40:
5250:
895:
4655:
realm of numbers expands to include the rational numbers, division is replaced by multiplication by certain rational numbers. In keeping with this change of viewpoint, the question, "Why can't we divide by zero?", becomes "Why can't a rational number have a zero denominator?". Answering this revised question precisely requires close examination of the definition of rational numbers.
2334:
A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite
4654:
As the realm of numbers to which these operations can be applied expands there are also changes in how the operations are viewed. For instance, in the realm of integers, subtraction is no longer considered a basic operation since it can be replaced by addition of signed numbers. Similarly, when the
4630:
The four basic operations – addition, subtraction, multiplication and division – as applied to whole numbers (positive integers), with some restrictions, in elementary arithmetic are used as a framework to support the extension of the realm of numbers to which they apply. For instance, to make it
4277:
applied as a new definition of distance between complex numbers; and in general the geometry of the sphere can be studied using complex arithmetic, and conversely complex arithmetic can be interpreted in terms of spherical geometry. As a consequence, the set of extended complex numbers is often
4939:
in which division by zero is always possible. However, the resulting mathematical structure is no longer a commutative ring, as multiplication no longer distributes over addition. Furthermore, in a wheel, division of an element by itself no longer results in the multiplicative identity element
580:
or it can be defined to result in signed infinity, with positive or negative sign depending on the sign of the dividend. In these number systems division by zero is no longer a special exception per se, but the point or points at infinity involve their own new types of exceptional behavior.
2307:
A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is
3268:
520:
As an alternative to the common convention of working with fields such as the real numbers and leaving division by zero undefined, it is possible to define the result of division by zero in other ways, resulting in different number systems. For example, the quotient
5214:
In IEEE arithmetic, division of 0/0 or ∞/∞ results in NaN, but otherwise division always produces a well-defined result. Dividing any non-zero number by positive zero (+0) results in an infinity of the same sign as the dividend. Dividing any non-zero number by
4658:
In the modern approach to constructing the field of real numbers, the rational numbers appear as an intermediate step in the development that is founded on set theory. First, the natural numbers (including zero) are established on an axiomatic basis such as
3660:
3129:
6203:
With appropriate care to be certain that the algebraic signs are not determined by rounding error, the affine mode preserves order relations while fixing up overflow. Thus, for example, the reciprocal of a negative number which underflows is still
4894:, which is a mathematical structure where addition, subtraction, and multiplication behave as they do in the more familiar number systems, but division may not be defined. Adjoining a multiplicative inverses to a commutative ring is called
4639:. During this gradual expansion of the number system, care is taken to ensure that the "extended operations", when applied to the older numbers, do not produce different results. Loosely speaking, since division by zero has no meaning (is
4667:. The next step is to define the rational numbers keeping in mind that this must be done using only the sets and operations that have already been established, namely, addition, multiplication and the integers. Starting with the set of
773:: dividing entails counting how many times the divisor can be subtracted before the dividend runs out. Because no finite number of subtractions of zero will ever exhaust a non-zero dividend, calculating division by zero in this way
1110:. The slope is defined to be the "rise" (change in vertical coordinate) divided by the "run" (change in horizontal coordinate) along the line. When this is written using the symmetrical ratio notation, a horizontal line has slope
1036:
could be maintained, for instance one cup of flour and one-fifth cup of sugar, or fifty cups of flour and ten cups of sugar. Now imagine a sugar-free cake recipe calls for ten cups of flour and zero cups of sugar. The ratio
3363:
results from dividing two functions whose limits both tend to infinity.) Such a limit may equal any real value, may tend to infinity, or may not converge at all, depending on the particular functions. For example, in
3138:
4334:
5470:, pp. 53–54, give an example of a king's heirs equally dividing their inheritance of 12 diamonds, and ask what would happen in the case that all of the heirs died before the king's will could be executed.
4732:
are then defined to be the rational numbers. It is in the formal proof that this relation is an equivalence relation that the requirement that the second coordinate is not zero is needed (for verifying
3493:
3435:
5440:
Some other operations, including division, can also be performed by the desk calculator (but don't try to divide by zero; the calculator never will stop trying to divide until stopped manually).
