1575:
2366:
769:
50:
1764:
147:
1052:
858:
1560:, respectively. In some contexts, the choice of this assignment (i.e., which range of values is considered positive and which negative) is natural, whereas in other contexts, the choice is arbitrary, making an explicit sign convention necessary, the only requirement being consistent use of the convention.
792:
When dealing with numbers, it is often convenient to have their sign available as a number. This is accomplished by functions that extract the sign of any number, and map it to a predefined value before making it available for further calculations. For example, it might be advantageous to formulate
2356:
values. The floating point values are represented using three separate values, mantissa, exponent, and sign. Given this separate sign bit, it is possible to represent both positive and negative zero. Most programming languages normally treat positive zero and negative zero as equivalent values,
685:
Since the real and complex numbers both form a field and contain the positive reals, they also contain the reciprocals of the magnitudes of all non-zero numbers. This means that any non-zero number may be multiplied with the reciprocal of its magnitude, that is, divided by its magnitude. It is
1530:
1047:{\displaystyle {\begin{aligned}\operatorname {sgn} :{}&\mathbb {R} \to \{-1,0,1\}\\&x\mapsto \operatorname {sgn}(x)={\begin{cases}-1&{\text{if }}x<0,\\~~\,0&{\text{if }}x=0,\\~~\,1&{\text{if }}x>0.\end{cases}}\end{aligned}}}
677:
In general, any arbitrary real value can be specified by its magnitude and its sign. Using the standard encoding, any real value is given by the product of the magnitude and the sign in standard encoding. This relation can be generalized to define a
628:
Complex numbers are impossible to order, so they cannot carry the structure of an ordered ring, and, accordingly, cannot be partitioned into positive and negative complex numbers. They do, however, share an attribute with the reals, which is called
1412:
1758:
is usually drawn with positive numbers to the right, and negative numbers to the left, a common convention is for motions to the right to be given a positive sign, and for motions to the left to be given a negative sign.
1165:
716:
The sign of a complex number is the exponential of the product of its argument with the imaginary unit. represents in some sense its complex argument. This is to be compared to the sign of real numbers, except with
193:, negation), an operation which is not restricted to real numbers. It applies among other objects to vectors, matrices, and complex numbers, which are not prescribed to be only either positive, negative, or zero.
604:
of a real number is always "non-negative", but is not necessarily "positive" in the first interpretation, whereas in the second interpretation, it is called "positive"—though not necessarily "strictly positive".
1294:
1702:
178:. Depending on local conventions, zero may be considered as having its own unique sign, having no sign, or having both positive and negative sign. In some contexts, it makes sense to distinguish between
1538:. If the original value was R,θ in polar form, then sign(R, θ) is 1 θ. Extension of sign() or signum() to any number of dimensions is obvious, but this has already been defined as normalizing a vector.
1791:, then the horizontal part will be positive for motion to the right and negative for motion to the left, while the vertical part will be positive for motion upward and negative for motion downward.
1379:
863:
390:). Without specific context (or when no explicit sign is given), a number is interpreted per default as positive. This notation establishes a strong association of the minus sign "
433:
853:
754:
339:
When a minus sign is used in between two numbers, it represents the binary operation of subtraction. When a minus sign is written before a single number, it represents the
359:), the additive inverse of a positive number is negative, and the additive inverse of a negative number is positive. A double application of this operation is written as
1085:
2333:, an integer value may be either signed or unsigned, depending on whether the computer is keeping track of a sign for the number. By restricting an integer
363:. The plus sign is predominantly used in algebra to denote the binary operation of addition, and only rarely to emphasize the positivity of an expression.
