 3840: 3832:program errors 3671:// Logical XOR 3644:// Logical AND 3549: 3486:microprocessor 3405: 3394: 3358:adding machine 3243: 3240: 3235: 3231: 3226: 3225:1 + 1 + 1 = 11 3219: 3218:1 + 0 + 1 = 10 3212: 3206: 3202: 3198: 3194: 3186: 3178: 3177: 3174: 3166: 3165: 3162: 3159: 3156: 3145:Main article: 3142: 3139: 3138: 3137: 3124: 3121: 3117: 3113: 3110: 3107: 3102: 3099: 3095: 3091: 3088: 3085: 3080: 3077: 3073: 3069: 3066: 3063: 3058: 3055: 3051: 3047: 3044: 3041: 3036: 3033: 3029: 3025: 3022: 2994: 2990: 2969: 2947: 2943: 2939: 2936: 2933: 2930: 2911:Main article: 2908: 2905: 2902: 2890: 2887: 2876: 2861:Main article: 2858: 2855: 2850: 2849: 2836: 2814: 2800: 2797:multiplication 2790: 2780: 2770: 2743: 2742:Decimal system 2740: 2737: 2736: 2733: 2730: 2727: 2724: 2721: 2718: 2715: 2712: 2709: 2706: 2702: 2701: 2698: 2695: 2692: 2689: 2686: 2683: 2680: 2677: 2674: 2671: 2667: 2666: 2663: 2660: 2657: 2654: 2651: 2648: 2645: 2642: 2639: 2636: 2632: 2631: 2628: 2625: 2622: 2619: 2616: 2613: 2610: 2607: 2604: 2601: 2597: 2596: 2593: 2590: 2587: 2584: 2581: 2578: 2575: 2572: 2569: 2566: 2562: 2561: 2558: 2555: 2552: 2549: 2546: 2543: 2540: 2537: 2534: 2531: 2527: 2526: 2523: 2520: 2517: 2514: 2511: 2508: 2505: 2502: 2499: 2496: 2492: 2491: 2488: 2485: 2482: 2479: 2476: 2473: 2470: 2467: 2464: 2461: 2457: 2456: 2453: 2450: 2447: 2444: 2441: 2438: 2435: 2432: 2429: 2426: 2422: 2421: 2418: 2415: 2412: 2409: 2406: 2403: 2400: 2397: 2394: 2391: 2387: 2386: 2383: 2380: 2377: 2374: 2371: 2368: 2365: 2362: 2359: 2356: 2346: 2343: 2310: 2307: 2287:rhesus macaque 2236: 2235:Innate ability 2233: 2231: 2228: 2215: 2212: 2168:, the integer 2157: 2154: 2103: 2102: 2055: 2052: 2044:exponentiation 1933: 1930: 1922: 1921: 1878: 1875: 1873: 1870: 1804:more units to 1792:that combines 1778: 1777: 1753: 1750: 1738: 1737: 1717: 1716:Combining sets 1714: 1705: 1702: 1671:Middle English 1643:ancient Greeks 1509: 1506: 1505: 1504: 1493: 1490: 1485: 1481: 1477: 1472: 1468: 1464: 1459: 1455: 1451: 1446: 1442: 1438: 1433: 1429: 1425: 1420: 1416: 1410: 1405: 1402: 1399: 1395: 1368: 1367: 1364:multiplication 1347: 1344: 1339: 1336: 1331: 1328: 1325: 1320: 1317: 1312: 1303:. For example, 1277: 1276: 1260: 1257: 1254: 1251: 1248: 1245: 1242: 1239: 1236: 1226: 1210: 1207: 1204: 1201: 1198: 1195: 1192: 1182: 1170: 1167: 1164: 1161: 1158: 1140:infix notation 1123: 1120: 1117: 1116: 1114: 1113: 1106: 1099: 1091: 1088: 1087: 1084: 1083: 1058: 1045: 1041: 1036:anti-logarithm 1033: 1030: 1021: 1009: 1006: 1005: 998: 997: 972: 959: 931: 928: 927: 917: 916: 891: 878: 873: 852: 851: 834: 833: 830: 819: 816: 815: 812:Exponentiation 808: 807: 793: 780: 779: 769: 768: 758: 757: 754: 741: 728: 723: 696: 695: 672: 671: 668: 657: 654: 653: 646: 645: 618: 605: 600: 586: 576: 575: 565: 555: 554: 551: 540: 537: 536: 533:Multiplication 529: 528: 503: 490: 485: 471: 461: 460: 450: 440: 439: 436: 425: 422: 421: 414: 413: 388: 375: 370: 356: 346: 345: 335: 325: 324: 314: 304: 303: 293: 283: 282: 279: 268: 265: 264: 253: 252: 250: 249: 242: 235: 227: 212:to the modern 82:multiplication 26: 9: 6: 4: 3: 2: 10599: 10588: 10585: 10583: 10580: 10578: 10575: 10574: 10572: 10557: 10554: 10552: 10549: 10547: 10544: 10542: 10539: 10537: 10534: 10533: 10531: 10527: 10521: 10518: 10516: 10515:Logarithm (3) 10513: 10511: 10508: 10506: 10503: 10501: 10498: 10497: 10495: 10491: 10485: 10482: 10480: 10477: 10475: 10472: 10470: 10467: 10465: 10462: 10461: 10459: 10456: 10452: 10446: 10443: 10441: 10440:Pentation (5) 10438: 10436: 10435:Tetration (4) 10433: 10431: 10428: 10426: 10423: 10421: 10418: 10416: 10415:Successor (0) 