3026:
6216:
4195:
3887:
3361:
4594:
4740:
Although division by zero cannot be sensibly defined with real numbers and integers, it is possible to consistently define it, or similar operations, in other mathematical structures.
4434:
4397:
2690:
4536:
723:
is the number of resulting parts. For example, imagine ten slices of bread are to be made into sandwiches, each requiring two slices of bread. A total of five sandwiches can be made
4020:
3981:
471:
5166:
is excluded from being a field. In the zero ring, division by zero is possible, which shows that the other field axioms are not sufficient to exclude division by zero in a field.
1983:
1777:
1678:
6163:
1484:
4502:
4468:
1636:
887:
758:
2454:
1199:
274:
204:
5541:
3948:
3329:
1850:
1402:
4565:
4139:
3486:
2194:
1447:
1257:
1228:
548:
393:
134:
4979:. In a field, every nonzero element is invertible under multiplication; as above, division poses problems only when attempting to divide by zero. This is likewise true in a
4620:
1712:
1521:
1095:
is perfectly sensible: it just means that the cake has no sugar. However, the question "How many parts flour for each part sugar?" still has no meaningful numerical answer.
1812:
891:
Now imagine instead that the ten cookies are to be divided among zero friends. How many cookies will each friend receive? Since there are no friends, this is an absurdity.
3845:
3780:
3701:
2366:
2132:
1601:
5140:
5106:
5020:
4797:
1945:
1913:
239:
2918:
2881:
3750:
2977:
2806:
4259:
3724:
300:
4218:
4046:
3915:
2776:
2083:
1339:
576:
4890:
In abstract algebra, the integers, the rational numbers, the real numbers, and the complex numbers can be abstracted to more general algebraic structures, such as a
2233:
1064:
654:
5053:
2753:
2266:
1160:
1093:
1008:
982:
956:
5506:
4926:
3808:
2782:, such limits are informally said to "equal infinity". If the value of the function decreases without bound, the function is said to "tend to negative infinity",
2720:
2013:
1734:
1543:
1318:
1134:
1034:
4359:
3782:
the concept of a "limit at infinity" can be made to work like a finite limit. When dealing with both positive and negative extended real numbers, the expression
3293:
2628:
2605:
2537:
1566:
1284:
511:
343:
4958:
3457:
3018:
2997:
2846:
2826:
2582:
2562:
2514:
2494:
2474:
924:
850:
830:
810:
721:
701:
681:
491:
320:
154:
5241:/2) = −∞. The exact result −2 is too large to represent as a single-precision number, so an infinity of the same sign is used instead to indicate overflow.
5621:
Clark, Matthew R.; Berenson, Sarah B.; Cavey, Laurie O. (2003), "A comparison of ratios and fractions and their roles as tools in proportional reasoning",
5749:
Robinson, K. M.; LeFevre, J. A. (2012), "The inverse relation between multiplication and division: Concepts, procedures, and a cognitive framework",
588:, an error may result from an attempt to divide by zero. Depending on the context and the type of number involved, dividing by zero may evaluate to
4635:
in order to incorporate the negative integers. Similarly, to support division of any integer by any other, the realm of numbers must expand to the
6610:
2270:
This is essentially the same fallacious computation as the previous numerical version, but the division by zero was obfuscated because we wrote
1820:
value can be substituted for the unknown quantity to yield a true statement, so there is no single number which can be assigned as the quotient
852:
is the resulting size of each part. For example, imagine ten cookies are to be divided among two friends. Each friend will receive five cookies
6360:
6282:
6240:
Wang, Xi; Chen, Haogang; Cheung, Alvin; Jia, Zhihao; Zeldovich, Nickolai; Kaashoek, M. Frans, "Undefined behavior: what happened to my code?",
1359:, meaning that multiplying and then dividing by the same non-zero quantity, or vice versa, leaves an original quantity unchanged; for example
6576:
6550:
6484:
6267:
6117:
5971:
5859:
5433:
3369:
365:, and situations where division by zero might occur must be treated with care. Since any number multiplied by zero is zero, the expression
4928:, so nontrivial commutative rings do not have inverses at zero, and thus division by zero is undefined for nontrivial commutative rings.