1214:
267:
numbers. Another property required for a ring to be ordered is that, for each positive number, there exists a unique corresponding number less than
1655:
514:(right-sided limit and left-sided limit, respectively). This notation refers to the behaviour of a function as its real input variable approaches
1525:{\displaystyle \operatorname {sgn}(z)={\begin{cases}0&{\text{for }}z=0\\{\dfrac {z}{|z|}}=e^{i\varphi }&{\text{otherwise}}.\end{cases}}}
351:) of the operand. Abstractly then, the difference of two number is the sum of the minuend with the additive inverse of the subtrahend. While
17:
1534:
This definition may also be recognized as a normalized vector, that is, a vector whose direction is unchanged, and whose length is fixed to
1302:
extracts the complex sign of a complex number by mapping the set of non-zero complex numbers to the set of unimodular complex numbers, and
251:
contains a unique number that when added with any number leaves the latter unchanged. This unique number is known as the system's additive
1193:
While a real number has a 1-dimensional direction, a complex number has a 2-dimensional direction. The complex sign function requires the
2376:
In addition to the sign of a real number, the word sign is also used in various related ways throughout mathematics and other sciences:
2533:
are sometimes used when it is convenient for certain areas or volumes to count as negative. This is particularly true in the theory of
1552:
In situations where there are exactly two possibilities on equal footing for an attribute, these are often labelled by convention as
2667:
114:
2561:
comes with a sign, either positive or negative. By convention, a positive charge is a charge with the same sign as that of a
86:
1313:
247:, ... may have multiple attributes, that fix certain properties of a number. A number system that bears the structure of an
196:
The word "sign" is also often used to indicate binary aspects of mathematical or scientific objects, such as odd and even (
67:
639:. Magnitudes are always non-negative real numbers, and to any non-zero number there belongs a positive real number, its
449:. There is generally no danger of confusing the value with its sign, although the convention of assigning both signs to
93:
133:
2650:
686:
immediate that the quotient of any non-zero real number by its magnitude yields exactly its sign. By analogy, the
100:
1625:
rotation around an oriented axis typically counts as positive, while a left-handed rotation counts as negative.
2341:
can be used for storing the value of a number. Because of the way integer arithmetic is done within computers,
71:
480:
472:
82:
1775:, the rightward and upward directions are usually thought of as positive, with rightward being the positive
2342:
1629:
38:
418:
2735:
2547:
2541:, each ordered basis for the vector space can be classified as either positively or negatively oriented.
185:
In mathematics and physics, the phrase "change of sign" is associated with exchanging an object for its
2538:
1617:
It is also possible to associate a sign to an angle of rotation in three dimensions, assuming that the
201:
567:
is said to be both positive and negative, modified phrases are used to refer to the sign of a number:
530:
is said to be neither positive nor negative, the following phrases may refer to the sign of a number:
255:. For example, the integers has the structure of an ordered ring. This number is generally denoted as
812:
809:
extracts the sign of a real number, by mapping the set of real numbers to the set of the three reals
1439:
942:
720:
2718:
2599:
2482:
197:
60:
2334:
1803:
1194:
609:
107:
2508:
2500:
2485:
is defined to be positive if the permutation is even, and negative if the permutation is odd.
1735:
507:
793:
an intricate algorithm for positive values only, and take care of the sign only afterwards.
2714:
1836:
1751:
1614:
to have counterclockwise angles count as positive, and clockwise angles count as negative.
375:
151:
8:
2584:
2346:
2314:
1610:
or counterclockwise direction. Though different conventions can be used, it is common in
475:
of real numbers within computers, it is useful to consider signed versions of zero, with
329:
2448:
1818:
1723:
2691:
1743:
1645:
1603:
328:
is defined). Since rational and real numbers are also ordered rings (in fact ordered
2589:
1788:
1784:
1618:
1587:
465:
344:
284:
252:
186:
612:
that yield real or other signed values. For example, a function would be called a
2558:
2515:
2447:, a quantity known up to sign is a stronger condition than a quantity with known
2440:
2369:
1772:
1622:
1547:
518:
along positive (resp., negative) values; the two limits need not exist or agree.
511:
340:
232:
209:
171:
1726:
has positive derivative, while any decreasing function has negative derivative.
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is a graph in which each edge has been marked with a positive or negative sign.
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2353:
1178:
640:
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601:
367:
236:
228:
2345:
usually do not store the sign as a single independent bit, instead using e.g.
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1822:
1739:
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415:
325:
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2579:
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2489:
2444:
1807:
248:
2624:
1628:
An angle which is the negative of a given angle has an equal arc, but the
1598:
In many contexts, it is common to associate a sign with the measure of an
2717:(1894) "Fundamental theorems of analysis generalized for space", page 3,
2534:
2526:
1755:
1611:
1583:
1535:
1160:{\displaystyle \operatorname {sgn}(x)={\frac {x}{|x|}}={\frac {|x|}{x}},}
476:
260:
179:
167:
159:
2594:
2365:
2324:
1719:
403:
374:
and elsewhere), the sign of a number is often made explicit by placing
371:
240:
2692:"Sign of Angles | What is An Angle? | Positive Angle | Negative Angle"
1606:. In such a situation, the sign indicates whether the angle is in the
394:" with negative numbers, and the plus sign "+" with positive numbers.
