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
1747:
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
5079:
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.
2896:
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
6777:
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
4269:
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
544:
2880:
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.
3443:
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
823:
746:
2340:
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.
3261:
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
7124:
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:
7212:
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.
7608:
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
7436:
4878:
3439:
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
3151:
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:
2867:
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.
642:
5340:
914:
7203:
995:
9842:
8745:
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."
411:
3016:
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3007:
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.
8607:
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!"
6895:
2337:
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.
98:
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
3476:
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.
1760:
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.
8758:
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.
9883:
Proceedings of the
International Congress of Mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume II: Invited lectures. Tropical Geometry and its Applications
3981:
1269:
1219:
1724:
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).
3005:
9778:
Akian, Marianne; Bapat, Ravindra; Gaubert, Stephane (2005). "Min-plus methods in eigenvalue perturbation theory and generalised
Lidskii-Vishik-Ljusternik theorem".
1179:
4957:
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
8272:
Beckmann, S. (2014). The twenty-third ICMI study: primary mathematics study on whole numbers. International
Journal of STEM Education, 1(1), 1-8. Chicago
5345:
Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers
8736:
The intuitive approach, inverting every element of a cut and taking its complement, works only for irrational numbers; see
Enderton p. 117 for details.
8696:
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.
7739:
Department of the Army (1961) Army
Technical Manual TM 11-684: Principles and Applications of Mathematics for Communications-Electronics . Section 5.1
3480:
and adding board, both addends are destroyed, leaving only the sum. The influence of the abacus on mathematical thinking was strong enough that early
2266:
is either 1 or 3. This finding has since been affirmed by a variety of laboratories using different methodologies. Another 1992 experiment with older
8946:
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.
2142:
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.
8019:
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,
10206:
Addition and Subtraction: A Cognitive Perspective. Interpretations of Number Operations and Symbolic Representations of Addition and Subtraction
935:
8973:
Enderton calls this statement the "Absorption Law of Cardinal Arithmetic"; it depends on the comparability of cardinals and therefore on the
4977:
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
7513:
number. Linear combinations are especially useful in contexts where straightforward addition would violate some normalization rule, such as
1388:
5116:
3364:
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
1306:
1776:
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
3484:
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:
3846:
To prove the usual properties of addition, one must first define addition for the context in question. Addition is first defined on the
9925:
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).
1924:
The fact that addition is commutative is known as the "commutative law of addition" or "commutative property of addition". Some other
193:
does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication.
10391:
7330:
4578:
was, more specifically, the result of this construction applied to the equivalences classes under isomorphisms of the objects of an
1744:
below). However, it is not obvious how one should extend this version of addition to include fractional numbers or negative numbers.
9397:
4822:
10039:
5417:
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
6943:
in both ways, one concludes that addition is forced to be commutative. For this reason, ring addition is commutative in general.
6727:
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
208:
system, starting with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient
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10085:
10049:
10008:
9987:
9951:
9904:
9759:
9738:
9715:
9691:
9672:
9653:
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9598:
9562:
9532:
9513:
9490:
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9450:
9427:
9350:
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9301:
9265:
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9200:
9172:
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9102:
8869:
8823:
8795:
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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.
10237:
9141:
7801:
7774:
2274:
balls from a box; the youngest responded well for small numbers, while older subjects were able to compute sums up to 5.
1101:
237:
9243:
8529:
8317:
8086:
6743:
Addition, along with subtraction, multiplication and division, is considered one of the basic operations and is used in
1768:
A number-line visualization of the unary addition 2 + 4 = 6. A translation by 4 is equivalent to four translations by 1.
1649:
to add upward, contrary to the modern practice of adding downward, so that a sum was literally higher than the addends.
10499:
10463:
9417:
1658:
8562:
507:
10213:
10127:
9827:
9369:
7792:
3230:. This time, a 1 is carried, and a 1 is written in the bottom row. Proceeding like this gives the final answer 100100
1062:
5460:
and for scaling vectors. A familiar vector space is the set of all ordered pairs of real numbers; the ordered pair (
5004:
9285:
8030:
4972:
623:
6923:
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.).