4285:
4282:. The set is usually denoted by the symbol for the complex numbers decorated by an asterisk, overline, tilde, or circumflex, for example
6348:
The standard division function on natural numbers in Coq, div, is total and pure, but incorrect: when the divisor is 0, the result is 0.
4238:
This can intuitively be thought of as wrapping up the infinite edges of the complex plane and pinning them together at the single point
1545:
represents the same unknown quantity, and then finding the value for which the statement is true; in this case the unknown quantity is
6529:
6511:
5751:
4895:
414:
can be expressed as a fraction whose denominator tends to zero, the output of the function becomes arbitrarily large, and is said to "
517:, as the resulting limit depends on the specific functions forming the fraction and cannot be determined from their separate limits.
6619:
6462:
6082:
31:
6412:
6690:
5308:
5187:
specification. In IEEE floating-point arithmetic, numbers are represented using a sign (positive or negative), a fixed-precision
5154:. Since the field axioms only guarantee the existence of such inverses for nonzero elements, this expression has no meaning when
4269:. This equivalence can be extended to a metrical equivalence by mapping each complex number to a point on the sphere via inverse
3810:
is usually left undefined. However, in contexts where only non-negative values are considered, it is often convenient to define
3263:{\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}={\frac {\displaystyle \lim _{x\to c}f(x)}{\displaystyle \lim _{x\to c}g(x)}}.}
2291:
6705:
6680:
5300:
3890:
5219:(−0) results in an infinity of the opposite sign as the dividend. This definition preserves the sign of the result in case of
6700:
5264:
division by zero is usually handled differently from floating point since there is no integer representation for the result.
4225:
3675:
2313:
5304:
4823:
algebra, square or rectangular blocks of numbers are manipulated as though they were numbers themselves: matrices can be
898:
The slope of line in the plane is a ratio of vertical to horizontal coordinate differences. For a vertical line, this is
6675:
5332:
3133:
In most cases, the limit of a quotient of functions is equal to the quotient of the limits of each function separately,
4831:, and in some cases, a version of division also exists. Dividing by a matrix means, more precisely, multiplying by its
6375:
5284:
processors can either cause a hardware exception or return zero. Because of this inconsistency between platforms, the
2137:
6639:, Princeton: D. Van Nostrand, §8.5 "The Problem of Division by Zero" and §8.7 "Five Approaches to Division by Zero"
6429:
Desperately Needed
Remedies for the Undebuggability of Large Floating-Point Computations in Science and Engineering
5261:
4769:
4262:
4163:
3894:
3655:{\displaystyle \lim _{x\to 1}{\frac {x^{2}-1}{x-1}}=\lim _{x\to 1}{\frac {(x-1)(x+1)}{x-1}}=\lim _{x\to 1}(x+1)=2.}
2335:
and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.
1714:
However, any number multiplied by zero is zero rather than six, so there exists no number which can substitute for
4859:
3858:
3334:
2092:
5959:
5371:
5180:
4570:
589:
5888:"Babylonian Pythagoras' Theorem, the Early History of Zero and a Polemic on the Study of the History of Science"
5376:
2339:
Historically, one of the earliest recorded references to the mathematical impossibility of assigning a value to
6685:
6354:
6276:
5285:
4971:
The concepts applied to standard arithmetic are similar to those in more general algebraic structures, such as
4631:
possible to subtract any whole number from another, the realm of numbers must be expanded to the entire set of
4270:
4158:
2631:
419:
4401:
4364:
513:
When both the numerator and the denominator tend to zero at the same input, the expression is said to take an
4506:
2641:
5281:
5265:
4232:
3992:
3953:
2206:
1260:
614:
362:
3124:{\displaystyle \lim _{x\to 0^{+}}{\frac {1}{x}}=+\infty ,\qquad \lim _{x\to 0^{-}}{\frac {1}{x}}=-\infty .}
429:
5328:
4965:
4800:
3272:
However, when a function is constructed by dividing two functions whose separate limits are both equal to
2398:
2245:
1950:
1742:
1643:
620:
403:
96:
1452:
5109:
4828:
4472:
4438:
1606:
1010:
To scale this recipe to larger or smaller quantities of cake, a ratio of flour to sugar proportional to
857:
782:
728:
164:
4987:). However, in other rings, division by nonzero elements may also pose problems. For example, the ring
4111:
This definition leads to many interesting results. However, the resulting algebraic structure is not a
2405:, the value to which a function's output tends as its input tends to some specific value. The notation
1169:
244:
174:
6296:
5514:
3924:
3302:
2408:
1823:
1362:
766:). Arbitrarily many such sandwiches can be made from ten slices of bread, as the bread is irrelevant.