2565:, and a negative charge is a charge with the same sign as that of an
2357:
albeit, they provide means by which the distinction can be detected.
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1607:
1591:
205:
2511:
in which the measure of a set may have positive or negative values.
49:
2566:
1814:
1799:
1715:
503:
244:
1569:
768:
2554:
1747:
616:
if its values are positive for all arguments of its domain, or a
386:
denotes "negative three" (algebraically: the additive inverse of
1289:{\displaystyle |z|={\sqrt {z{\bar {z}}}}={\sqrt {x^{2}+y^{2}}}.}
2562:
1579:
224:
190:
2550:, each digit of a number may have a positive or negative sign.
1763:
263:
in this ring, there are numbers greater than zero, called the
2382:
1806:, i.e., receding instead of advancing; a special case is the
1795:
1697:{\displaystyle \Delta x=x_{\text{final}}-x_{\text{initial}}.}
1599:
479:
referring to different, discrete number representations (see
1388:
be also expressed by its magnitude and one of its arguments
464:
positive and negative following the convention set forth by
283:
numbers. The numbers in each such pair are their respective
146:
1518:
1036:
456:
In certain
European countries, e.g. in Belgium and France,
406:
being neither positive nor negative, a specific sign-value
31:
2421:
2338:
2352:
In contrast, real numbers are stored and manipulated as
1718:, this same convention is used in the definition of the
287:. This attribute of a number, being exclusively either
1374:{\displaystyle \{z\in \mathbb {C} :|z|=1\}\cup \{0\}.}
1658:
1466:
1415:
1316:
1217:
1088:
861:
815:
723:
421:
175:
453:
does not immediately allow for this discrimination.
1082:, this function can also be defined by the formula
756:For the definition of a complex sign-function. see
74:. Unsourced material may be challenged and removed.
1817:, notions related to sign can be found in the two
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1524:
1373:
1288:
1159:
1046:
847:
748:
427:
2727:
336:attribute also applies to these number systems.
1710:counts as positive change, while a decrease of
271:whose sum with the original positive number is
437:, as defined for real numbers. In arithmetic,
27:Number property of being positive or negative
1365:
1359:
1353:
1317:
905:
884:
839:
816:
2428:. For real numbers, it means that only the
1798:(rate of change of displacement) implies a
608:The same terminology is sometimes used for
312:, and is often encoded to the real numbers
2462:, there are many other possible values of
1828:
1779:-direction, and upward being the positive
835:
828:
757:
170:is its property of being either positive,
1750:, it is common to label the two possible
1327:
1013:
980:
877:
134:Learn how and when to remove this message
2364:
1840:
1573:
1188:
767:
521:
145:
1602:, particularly an oriented angle or an
1541:
620:if all of its values are non-negative.
589:if it is greater than or equal to zero.
552:if it is greater than or equal to zero.
324:, respectively (similar to the way the
14:
2728:
2507:is a generalization of the concept of
2337:to non-negative values only, one more
1754:as positive and negative. Because the
1729:
1706:Using this convention, an increase in
154:are used to show the sign of a number.
2622:
2317:to represent the sign of an integer.
1652:is typically defined by the equation
796:
2647:
2618:
2616:
2614:
1078:is negative. For non-zero values of
596:if it is less than or equal to zero.
559:if it is less than or equal to zero.
428:{\displaystyle \operatorname {sgn} }
410:may be assigned to the number value
72:adding citations to reliable sources
43:
1635:
1621:has been oriented. Specifically, a
1563:
646:For example, the absolute value of
219:
213:
24:
1762:
1659:
623:
227:from various number systems, like
212:, and other concepts described in
37:For symbols named "... sign", see
25:
2747:
2611:
2360:
763:
494:rarely appear as substitutes for
658:. This is written in symbols as
378:before the number. For example,
48:
1590:direction, and negative in the
848:{\displaystyle \{-1,\;0,\;1\}.}
695:can be defined as the quotient
397:
59:needs additional citations for
2708:
2684:
2660:
2641:
2439:of the quantity is known. For
1783:-direction. If a displacement
1734:When studying one-dimensional
1714:counts as negative change. In
1481:
1473:
1428:
1422:
1381:It may be defined as follows:
1343:
1335:
1245:
1227:
1219:
1144:
1136:
1122:
1114:
1101:
1095:
931:
925:
916:
881:
855:It can be defined as follows:
473:floating-point representations
382:denotes "positive three", and
180:a positive and a negative zero
13:
1:
2605:
2343:signed number representations
1211:, which can be calculated as
749:{\displaystyle e^{i\pi }=-1.}
481:signed number representations
355:is its own additive inverse (
18:Negative and positive numbers
2372:may be positive or negative.