7220:
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
7856:
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.
7967:
See Viro 2001 for an example of the sophistication involved in adding with sets of "fractional cardinality".
6656:
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.
6841:
6778:
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
9480:
8937:
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:
2924:
10282:
8020:
7941:
7214:
3445:
2210:
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
4599:
3529:
3274:
2124:
1583:
99:
10353:
5241:
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
1571:
1512:
The numbers or the objects to be added in general addition are collectively referred to as the
1375:
1296:
649:
110:
85:
17:
10078:
Histoire des Instruments et Machines à Calculer, Trois Siècles de Mécanique Pensante 1642–1942
6708:
in set theory. These give two different generalizations of addition of natural numbers to the
3838:
was a series of bugs where overflow errors occurred due to use of a 2-digit format for years.
2750:
system is the fluent recall or derivation of the 100 single-digit "addition facts". One could
2119:
is negative, positive, or zero itself, and he used words rather than algebraic symbols. Later
1772:
A second interpretation of addition comes from extending an initial length by a given length:
1283:
Columnar addition – the numbers in the column are to be added, with the sum written below the
1186:
10545:
10253:
7954:
7526:
6744:
5406:
4560:
4093:
3867:
3469:
3365:
3305:
2776:
2120:
2111:
620:
223:
8202:
2983:
2262:
to be 2, and they are comparatively surprised when a physical situation seems to imply that
2127:
wrote, "zero becomes the same as what is added to it", corresponding to the unary statement
10115:
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
5237:
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:×
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7936:"plus"
7892:Romans
7888:Greeks
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7477:series
6658:circle
5579:forces
5405:, are
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3866:. (In
3858:, the
3743:return
3698:return
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3454:memory
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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
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1508:Terms
1372:series
951:degree
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559:factor
360:addend
350:augend
339:addend
329:addend
210:abacus
52:apples
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10283:+
10273:+
9930:arXiv
9891:arXiv
9860:arXiv
9814:arXiv
9784:arXiv
9258:Wiley
8433:p. 11
8256:JSTOR
8029:(and
7951:
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7583:Notes
7320:" as
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3109:2.907
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2857:Carry
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2264:1 + 1
2260:1 + 1
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1690:ASCII
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1273:below
1223:below
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1024:base
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4187:b
4183:a
4179:b
4175:a
4171:b
4167:a
4163:n
4159:n
4113:a
4109:a
4105:b
4101:a
4097:N
4084:.
4075:)
4073:b
4069:a
4065:b
4061:a
4055:a
4051:a
4038:n
4034:n
4030:n
4015:B
4011:A
4007:B
4003:A
3994:B
3990:A
3983:.
3971:)
3968:B
3962:A
3959:(
3956:N
3945:b
3941:a
3935:b
3931:B
3924:a
3920:A
3914:B
3910:A
3906:S
3902:S
3890:b
3886:a
3821:}
3815:1
3809:)
3806:y
3803:,
3800:x
3797:(
3788:y
3785:,
3782:x
3779:(
3773:(
3764:)
3761:0
3755:y
3752:(
3746:x
3740:{
3737:)
3734:y
3728:,
3725:x
3719:(
3707:}
3704:;
3701:x
3695:}
3689:;
3686:1
3677:=
3674:y
3665:y
3662:,
3659:x
3656:(
3650:=
3647:x
3638:y
3635:,
3632:x
3629:(
3623:=
3617:{
3614:)
3611:0
3605:y
3602:(
3596:;
3593:0
3590:=
3581:{
3578:)
3575:y
3569:,
3566:x
3560:(
3543:b
3539:a
3530:C
3525:b
3521:a
3517:a
3512:a
3508:b
3504:a
3499:b
3495:a
3409:.
3404:C
3400:S
3393:C
3389:B
3385:A
3379:"
3232:2
3227:2
3220:2
3213:2
3203:2
3195:2
3123:5
3106:=
3101:5
3084:+
3079:5
3062:=
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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