10413: 10412: 10410: 10406: 10402: 10395: 10390: 10388: 10383: 10381: 10376: 10375: 10372: 10357: 10355: 10351: 10346: 10341: 10335: 10332: 10330: 10326: 10321: 10316: 10310: 10307: 10305: 10300: 10295: 10289: 10286: 10284: 10279: 10274: 10268: 10265: 10264: 10259: 10255: 10248: 10243: 10241: 10236: 10234: 10229: 10228: 10225: 10217: 10215:0-89859-171-6 10211: 10207: 10202: 10198: 10194: 10190: 10186: 10182: 10177: 10173: 10167: 10162: 10161: 10154: 10150: 10144: 10140: 10135: 10131: 10129:0-8058-3155-X 10125: 10121: 10117: 10116: 10110: 10109: 10098: 10093: 10089: 10083: 10079: 10074: 10070: 10066: 10062: 10057: 10053: 10047: 10044:. MIT Press. 10043: 10042: 10037: 10033: 10029: 10025: 10021: 10016: 10012: 10006: 10002: 10001: 9995: 9991: 9985: 9981: 9976: 9975: 9972: 9969: 9968: 9963: 9959: 9955: 9949: 9945: 9941: 9936: 9931: 9927: 9926: 9920: 9916: 9912: 9908: 9902: 9897: 9892: 9888: 9884: 9879: 9875: 9871: 9866: 9861: 9857: 9853: 9848: 9845: 9844: 9839: 9835: 9831: 9829:3-540-66913-2 9825: 9820: 9815: 9811: 9807: 9803: 9799: 9795: 9790: 9785: 9781: 9780:INRIA Reports 9776: 9775: 9772: 9769: 9768: 9763: 9757: 9753: 9752: 9746: 9742: 9736: 9731: 9730: 9723: 9719: 9713: 9709: 9708: 9703: 9699: 9695: 9689: 9685: 9680: 9676: 9670: 9666: 9661: 9657: 9651: 9647: 9643: 9638: 9634: 9630: 9626: 9620: 9616: 9615: 9610: 9606: 9602: 9596: 9592: 9587: 9583: 9579: 9575: 9570: 9566: 9560: 9556: 9555: 9549: 9548: 9545: 9542: 9541: 9536: 9530: 9526: 9521: 9517: 9511: 9506: 9505: 9498: 9494: 9488: 9484: 9483: 9477: 9473: 9467: 9463: 9458: 9454: 9448: 9443: 9442: 9435: 9431: 9425: 9421: 9420: 9415: 9411: 9399: 9395: 9391: 9386: 9385: 9382: 9379: 9378: 9373: 9371:0-86377-816-X 9367: 9363: 9358: 9354: 9348: 9345:. Heinemann. 9344: 9339: 9338: 9335: 9332: 9331: 9326: 9320: 9316: 9315: 9309: 9305: 9299: 9295: 9294:10.17226/9822 9291: 9287: 9283: 9282: 9277: 9273: 9269: 9263: 9259: 9255: 9254: 9248: 9245: 9242: 9238: 9232: 9228: 9224: 9223: 9217: 9216: 9213: 9210: 9209: 9204: 9198: 9194: 9189: 9188: 9185: 9182: 9181: 9176: 9170: 9165: 9164: 9157: 9153: 9147: 9143: 9138: 9137: 9130: 9126: 9122: 9118: 9114: 9110: 9106: 9100: 9095: 9094: 9087: 9086: 9083: 9080: 9079: 9064: 9055: 9046: 9037: 9028: 9019: 9010: 9001: 8992: 8983: 8976: 8970: 8961: 8952: 8943: 8936: 8932: 8926: 8917: 8908: 8899: 8893:Riehl, p. 100 8890: 8881: 8873: 8867: 8862: 8861: 8852: 8843: 8834: 8827: 8821: 8817: 8813: 8806: 8799: 8793: 8789: 8782: 8773: 8764: 8755: 8748: 8742: 8733: 8726: 8725: 8721: 8718: 8711: 8702: 8693: 8686: 8680: 8671: 8662: 8653: 8644: 8637: 8631: 8622: 8613: 8604: 8597: 8591: 8584: 8579: 8564: 8560: 8556: 8552: 8545: 8538: 8534: 8531: 8525: 8518: 8514: 8508: 8499: 8492: 8486: 8477: 8468: 8461: 8455: 8448: 8444: 8439: 8430: 8424: 8420: 8419: 8411: 8402: 8395: 8389: 8382: 8376: 8369: 8363: 8348: 8344: 8338: 8323: 8319: 8313: 8304: 8302: 8300: 8298: 8296: 8294: 8292: 8284: 8278: 8269: 8261: 8257: 8252: 8247: 8243: 8239: 8235: 8228: 8226: 8217: 8211: 8206: 8205: 8199: 8198:Empson, Susan 8195: 8188: 8179: 8163: 8159: 8155: 8148: 8139: 8130: 8121: 8112: 8104: 8098: 8094: 8090: 8089: 8081: 8074: 8068: 8059: 8050: 8042: 8036: 8032: 8028: 8024: 8023: 8015: 8008: 8004: 7998: 7991: 7985: 7978: 7973: 7964: 7956: 7948: 7944: 7943: 7937: 7930: 7922: 7921: 7913: 7904: 7897: 7893: 7889: 7883: 7874: 7865: 7863: 7853: 7839: 7835: 7828: 7826: 7817: 7811: 7807: 7803: 7796: 7794:0-471-76180-X 7790: 7786: 7782: 7781: 7776: 7770: 7768: 7760:. p. 80. 