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758:§ Complex sign function
445:both denote the same number
39:List of mathematical symbols
7:
2573:
2548:signed-digit representation
2415:. It is often expressed as
2305:
2253:
2201:
2149:
2097:
2045:
1993:
1941:
1889:
575:if it is greater than zero.
538:if it is greater than zero.
414:. This is exploited in the
10:
2752:
2391:, it is known that either
2387:mean that, for a quantity
2322:
1567:
1545:
785:
650:and the absolute value of
471:In some contexts, such as
36:
29:
2719:link via Internet Archive
2312:
1586:count as positive in the
402:Within the convention of
2651:Éléments de mathématique
1298:Analogous to above, the
689:sign of a complex number
582:if it is less than zero.
545:if it is less than zero.
275:These numbers less than
30:Not to be confused with
2600:Symmetry in mathematics
1829:Signedness in computing
1644:changes over time, the
83:"Sign" mathematics
2373:
1787:is separated into its
1767:
1698:
1595:
1526:
1375:
1290:
1161:
1048:
849:
783:
750:
429:
376:a plus or a minus sign
155:
152:plus and minus symbols
2629:mathworld.wolfram.com
2539:oriented vector space
2501:mathematical analysis
2483:sign of a permutation
2368:
1794:Likewise, a negative
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1699:
1577:
1527:
1376:
1300:complex sign function
1291:
1189:Complex sign function
1162:
1049:
850:
771:
751:
682:for complex numbers.
618:non-negative function
522:Terminology for signs
508:mathematical analysis
430:
214:§ Other meanings
198:sign of a permutation
189:(multiplication with
149:
2715:Alexander Macfarlane
1837:most-significant bit
1656:
1542:Signs per convention
1413:
1314:
1215:
1086:
859:
813:
721:
460:is considered to be
419:
68:improve this article
2648:Bourbaki, Nicolas.
2623:Weisstein, Eric W.
2537:. In an (abstract)
2313:Most computers use
1819:normal orientations
1730:Sign of a direction
1724:increasing function
1722:. As a result, any
1578:Measuring from the
772:Real sign function
2736:Sign (mathematics)
2518:is used to convey
2374:
1804:opposite direction
1768:
1694:
1596:
1522:
1517:
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1371:
1286:
1157:
1044:
1042:
1035:
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797:Real sign function
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746:
654:are both equal to
425:
347:(sometimes called
156:
2655:. p. A VI.4.
2321:
2320:
1789:vector components
1744:analytic geometry
1688:
1675:
1604:angle of rotation
1570:Angle § Sign
1510:
1486:
1450:
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1251:
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1152:
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1066:is positive, and
1022:
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1009:
989:
979:
976:
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614:positive function
600:For example, the
580:strictly negative
573:strictly positive
285:additive inverses
144:
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136:
118:
16:(Redirected from
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2706:
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2702:
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2668:"SignumFunction"
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2639:
2638:
2636:
2635:
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2590:Positive element
2522:, inside or out.
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2465:
2461:
2454:
2438:
2427:
2414:
2410:
2400:
2390:
2347:two's complement
2315:two's complement
1833:
1832:
1703:
1701:
1700:
1695:
1690:
1689:
1686:
1677:
1676:
1673:
1648:in the value of
1640:When a quantity
1636:Sign of a change
1619:axis of rotation
1588:counterclockwise
1582:, angles on the
1564:Sign of an angle
1531:
1529:
1528:
1523:
1521:
1520:
1511:
1508:
1504:
1503:
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1197:of its argument
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529:
517:
512:one-sided limits
501:
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489:
466:Nicolas Bourbaki
459:
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448:
444:
440:
434:
432:
431:
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409:
393:
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385:
381:
368:numeral notation
362:
358:
354:
345:additive inverse
343:of yielding the
323:
319:
315:
308:, is called its
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293:
278:
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270:
258:
253:identity element
220:Sign of a number
210:one sided limits
187:additive inverse
139:
132:
128:
125:
119:
117:
76:
52:
44:
34:in trigonometry.