7759: 7755: 7748: 7746: 7736: 7729: 7723: 7714: 7700: 7696: 7690: 7682: 7676: 7672: 7668: 7664: 7663: 7658: 7651: 7644: 7641:= 2 and card 7640: 7636: 7632: 7626: 7622: 7605: 7598: 7592: 7588: 7577: 7574: 7572: 7569: 7567: 7564: 7562: 7559: 7558: 7552: 7550: 7546: 7542: 7538: 7536: 7532: 7528: 7527:superposition 7524: 7520: 7516: 7512: 7508: 7504: 7500: 7498: 7494: 7490: 7486: 7484: 7480: 7478: 7474: 7470: 7466: 7462: 7461:to a number. 7460: 7456: 7446: 7444: 7425: 7417: 7413: 7409: 7405: 7401: 7396: 7392: 7388: 7384: 7377: 7374: 7371: 7366: 7360: 7352: 7346: 7343: 7340: 7327: 7326: 7325: 7323: 7319: 7315: 7311: 7307: 7288: 7282: 7279: 7276: 7273: 7270: 7267: 7264: 7255: 7249: 7246: 7243: 7237: 7234: 7227: 7226: 7225: 7223: 7218: 7216: 7211: 7192: 7186: 7183: 7180: 7177: 7174: 7171: 7168: 7159: 7153: 7150: 7147: 7138: 7135: 7128: 7127: 7126: 7122: 7120: 7116: 7112: 7107: 7105: 7104: 7099: 7094: 7090: 7086: 7080: 7076: 7072: 7068: 7067:Taylor series 7064: 7060: 7056: 7052: 7048: 7021: 7017: 7008: 6997: 6993: 6989: 6985: 6981: 6977: 6973: 6965: 6961: 6957: 6953: 6948: 6944: 6940: 6936: 6930: 6926: 6921: 6919: 6915: 6911: 6907: 6903: 6884: 6879: 6875: 6869: 6865: 6861: 6856: 6853: 6850: 6846: 6838: 6837: 6836: 6834: 6825: 6821: 6820:of a number. 6819: 6815: 6811: 6807: 6799: 6795: 6787: 6783: 6779: 6775: 6773: 6761: 6757: 6748: 6746: 6736: 6734: 6730: 6726: 6722: 6717: 6715: 6711: 6707: 6703: 6693: 6691: 6687: 6683: 6679: 6675: 6665: 6663: 6659: 6655: 6651: 6647: 6643: 6642:Boolean logic 6639: 6635: 6629: 6603: 6597: 6592: 6585: 6580: 6573: 6568: 6562: 6557: 6552: 6546: 6543: 6540: 6535: 6532: 6529: 6522: 6519: 6516: 6511: 6508: 6505: 6498: 6495: 6492: 6487: 6484: 6481: 6475: 6470: 6465: 6459: 6454: 6447: 6442: 6435: 6430: 6424: 6419: 6414: 6408: 6403: 6396: 6391: 6384: 6379: 6373: 6364: 6363: 6362: 6361:For example: 6339: 6331: 6328: 6324: 6320: 6315: 6312: 6308: 6302: 6295: 6292: 6288: 6284: 6279: 6276: 6272: 6264: 6261: 6257: 6253: 6248: 6245: 6241: 6233: 6228: 6223: 6218: 6209: 6206: 6202: 6198: 6193: 6190: 6186: 6180: 6173: 6169: 6165: 6160: 6156: 6148: 6144: 6140: 6135: 6131: 6121: 6118: 6114: 6110: 6105: 6102: 6098: 6092: 6085: 6081: 6077: 6072: 6068: 6060: 6056: 6052: 6047: 6043: 6036: 6031: 6029: 6019: 6011: 6008: 6004: 5998: 5991: 5988: 5984: 5976: 5973: 5969: 5961: 5956: 5951: 5946: 5937: 5934: 5930: 5924: 5917: 5913: 5905: 5901: 5891: 5888: 5884: 5878: 5871: 5867: 5859: 5855: 5848: 5843: 5838: 5830: 5827: 5823: 5817: 5810: 5807: 5803: 5795: 5792: 5788: 5780: 5775: 5770: 5765: 5756: 5753: 5749: 5743: 5736: 5732: 5724: 5720: 5710: 5707: 5703: 5697: 5690: 5686: 5678: 5674: 5667: 5662: 5660: 5650: 5634: 5633: 5632: 5629: 5625: 5619: 5615: 5611:, denoted by 5610: 5606: 5602: 5598: 5592: 5582: 5580: 5576: 5575:accelerations 5572: 5568: 5549: 5543: 5540: 5537: 5534: 5531: 5528: 5525: 5519: 5513: 5510: 5507: 5501: 5495: 5492: 5489: 5479: 5478: 5477: 5475: 5471: 5467: 5463: 5459: 5455: 5451: 5445: 5430: 5428: 5424: 5420: 5410: 5408: 5404: 5400: 5396: 5392: 5388: 5384: 5380: 5376: 5372: 5368: 5364: 5360: 5359:parallelogram 5356: 5352: 5348: 5329: 5326: 5320: 5317: 5314: 5308: 5302: 5299: 5296: 5290: 5284: 5281: 5278: 5275: 5269: 5263: 5260: 5257: 5254: 5244: 5243: 5242: 5235: 5226: 5224: 5205: 5197: 5193: 5189: 5184: 5180: 5171: 5163: 5158: 5154: 5148: 5140: 5135: 5131: 5125: 5112: 5111: 5110: 5107: 5103: 5099: 5090: 5085: 5081: 5078: 5059: 5053: 5050: 5047: 5044: 5041: 5038: 5035: 5032: 5029: 5026: 5023: 5017: 5014: 5011: 5008: 5000: 4999: 4998: 4996: 4992: 4988: 4984: 4983:non-empty set 4980: 4974: 4964: 4962: 4961: 4955: 4939: 4936: 4931: 4926: 4922: 4919: 4916: 4910: 4905: 4902: 4897: 4892: 4889: 4865: 4861: 4858: 4855: 4849: 4844: 4841: 4836: 4831: 4828: 4818: 4813: 4797: 4794: 4789: 4784: 4781: 4776: 4771: 4767: 4764: 4761: 4755: 4749: 4746: 4743: 4738: 4735: 4732: 4729: 4726: 4723: 4720: 4714: 4709: 4706: 4701: 4696: 4693: 4668: 4662: 4659: 4654: 4651: 4648: 4645: 4642: 4636: 4631: 4628: 4623: 4618: 4615: 4605: 4604: 4603: 4601: 4597: 4587: 4585: 4581: 4577: 4573: 4569: 4564: 4562: 4558: 4554: 4549: 4531: 4525: 4522: 4519: 4516: 4513: 4510: 4507: 4501: 4495: 4492: 4489: 4483: 4477: 4474: 4471: 4461: 4460: 4459: 4456: 4453: 4441: 4434: 4426: 4419: 4407: 4403: 4396: 4392: 4385: 4381: 4373: 4369: 4358: 4354: 4350: 4346: 4339: 4335: 4331: 4327: 4322: 4321: 4320: 4319: 4315: 4314:ordered pairs 4311: 4306: 4302: 4298: 4294: 4288: 4284: 4278: 4274: 4270:differences, 4260: 4256: 4251: 4247: 4242: 4238: 4232: 4228: 4224: 4220: 4215: 4211: 4205: 4201: 4197: 4193: 4188: 4184: 4180: 4176: 4172: 4168: 4164: 4160: 4156: 4155: 4154: 4152: 4148: 4144: 4140: 4134: 4124: 4122: 4116: 4114: 4110: 4106: 4102: 4098: 4095: 4091: 4074: 4070: 4066: 4062: 4056: 4052: 4039: 4035: 4031: 4027: 4026: 4025: 4022: 4020: 4016: 4012: 4008: 4004: 4000: 3995: 3991: 3967: 3964: 3961: 3955: 3946: 3942: 3936: 3932: 3925: 3921: 3915: 3911: 3907: 3903: 3899: 3898: 3897: 3895: 3894:cardinalities 3891: 3887: 3881: 3871: 3869: 3865: 3861: 3857: 3853: 3849: 3839: 3837: 3833: 3829: 3547: 3544: 3540: 3535: 3531: 3526: 3522: 3518: 3513: 3509: 3505: 3500: 3496: 3492:, evaluating 3491: 3487: 3483: 3479: 3475: 3471: 3467: 3463: 3459: 3455: 3451: 3447: 3442: 3438: 3434: 3429: 3423: 3419: 3415: 3408: 3401: 3397: 3390: 3386: 3382: 3377: 3373: 3371: 3367: 3363: 3359: 3355: 3354:Blaise Pascal 3351: 3349: 3345: 3341: 3337: 3329: 3324: 3320: 3318: 3314: 3309: 3307: 3303: 3300: 3296: 3292: 3288: 3284: 3280: 3276: 3272: 3268: 3265: 3260: 3253: 3248: 3239: 3185: 3183: 3175: 3172: 3171: 3170: 3163: 3160: 3157: 3154: 3153: 3152: 3148: 3122: 3119: 3115: 3111: 3108: 3105: 3100: 3097: 3093: 3089: 3086: 3083: 3078: 3075: 3071: 3067: 3064: 3061: 3056: 3053: 3049: 3045: 3042: 3039: 3034: 3031: 3027: 3023: 3020: 3013: 3012: 3011: 3010:For example: 3008: 2992: 2988: 2967: 2945: 2941: 2937: 2934: 2931: 2928: 2920: 2914: 2901: 2898: 2895: 2886: 2882: 2875: 2870: 2864: 2854: 2840: 2837: 2818: 2815: 2804: 2801: 2798: 2794: 2791: 2788: 2787:word problems 2784: 2781: 2778: 2774: 2771: 2768: 2767:a + b = b + a 2764: 2761: 2760: 2759: 2757: 2753: 2749: 2734: 2731: 2728: 2725: 2722: 2719: 2716: 2713: 2710: 2707: 2703: 2699: 2696: 2693: 2690: 2687: 2684: 2681: 2678: 2675: 2672: 2668: 2664: 2661: 2658: 2655: 2652: 2649: 2646: 2643: 2640: 2637: 2633: 2629: 2626: 2623: 2620: 2617: 2614: 2611: 2608: 2605: 2602: 2598: 2594: 2591: 2588: 2585: 2582: 2579: 2576: 2573: 2570: 2567: 2563: 2559: 2556: 2553: 2550: 2547: 2544: 2541: 2538: 2535: 2532: 2528: 2524: 2521: 2518: 2515: 2512: 2509: 2506: 2503: 2500: 2497: 2493: 2489: 2486: 2483: 2480: 2477: 2474: 2471: 2468: 2465: 2462: 2458: 2454: 2451: 2448: 2445: 2442: 2439: 2436: 2433: 2430: 2427: 2423: 2419: 2416: 2413: 2410: 2407: 2404: 2401: 2398: 2395: 2392: 2388: 2384: 2381: 2378: 2375: 2372: 2369: 2366: 2363: 2360: 2354: 2353: 2350: 2342: 2338: 2328: 2324: 2320: 2316: 2306: 2304: 2300: 2296: 2292: 2288: 2284: 2280: 2275: 2273: 2269: 2258: 2254: 2250: 2246: 2242: 2227: 2225: 2221: 2211: 2205: 2201: 2196: 2192: 2187: 2183: 2179: 2173: 2167: 2163: 2153: 2150: 2146: 2141: 2136: 2132: 2126: 2122: 2118: 2114: 2113: 2108: 2099: 2095: 2091: 2087: 2086: 2085: 2082: 2077: 2073: 2069: 2060: 2051: 2049: 2045: 2041: 2036: 2028: 2024: 2020: 2016: 2012: 2008: 2002: 1998: 1994: 1990: 1986: 1982: 1978: 1974: 1970: 1966: 1962: 1958: 1953: 1951: 1947: 1938: 1932:Associativity 1929: 1927: 1919: 1915: 1911: 1907: 