21:
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2612:
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2585:Plus–minus sign
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2559:electric charge
2516:signed distance
2514:The concept of
2467:
2463:
2456:
2452:
2441:complex numbers
2432:
2416:
2412:
2402:
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2370:Electric charge
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1831:
1773:Cartesian plane
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1548:Sign convention
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807:signum function
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719:
718:
707:
705:
703:
698:
696:
691:
687:
669:
667:
661:
659:
655:
651:
647:
626:
624:Complex numbers
564:
527:
524:
515:
499:
495:
491:
487:
457:
450:
446:
442:
438:
420:
417:
416:
411:
407:
400:
391:
387:
383:
379:
360:
356:
352:
341:unary operation
321:
317:
313:
305:
298:
291:
279:are called the
276:
272:
268:
259:Because of the
256:
237:complex numbers
222:
140:
129:
123:
120:
77:
75:
65:
53:
42:
35:
28:
23:
22:
15:
12:
11:
5:
2749:
2739:
2738:
2723:
2722:
2707:
2696:Math Only Math
2683:
2659:
2640:
2609:
2607:
2604:
2603:
2602:
2597:
2592:
2587:
2582:
2575:
2572:
2571:
2570:
2551:
2544:
2543:
2542:
2523:
2505:signed measure
2497:
2486:
2479:
2430:absolute value
2362:
2361:Other meanings
2359:
2354:floating point
2323:Main article:
2319:
2318:
2310:
2309:
2304:
2299:
2294:
2289:
2284:
2279:
2274:
2269:
2264:
2258:
2257:
2252:
2247:
2242:
2237:
2232:
2227:
2222:
2217:
2212:
2206:
2205:
2200:
2195:
2190:
2185:
2180:
2175:
2170:
2165:
2160:
2154:
2153:
2148:
2143:
2138:
2133:
2128:
2123:
2118:
2113:
2108:
2102:
2101:
2096:
2091:
2086:
2081:
2076:
2071:
2066:
2061:
2056:
2050:
2049:
2044:
2039:
2034:
2029:
2024:
2019:
2014:
2009:
2004:
1998:
1997:
1992:
1987:
1982:
1977:
1972:
1967:
1962:
1957:
1952:
1946:
1945:
1940:
1935:
1930:
1925:
1920:
1915:
1910:
1905:
1900:
1894:
1893:
1888:
1883:
1878:
1873:
1868:
1863:
1858:
1853:
1848:
1842:
1841:
1839:
1830:
1827:
1761:
1731:
1728:
1693:
1684:
1680:
1671:
1667:
1664:
1661:
1637:
1634:
1568:Main article:
1565:
1562:
1546:Main article:
1543:
1540:
1519:
1514:
1506:
1502:
1499:
1495:
1491:
1483:
1479:
1475:
1470:
1464:
1463:
1460:
1457:
1454:
1446:
1444:
1441:
1440:
1438:
1433:
1430:
1427:
1424:
1421:
1418:
1370:
1367:
1364:
1361:
1358:
1355:
1352:
1349:
1345:
1341:
1337:
1333:
1329:
1325:
1322:
1319:
1285:
1278:
1274:
1270:
1265:
1261:
1255:
1247:
1244:
1238:
1233:
1229:
1225:
1221:
1190:
1187:
1179:absolute value
1156:
1151:
1146:
1142:
1138:
1131:
1124:
1120:
1116:
1111:
1106:
1103:
1100:
1097:
1094:
1091:
1037:
1032:
1029:
1026:
1018:
1016:
1006:
1005:
1002:
999:
996:
993:
985:
983:
973:
972:
969:
966:
963:
960:
952:
950:
947:
944:
943:
941:
936:
933:
930:
927:
924:
921:
918:
915:
912:
910:
907:
904:
901:
898:
895:
892:
889:
886:
883:
879:
875:
871:
868:
865:
864:
844:
841:
838:
834:
831:
827:
824:
821:
818:
798:
795:
786:Main article:
765:
764:Sign functions
762:
745:
742:
739:
734:
731:
727:
641:absolute value
632:absolute value
625:
622:
602:absolute value
598:
597:
590:
583:
576:
561:
560:
553:
546:
539:
523:
520:
424:
399:
396:
221:
218:
142:
141:
56:
54:
47:
26:
9:
6:
4:
3:
2:
2748:
2737:
2734:
2733:
2731:
2720:
2716:
2711:
2697:
2693:
2687:
2673:
2672:www.cs.cas.cz
2669:
2663:
2654:
2652:
2644:
2630:
2626:
2619:
2617:
2615:
2610:
2601:
2598:
2596:
2593:
2591:
2588:
2586:
2583:
2581:
2578:
2577:
2568:
2564:
2560:
2556:
2552:
2549:
2545:
2540:
2536:
2532:
2531:signed volume
2528:
2525:The ideas of
2524:
2521:
2517:
2513:
2512:
2510:
2506:
2502:
2498:
2495:
2491:
2487:
2484:
2480:
2475:
2471:
2460:
2450:
2446:
2442:
2436:
2431:
2426:
2423:
2419:
2409:
2405:
2399:
2395:
2386:
2384:
2379:
2378:
2377:
2371:
2367:
2358:
2355:
2350:
2348:
2344:
2340:
2336:
2332:
2326:
2316:
2311:
2308:
2303:
2298:
2293:
2288:
2283:
2278:
2273:
2268:
2263:
2259:
2256:
2251:
2246:
2241:
2236:
2231:
2226:
2221:
2216:
2211:
2207:
2204:
2199:
2194:
2189:
2184:
2179:
2174:
2169:
2164:
2159:
2155:
2152:
2147:
2142:
2137:
2132:
2127:
2122:
2117:
2112:
2107:
2103:
2100:
2095:
2090:
2085:
2080:
2075:
2070:
2065:
2060:
2055:
2051:
2048:
2043:
2038:
2033:
2028:
2023:
2018:
2013:
2008:
2003:
1999:
1996:
1991:
1986:
1981:
1976:
1971:
1966:
1961:
1956:
1951:
1947:
1944:
1939:
1934:
1929:
1924:
1919:
1914:
1909:
1904:
1899:
1895:
1892:
1887:
1882:
1877:
1872:
1867:
1862:
1857:
1852:
1847:
1843:
1838:
1834:
1826:
1824:
1823:orientability
1820:
1816:
1811:
1809:
1805:
1801:
1797:
1792:
1790:
1786:
1782:
1778:
1774:
1765:
1760:
1757:
1753:
1749:
1745:
1741:
1737:
1736:displacements
1727:
1725:
1721:
1717:
1713:
1709:
1704:
1691:
1682:
1678:
1669:
1665:
1662:
1651:
1647:
1643:
1633:
1631:
1630:opposite axis
1626:
1624:
1620:
1615:
1613:
1609:
1605:
1601:
1593:
1589:
1585:
1581:
1576:
1571:
1561:
1559:
1555:
1549:
1539:
1537:
1532:
1512:
1500:
1497:
1493:
1489:
1477:
1468:
1458:
1455:
1452:
1442:
1436:
1431:
1425:
1419:
1416:
1406:
1401:
1396:
1382:
1368:
1362:
1356:
1350:
1347:
1339:
1331:
1323:
1320:
1301:
1296:
1283:
1276:
1272:
1268:
1263:
1259:
1253:
1242:
1236:
1231:
1223:
1209:
1205:
1201:
1196:
1186:
1180:
1173:
1154:
1149:
1140:
1129:
1118:
1109:
1104:
1098:
1092:
1089:
1071:
1059:
1030:
1027:
1024:
1014:
1000:
997:
994:
991:
981:
967:
964:
961:
958:
948:
945:
939:
934:
928:
922:
919:
913:
902:
899:
896:
893:
890:
887:
869:
866:
842:
836:
832:
829:
825:
822:
819:
808:
804:
803:sign function
794:
789:
788:sign function
780:
776:
770:
761:
759:
743:
740:
737:
732:
729:
725:
710:
690:
683:
681:
675:
644:
642:
638:
634:
633:
621:
619:
615:
611:
606:
603:
595:
591:
588:
584:
581:
577:
574:
570:
569:
568:
558:
554:
551:
547:
544:
540:
537:
533:
532:
531:
519:
513:
509:
505:
484:
482:
478:
474:
469:
467:
463:
454:
436:
422:
405:
395:
377:
373:
369:
364:
350:
346:
342:
337:
335:
331:
327:
326:sign function
311:
304:
297:
290:
286:
282:
266:
262:
254:
250:
246:
242:
238:
234:
230:
226:
217:
215:
211:
207:
204:or rotation (
203:
199:
194:
192:
188:
183:
181:
177:
173:
169:
165:
161:
153:
148:
138:
135:
127:
116:
113:
109:
106:
102:
99:
95:
92:
88:
85: –
84:
80:
79:Find sources:
73:
69:
63:
62:
57:This article
55:
51:
46:
45:
40:
33:
32:sine function
19:
2710:
2699:. Retrieved
2695:
2686:
2675:. Retrieved
2671:
2662:
2649:
2643:
2632:. Retrieved
2628:
2580:Percent sign
2535:determinants
2519:
2494:signed graph
2490:graph theory
2473:
2469:
2458:
2434:
2424:
2417:
2411:for certain
2407:
2403:
2397:
2393:
2381:
2375:
2351:
2328:
2306:
2301:
2296:
2291:
2286:
2281:
2276:
2271:
2266:
2261:
2254:
2249:
2244:
2239:
2234:
2229:
2224:
2219:
2214:
2209:
2202:
2197:
2192:
2187:
2182:
2177:
2172:
2167:
2162:
2157:
2150:
2145:
2140:
2135:
2130:
2125:
2120:
2115:
2110:
2105:
2098:
2093:
2088:
2083:
2078:
2073:
2068:
2063:
2058:
2053:
2046:
2041:
2036:
2031:
2026:
2021:
2016:
2011:
2006:
2001:
1994:
1989:
1984:
1979:
1974:
1969:
1964:
1959:
1954:
1949:
1942:
1937:
1932:
1927:
1922:
1917:
1912:
1907:
1902:
1897:
1890:
1885:
1880:
1875:
1870:
1865:
1860:
1855:
1850:
1845:
1825:in general.