1904: 1903: 1902: 1900: 1896: 1892: 1883: 1877:Commutativity 1869: 1867: 1863: 1859: 1855: 1851: 1847: 1843: 1839: 1835: 1831: 1826: 1822: 1816: 1812: 1807: 1803: 1799: 1795: 1791: 1787: 1783: 1775: 1774: 1773: 1766: 1758: 1749: 1748:of the rods. 1745: 1743: 1735: 1734: 1733: 1731: 1722: 1713: 1711: 1701: 1699: 1691: 1687: 1683: 1678: 1676: 1672: 1668: 1664: 1660: 1656: 1652: 1648: 1644: 1640: 1636: 1633: 1625: 1620: 1616: 1614: 1610: 1607: 1604: 1600: 1596: 1590: 1585: 1581: 1577: 1573: 1569: 1566: 1562: 1558: 1554: 1550: 1545: 1543: 1539: 1535: 1531: 1527: 1523: 1519: 1515: 1491: 1488: 1483: 1479: 1475: 1470: 1466: 1462: 1457: 1453: 1449: 1444: 1440: 1436: 1431: 1427: 1423: 1418: 1414: 1408: 1403: 1400: 1397: 1393: 1385: 1384: 1383: 1381: 1377: 1373: 1370:The sum of a 1365: 1361: 1360:juxtaposition 1345: 1342: 1337: 1334: 1329: 1326: 1323: 1318: 1315: 1310: 1302: 1298: 1294: 1293: 1292: 1286: 1281: 1274: 1258: 1255: 1252: 1249: 1246: 1243: 1240: 1237: 1234: 1227: 1224: 1208: 1205: 1202: 1199: 1196: 1193: 1190: 1183: 1168: 1165: 1162: 1159: 1156: 1149: 1148: 1147: 1145: 1141: 1137: 1130:The plus sign 1128: 1112: 1107: 1105: 1100: 1098: 1093: 1092: 1090: 1089: 1059: 1043: 1028: 1019: 1010: 1007: 1003: 999: 973: 957: 932: 929: 925: 923: 918: 892: 876: 871: 820: 817: 813: 809: 806: 752: 742: 726: 721: 658: 655: 651: 647: 644: 619: 603: 598: 584: 563: 541: 538: 534: 530: 504: 488: 483: 469: 448: 426: 423: 419: 415: 389: 373: 368: 354: 333: 312: 291: 269: 266: 262: 258: 255: 254: 248: 243: 241: 236: 234: 229: 228: 225: 221: 220: 217: 215: 211: 207: 203: 194: 192: 188: 187: 182: 178: 174: 170: 165: 163: 159: 155: 151: 147: 143: 139: 135: 131: 127: 123: 119: 114: 112: 108: 105:(that is, "3 101: 97: 96: 91: 90:whole numbers 87: 83: 79: 75: 71: 64: 60: 53: 48: 44: 40: 33: 19: 10510:Division (2) 10474:Division (2) 10445:Hexation (6) 10420:Addition (1) 10419: 10337: 10312: 10291: 10277: 10272: 10270: 10205: 10188: 10184: 10159: 10138: 10114: 10096: 10077: 10060: 10040: 10019: 9999: 9979: 9935:math/0005163 9924: 9882: 9865:math/0112034 9855: 9851: 9841: 9809: 9779: 9750: 9728: 9706: 9702:Riehl, Emily 9683: 9667:. Springer. 9664: 9641: 9613: 9590: 9573: 9553: 9524: 9503: 9481: 9461: 9440: 9418: 9402:. Retrieved 9393: 9361: 9342: 9313: 9280: 9252: 9221: 9192: 9162: 9135: 9116: 9092: 9067:Gbur, p. 300 9063: 9054: 9049:Stewart p. 8 9045: 9040:Martin p. 49 9036: 9027: 9018: 9009: 9000: 8991: 8982: 8969: 8960: 8951: 8942: 8934: 8930: 8925: 8916: 8911:Rudin p. 178 8907: 8898: 8889: 8880: 8859: 8851: 8842: 8833: 8814:, New York: 8811: 8805: 8790:, Springer, 8787: 8781: 8772: 8763: 8754: 8746: 8741: 8732: 8715: 8710: 8701: 8692: 8684: 8679: 8670: 8661: 8652: 8643: 8630: 8621: 8612: 8603: 8595: 8590: 8578: 8567:. Retrieved 8558: 8554: 8544: 8524: 8507: 8498: 8490: 8485: 8476: 8467: 8454: 8443:Jean Marguin 8438: 8417: 8410: 8401: 8393: 8388: 8380: 8375: 8367: 8362: 8350:. Retrieved 8346: 8337: 8325:. Retrieved 8321: 8312: 8282: 8277: 8268: 8241: 8237: 8203: 8187: 8178: 8166:. Retrieved 8158:The Guardian 8157: 8147: 8138: 8129: 8120: 8111: 8087: 8080: 8072: 8067: 8058: 8049: 8021: 8014: 8006: 7997: 7989: 7984: 7977:Adding it up 7976: 7972: 7963: 7940: 7929: 7919: 7912: 7903: 7882: 7873: 7852: 7841:. Retrieved 7837: 7805: 7779: 7753: 7735: 7727: 7722: 7713: 7702:. Retrieved 7698: 7689: 7661: 7657:"Arithmetic" 7650: 7642: 7638: 7634: 7630: 7625: 7604: 7596: 7591: 7539: 7501: 7487: 7481: 7463: 7452: 7442: 7440: 7321: 7305: 7303: 7219: 7207: 7123: 7114: 7110: 7108: 7101: 7092: 7088: 7084: 7078: 7074: 7058: 7054: 7050: 7046: 7044: 7020:Log-log plot 6995: 6991: 6987: 6983: 6979: 6975: 6971: 6963: 6959: 6955: 6951: 6945: 6938: 6934: 6922: 6899: 6830: 6805: 6797: 6793: 6780: 6776: 6754: 6742: 6718: 6699: 6671: 6646:exclusive or 6631: 6360: 5627: 5623: 5617: 5613: 5608: 5604: 5600: 5596: 5594: 5564: 5473: 5469: 5465: 5461: 5454:vector space 5447: 5416: 5402: 5398: 5394: 5390: 5386: 5382: 5374: 5370: 5366: 5362: 5354: 5350: 5346: 5344: 5240: 5223:Georg Cantor 5220: 5105: 5101: 5094: 5088: 5074: 4994: 4990: 4979:Dedekind cut 4976: 4967:Real numbers 4958: 4956: 4817:denominators 4814: 4683: 4594:Addition of 4593: 4575: 4565: 4550: 4546: 4457: 4451: 4439: 4432: 4424: 4417: 4405: 4401: 4394: 4390: 4383: 4379: 4371: 4367: 4363: 4356: 4352: 4348: 4344: 4337: 4333: 4329: 4325: 4304: 4300: 4296: 4292: 4286: 4282: 4276: 4272: 4268: 4258: 4254: 4249: 4245: 4240: 4236: 4230: 4226: 4222: 4218: 4213: 4209: 4203: 4199: 4195: 4191: 4186: 4182: 4178: 4174: 4170: 4166: 4162: 4158: 4136: 4117: 4112: 4108: 4104: 4100: 4096: 4087: 4072: 4068: 4064: 4060: 4054: 4050: 4037: 4033: 4029: 4023: 4014: 4010: 4006: 4002: 3993: 3989: 3986: 3944: 3940: 3934: 3930: 3923: 3919: 3913: 3909: 3905: 3901: 3889: 3885: 3883: 3864:real numbers 3845: 3825: 3542: 3538: 3524: 3520: 3516: 3511: 3507: 3503: 3498: 3494: 3458:instructions 3430: 3412: 3403: 3399: 3392: 3388: 3384: 3352: 3333: 3310: 3275:differential 3257: 3254:for details. 3192: 3181: 3179: 3167: 3150: 3009: 2916: 2899: 2892: 2883: 2879: 2866: 2851: 2838: 2817:Five and ten 2816: 2803:Near-doubles 2802: 2792: 2782: 2772: 2766: 2762: 2745: 2348: 2339: 2327:number bonds 2322: 2318: 2312: 2276: 2256: 2253:Mickey Mouse 2238: 2217: 2203: 2199: 2194: 2190: 2185: 2177: 2171: 2165: 2159: 2148: 2144: 2134: 2130: 2116: 2110: 2104: 2097: 2093: 2089: 2080: 2065: 2037: 2026: 2022: 2018: 2014: 2010: 2006: 2000: 1996: 1992: 1988: 1984: 1980: 1976: 1972: 1968: 1964: 1960: 1956: 1954: 1944:Addition is 1943: 1923: 1917: 1913: 1909: 1905: 1898: 1894: 1889:Addition is 1888: 1865: 1857: 1853: 1849: 1845: 1841: 1837: 1833: 1824: 1820: 1814: 1810: 1805: 1801: 1797: 1793: 1785: 1781: 1779: 1771: 1746: 1739: 1727: 1707: 1697: 1679: 1654: 1650: 1638: 1634: 1629: 1623: 1612: 1608: 1601:. Using the 1598: 1594: 1588: 1579: 1575: 1567: 1546: 1533: 1528:, which are 1525: 1521: 1517: 1513: 1511: 1369: 1301:mixed number 1300: 1290: 1133: 921: 590:multiplicand 260: 195: 184: 166: 134:real numbers 115: 106: 93: 58: 57: 43: 10299:Subtraction 9227:McGraw-Hill 8519:see p. 767. 8347:nychold.com 7547:defined by 7541:Convolution 7523:game theory 7489:Integration 7471:, which is 7443:dequantized 7001:1 / (2 + 2) 6918:Lie algebra 6816:yields the 6756:Subtraction 6716:operation. 6710:transfinite 6678:commutative 6674:associative 5569:, in which 4582:, with the 4263:−6 + 4 = −2 3201:) and 10111 3141:Non-decimal 2846:10 + 4 = 14 2241:habituation 2107:Brahmagupta 1946:associative 1891:commutative 1862:subtraction 1538:Renaissance 1144:equals sign 710:denominator 418:Subtraction 177:associative 169:commutative 142:mathematics 103:"3 + 2 = 5" 78:subtraction 63:plus symbol 10571:Categories 10181:"Addition" 9962:1024.14026 9915:1103.14034 9885:. Zürich: 9609:Gbur, Greg 9404:3 February 9390:"Addition" 9075:References 8837:Gbur, p. 1 8569:2015-03-30 8447:René Taton 8142:Wynn p. 19 8133:Wynn p. 17 8124:Wynn p. 15 8003:Lappan, G. 7957:required.) 