1812:
1808:radial speed
1793:
1780:
1776:
1770:
1733:
1711:
1707:
1705:
1649:
1641:
1639:
1627:
1623:right-handed
1616:
1597:
1557:
1553:
1551:
1533:
1404:
1399:
1394:
1383:
1299:
1297:
1207:
1203:
1199:
1192:
1171:
1069:
1057:
806:
802:
800:
791:
778:
774:
708:
688:
684:
679:
676:
645:
636:
630:
627:
617:
613:
607:
599:
593:
592:A number is
586:
585:A number is
579:
578:A number is
572:
571:A number is
562:
557:non-positive
556:
555:A number is
550:non-negative
549:
548:A number is
542:
541:A number is
535:
534:A number is
525:
486:The symbols
485:
477:signed zeros
470:
461:
455:
401:
398:Sign of zero
365:
348:
338:
333:
309:
302:
295:
288:
280:
264:
249:ordered ring
223:
200:), sense of
195:
184:
163:
157:
130:
121:
111:
104:
97:
90:
78:
66:Please help
61:verification
58:
2527:signed area
1756:number line
1612:mathematics
1584:unit circle
1074:is −1 when
483:for more).
261:total order
241:quaternions
202:orientation
168:real number
160:mathematics
124:August 2020
2701:2020-08-26
2677:2020-08-26
2634:2020-08-26
2606:References
2595:Signedness
2466:such that
2325:Signedness
1752:directions
1720:derivative
1594:direction.
1062:is 1 when
704:magnitude
672:| = 3
664:| = 3
502:, used in
372:arithmetic
366:In common
94:newspapers
2653:: Algèbre
2449:magnitude
2331:computing
1679:−
1660:Δ
1608:clockwise
1592:clockwise
1509:otherwise
1501:φ
1449:for
1420:
1357:∪
1324:∈
1246:¯
1195:magnitude
1093:
946:−
923:
917:↦
888:−
882:→
820:−
741:−
733:π
637:magnitude
610:functions
435:-function
370:(used in
361:−(−3) = 3
245:octonions
233:rationals
2730:Category
2574:See also
2567:electron
2451:: aside
2335:variable
1815:3D space
1800:velocity
1716:calculus
1397:= |
1021:if
988:if
955:if
702:and its
594:negative
587:positive
543:negative
536:positive
504:calculus
349:negation
303:negative
296:positive
281:negative
265:positive
229:integers
172:negative
2555:physics
2509:measure
2445:vectors
1802:in the
1771:On the
1748:physics
1740:motions
1687:initial
1403:|⋅
1177:is the
760:below.
332:), the
225:Numbers
216:below.
108:scholar
2625:"Sign"
2563:proton
2557:, any
2380:Words
1785:vector
1646:change
1580:x-axis
1175:|
1169:|
1167:where
1011:
1008:
978:
975:
777:= sgn(
712:|
706:|
668:|
660:|
357:−0 = 0
330:fields
320:, and
206:cw/ccw
162:, the
110:
103:
96:
89:
81:
2546:In a
2472:| = |
2383:up to
1796:speed
1674:final
1600:angle
1558:minus
1536:unity
1409:then
1054:Thus
563:When
526:When
301:, or
174:, or
166:of a
115:JSTOR
101:books
2529:and
2520:side
2503:, a
2492:, a
2481:The
2455:and
2443:and
2385:sign
2307:−128
2255:−127
1821:and
1746:and
1738:and
1556:and
1554:plus
1384:Let
1068:sgn(
1056:sgn(
1028:>
962:<
801:The
680:sign
666:and
510:for
506:and
498:and
490:and
462:both
441:and
404:zero
334:sign
310:sign
289:zero
164:sign
150:The
87:news
2553:In
2499:In
2488:In
2406:= −
2401:or
2339:bit
2329:In
1943:126
1891:127
1813:In
1742:in
1417:sgn
1392:as
1306:to
1181:of
1090:sgn
920:sgn
867:sgn
805:or
697:of
643:.