7843:2020-08-25 7834:"Addition" 7704:2020-08-25 7695:"Addition" 7637:with card 7519:strategies 6925:distribute 6914:Lie groups 6910:slide rule 6906:logarithms 6751:Arithmetic 6729:direct sum 5571:velocities 5423:set theory 4584:direct sum 3862:, and the 3852:set theory 3472:, and the 3441:full adder 3422:carry skip 3381:Full adder 3211:1 + 1 = 10 2839:Making ten 2833:5 + 7 = 12 2829:6 + 7 = 13 2811:7 + 7 = 14 2807:6 + 6 = 12 2331:6 + 6 = 12 2299:chimpanzee 2249:Karen Wynn 2092:+ 0 = 0 + 2084:, one has 1872:Properties 1866:vice versa 1530:multiplied 1285:underlined 580:multiplier 514:difference 475:subtrahend 74:arithmetic 70:operations 10197:2151-5743 9982:. Wiley. 9970:Computing 9846:, Kluwer. 9710:. Dover. 9633:704518582 9582:QA248.B95 9485:. Wolfe. 9445:. Wiley. 9211:Education 9031:Viro p. 4 8352:April 20, 8327:April 20, 8115:Wynn p. 5 7804:(1983) . 7758:CRC Press 7617:Footnotes 7493:continuum 7469:empty sum 7465:Summation 7378:⁡ 7364:→ 7280:⁡ 7268:⁡ 7256:≈ 7238:⁡ 7222:logarithm 7005:1/2 + 1/2 6808:is not a 6733:wedge sum 6725:coproduct 6303:⋯ 6234:⋮ 6229:⋱ 6224:⋮ 6219:⋮ 6181:⋯ 6093:⋯ 5999:⋯ 5962:⋮ 5957:⋱ 5952:⋮ 5947:⋮ 5925:⋯ 5879:⋯ 5818:⋯ 5781:⋮ 5776:⋱ 5771:⋮ 5766:⋮ 5744:⋯ 5698:⋯ 5407:congruent 5379:triangles 5051:∈ 5039:∈ 5033:∣ 4747:× 4736:× 4724:× 4557:semigroup 4429:, and by 4048:. Define 3965:∪ 3462:branching 3279:pressures 3264:averaging 3242:Computers 3161:1 + 0 → 1 3158:0 + 1 → 1 3155:0 + 0 → 0 3120:− 3112:× 3098:− 3090:× 3076:− 3068:× 3054:− 3046:× 3032:− 3024:× 2938:× 2823:and 10 + 2777:intuition 2283:eggplants 2272:ping-pong 2182:successor 2156:Successor 2048:nth roots 1694:+ 1688::U+002B; 1682:plus sign 1667:Frontinus 1663:Vitruvius 1603:gerundive 1578:"to" and 1394:∑ 1380:iteration 1136:plus sign 1069:logarithm 1029:⁡ 1002:Logarithm 703:numerator 585:× 564:× 470:− 449:− 162:subgroups 158:subspaces 10577:Addition 10345:Division 10278:Addition 9806:Baez, J. 9704:(2016). 9611:(2011). 9416:(2017). 9398:Archived 9278:(2001). 9115:(1925). 8935:addition 8720:Archived 8583:Enderton 8563:Archived 8533:Archived 8260:30034895 8200:(1999). 8168:29 March 8162:Archived 8093:OpenStax 8005:(2014). 7777:(1974). 7555:See also 7483:Counting 7011:Ordering 6947:Division 6933:(1 + 1)( 6650:geometry 6644:as the " 5585:Matrices 4151:negative 4147:positive 4127:Integers 4077:. Hence 3856:integers 3812:<< 3683:<< 3466:parallel 3299:resistor 3291:voltages 3182:carrying 2960:, where 2752:memorize 2315:counting 2279:primates 2268:toddlers 2140:Bhaskara 2125:Mahavira 1780:The sum 1659:Boethius 1572:compound 1553:Addition 1522:summands 1366:instead. 1362:denotes 1297:fraction 945:radicand 844:exponent 773:quotient 762:fraction 679:dividend 650:Division 261:Addition 214:computer 173:operands 154:matrices 130:integers 118:counting 116:Besides 113:to 5"). 86:division 59:Addition 10455:Inverse 10408:Primary 10354:∕ 10304:− 10294:− 10191:(3–5). 9940:Bibcode 9858:: 275. 9794:Bibcode 9125:QA21.K3 9081:History 8749:(1995). 7511:complex 5458:vectors 5438:Vectors 5113:Define 5001:Define 4606:Define 4257:| and | 4161:, let | 4133:Integer 4092:on the 3997:is the 3938:. Then 3916:, with 3460:during 3450:address 3426:999 + 1 3302:network 3287:pistons 2873:27 + 59 2869:carried 2793:Doubles 2748:decimal 2245:infants 2066:Adding 1686:Unicode 1675:Chaucer 1655:summare 1639:summare 1599:give to 1561:English 1555:" and " 1526:factors 1520:or the 1518:addends 1287:number. 924:th root 686:divisor 630:product 465:minuend 318:summand 308:summand 206:decimal 150:vectors 146:algebra 126:numbers 10350:÷ 10340:÷ 10329:· 10325:× 10315:× 10212: 10195: 10168: 10145: 10126: 10084: 10067: 10048: 10026: 10007: 9986: 9960: 9950: 9913: 9903: 9826: 9758: 9737: 9714: 9690: 9671: 9652: 9631: 9621: 9597: 9580: 9561: 9531: 9512: 9489: 9468: 9449: 9426: 9368: 9349: 9321: 9300: 9264: 9233: 9199: 9171: 9148: 9123: 9101: 8868: 8822: 8794: 8561:(45). 8449:(1963) 8425: 8258: 8212: 8099: 8037: 7936:"plus" 7892:Romans 7888:Greeks 7812: 7791: 7677: 7531:states 7515:mixing 7477:series 6658:circle 5579:forces 5405:, are 5393:, and 4053:+ 0 = 3987:Here, 3900:Let N( 3866:. (In 3858:, the 3743:return 3698:return 3478:abacus 3454:memory 3414:Adders 3344:Africa 3336:abacus 3295:ground 3271:shafts 2257:expect 2147:+ 0 = 1999:, and 1854:augend 1651:Addere 1647:Romans 1613:augere 1606:suffix 1597:is to 1589:*deh₃- 1568:addere 1559:" are 1534:augend 1516:, the 1508:Terms 1372:series 951:degree 569:factor 559:factor 360:addend 350:augend 339:addend 329:addend 210:abacus 52:apples 18:Addend 10283:+ 10273:+ 9930:arXiv 9891:arXiv 9860:arXiv 9814:arXiv 9784:arXiv 9258:Wiley 8433:p. 11 8256:JSTOR 8029:(and 7951: 7609:term. 7583:Notes 7320:" as 7312:from 7037:from 7031:max ( 6902:table 6804:. If 4880:, so 4559:with 4553:group 4398:, or 4332:) ~ ( 4235:. If 4229:| + | 4225:= −(| 4208:. If 4202:| + | 4082:1 = 2 4067:) = ( 4046:1 = 2 4042:0 = 1 3999:union 3850:. 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5045:, 5042:a 5036:q 5030:r 5027:+ 5024:q 5021:{ 5018:= 5015:b 5012:+ 5009:a 4995:b 4991:a 4940:4 4937:3 4932:= 4927:4 4923:2 4920:+ 4917:1 4911:= 4906:4 4903:2 4898:+ 4893:4 4890:1 4866:c 4862:b 4859:+ 4856:a 4850:= 4845:c 4842:b 4837:+ 4832:c 4829:a 4798:8 4795:7 4790:= 4777:= 4768:4 4765:+ 4756:= 4750:8 4744:4 4739:1 4733:4 4730:+ 4727:8 4721:3 4715:= 4710:8 4707:1 4702:+ 4697:4 4694:3 4669:. 4663:d 4660:b 4655:c 4652:b 4649:+ 4646:d 4643:a 4637:= 4632:d 4629:c 4624:+ 4619:b 4616:a 4532:. 4529:) 4526:d 4523:+ 4520:b 4517:, 4514:c 4511:+ 4508:a 4505:( 4502:= 4499:) 4496:d 4493:, 4490:c 4487:( 4484:+ 4481:) 4478:b 4475:, 4472:a 4469:( 4452:n 4450:+ 4446:n 4442:) 4440:n 4433:n 4431:– 4425:n 4423:( 4418:n 4416:+ 4412:n 4408:) 4406:a 4402:b 4395:b 4391:a 4384:b 4380:a 4378:( 4374:) 4372:b 4368:a 4366:( 4360:. 4357:c 4353:b 4349:d 4345:a 4340:) 4338:d 4334:c 4330:b 4326:a 4324:( 4305:c 4301:b 4297:d 4293:a 4287:d 4283:c 4277:b 4273:a 4259:b 4255:a 4250:b 4246:a 4241:b 4237:a 4231:b 4227:a 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3057:6 3040:+ 3035:5 2993:b 2968:a 2946:b 2935:a 2932:= 2929:x 2848:. 2825:x 2821:x 2779:. 2708:9 2676:9 2673:8 2644:9 2641:8 2638:7 2612:9 2609:8 2606:7 2603:6 2580:9 2577:8 2574:7 2571:6 2568:5 2548:9 2545:8 2542:7 2539:6 2536:5 2533:4 2516:9 2513:8 2510:7 2507:6 2504:5 2501:4 2498:3 2484:9 2481:8 2478:7 2475:6 2472:5 2469:4 2466:3 2463:2 2452:9 2449:8 2446:7 2443:6 2440:5 2437:4 2434:3 2431:2 2428:1 2417:8 2414:7 2411:6 2408:5 2405:4 2402:3 2399:2 2396:1 2393:0 2382:8 2379:7 2376:6 2373:5 2370:4 2367:3 2364:2 2361:1 2358:0 2204:a 2200:b 2195:b 2191:a 2186:a 2178:a 2172:a 2170:( 2166:a 2149:a 2145:a 2135:a 2131:a 2117:a 2101:. 2098:a 2094:a 2090:a 2081:a 2029:) 2027:c 2023:b 2019:a 2015:c 2011:b 2007:a 2005:( 2001:c 1997:b 1993:a 1989:c 1985:b 1981:a 1977:c 1973:b 1969:a 1965:c 1961:b 1957:a 1920:. 1918:a 1914:b 1910:b 1906:a 1899:b 1895:a 1858:a 1850:a 1846:b 1842:a 1838:a 1834:b 1832:+ 1825:b 1821:a 1815:b 1811:a 1806:a 1802:b 1798:b 1794:a 1786:b 1782:a 1489:= 1484:2 1480:5 1476:+ 1471:2 1467:4 1463:+ 1458:2 1454:3 1450:+ 1445:2 1441:2 1437:+ 1432:2 1428:1 1424:= 1419:2 1415:k 1409:5 1404:1 1401:= 1398:k 1343:= 1338:2 1335:1 1330:+ 1327:3 1324:= 1319:2 1316:1 1311:3 1275:) 1256:= 1253:3 1250:+ 1247:3 1244:+ 1241:3 1238:+ 1235:3 1225:) 1206:= 1203:2 1200:+ 1197:4 1194:+ 1191:5 1169:3 1166:= 1163:2 1160:+ 1157:1 1110:e 1103:t 1096:v 1044:= 1040:) 1032:( 958:= 922:n 877:= 872:} 753:{ 727:= 722:} 604:= 599:} 489:= 484:} 374:= 369:} 355:+ 334:+ 313:+ 292:+ 246:e 239:t 232:v 191:0 181:1 66:+ 41:. 34:. 20:)

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