635:or
423:sgn
306:(−)
299:(+)
292:(0)
208:),
182:.
158:In
70:by
2732::
2694:.
2670:.
2627:.
2613:^
2420:=
2396:=
2349:.
2203:−2
2151:−1
1810:.
1632:.
1310::
1208:iy
1206:+
1202:=
1185:.
1031:0.
744:1.
674:.
662:−3
648:−3
492:−0
488:+0
468:.
443:−0
439:+0
384:−3
380:+3
322:−1
316:,
294:,
273:0.
257:0.
243:,
239:,
235:,
231:,
191:−1
2704:.
2680:.
2657:.
2637:.
2569:.
2478:.
2476:|
2474:Q
2470:q
2468:|
2464:q
2459:Q
2457:−
2453:Q
2437:|
2435:q
2433:|
2425:Q
2422:±
2418:q
2413:Q
2408:Q
2404:q
2398:Q
2394:q
2389:q
2302:=
2297:0
2292:0
2287:0
2282:0
2277:0
2272:0
2267:0
2262:1
2250:=
2245:1
2240:0
2235:0
2230:0
2225:0
2220:0
2215:0
2210:1
2198:=
2193:0
2188:1
2183:1
2178:1
2173:1
2168:1
2163:1
2158:1
2146:=
2141:1
2136:1
2131:1
2126:1
2121:1
2116:1
2111:1
2106:1
2099:0
2094:=
2089:0
2084:0
2079:0
2074:0
2069:0
2064:0
2059:0
2054:0
2047:1
2042:=
2037:1
2032:0
2027:0
2022:0
2017:0
2012:0
2007:0
2002:0
1995:2
1990:=
1985:0
1980:1
1975:0
1970:0
1965:0
1960:0
1955:0
1950:0
1938:=
1933:0
1928:1
1923:1
1918:1
1913:1
1908:1
1903:1
1898:0
1886:=
1881:1
1876:1
1871:1
1866:1
1861:1
1856:1
1851:1
1846:0
1781:y
1777:x
1712:x
1708:x
1692:.
1683:x
1670:x
1666:=
1663:x
1650:x
1642:x
1513:.
1498:i
1494:e
1490:=
1482:|
1478:z
1474:|
1469:z
1459:0
1456:=
1453:z
1443:0
1437:{
1432:=
1429:)
1426:z
1423:(
1407:,
1405:e
1400:z
1395:z
1390:φ
1386:z
1369:.
1366:}
1363:0
1360:{
1354:}
1351:1
1348:=
1344:|
1340:z
1336:|
1332::
1328:C
1321:z
1318:{
1308:0
1304:0
1284:.
1277:2
1273:y
1269:+
1264:2
1260:x
1254:=
1243:z
1237:z
1232:=
1228:|
1224:z
1220:|
1204:x
1200:z
1183:x
1172:x
1155:,
1150:x
1145:|
1141:x
1137:|
1130:=
1123:|
1119:x
1115:|
1110:x
1105:=
1102:)
1099:x
1096:(
1080:x
1076:x
1072:)
1070:x
1064:x
1060:)
1058:x
1025:x
1015:1
1001:,
998:0
995:=
992:x
982:0
968:,
965:0
959:x
949:1
940:{
935:=
932:)
929:x
926:(
914:x
906:}
903:1
900:,
897:0
894:,
891:1
885:{
878:R
870::
843:.
840:}
837:1
833:,
830:0
826:,
823:1
817:{
781:)
779:x
775:y
738:=
730:i
726:e
714:.
709:z
699:z
692:z
670:3
656:3
652:3
565:0
528:0
516:0
500:0
496:0
458:0
451:0
447:0
412:0
408:0
392:−
388:3
353:0
318:1
314:0
277:0
269:0
176:0
137:)
131:(
126:)
122:(
112:·
105:·
98:·
91:·
64:.
41:.
20:)
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