6105:
5349:
5292:
5220:
5200:
4820:
4725:
4541:
4118:
3462:
1867:
1406:
1233:
1204:
524:
369:
350:
110:
5545:
Lo, Jane-Jane; Watanabe, Tad; Cai, Jinfa (2004), "Developing Ratio
Concepts: An Asian Perspective",
4599:
1683:
1489:
6670:
4976:
4734:
4112:
2930:
2693:
2635:
2402:
1783:
1287:
423:
415:
407:
358:
5600:
3813:
3759:
3684:
3295:
then the limit of the result cannot be determined from the separate limits, so is said to take an
1571:
6329:
6194:
5936:
5768:
5732:
5698:
5664:
5604:
5596:
5562:
5296:
4972:
4274:
3296:
2369:
2342:
2031:
1918:
1886:
1854:
Because of these difficulties, quotients where the divisor is zero are traditionally taken to be
514:
346:
209:
5118:
5084:
4998:
4775:
2890:
2853:
2303:
as a number in its own right and to define operations involving zero. According to
Brahmagupta,
5887:
5311:
is raised for attempted division by zero, which can be handled in another part of the program.
3729:
2936:
2785:
6695:
6615:
6572:
6546:
6525:
6507:
6480:
6458:
6263:
6157:
6113:
6078:
5977:
5967:
5855:
5195:. Numbers whose exponent is too large to represent instead "overflow" to positive or negative
4832:
4729:
4241:
4221:
3918:
3753:
3706:
1291:
958:
For example, a cake recipe might call for ten cups of flour and two cups of sugar, a ratio of
597:
551:
279:
5851:
4203:
4031:
3900:
2761:
2055:
1324:
561:
6649:
6632:
6593:
6564:
6538:
6450:
6339:
6255:
6245:
6186:
6145:
5928:
5899:
5760:
5724:
5690:
5656:
5630:
5588:
5554:
5543:
By contrast in the USA textbooks typically treat them as two notations for the same thing.
4932:
4891:
4808:
4749:
4664:
4197:
the set of complex numbers with a single additional number appended, usually denoted by the
4150:
4049:
1879:
When working with numbers, it is easy to identify an illegal division by zero. For example:
763:
5356:
brought down all the machines on the network, causing the ship's propulsion system to fail.
5249:
1201:
while a vertical line has an undefined slope, since in real-number arithmetic the quotient
1040:
625:
5681:
Cavey, Laurie O.; Mahavier, W. Ted (2010), "Seeing the potential in students' questions",
5320:
5029:
4961:
4824:
4691:
4636:
4198:
4154:
2923:
2756:
2729:
2373:
1139:
1107:
1069:
987:
961:
932:
555:
6656:, Oxford University Press, §53 "Definitions whose definiendum contains the identity sign"
5485:
4905:
3785:
2699:
1992:
1717:
1526:
1297:
1113:
1013:
6633:
6594:
6451:
6109:
5348:
On 21 September 1997, a division by zero error in the "Remote Data Base
Manager" aboard
4341:
3275:
2610:
2587:
2519:
1548:
1266:
496:
325:
6650:
6628:
5955:
5916:
5592:
5336:
5324:
5158:
is zero. Modern texts, that define fields as a special type of ring, include the axiom
4943:
4757:
4648:
4279:
3442:
3003:
2982:
2831:
2811:
2567:
2547:
2499:
2479:
2459:
2323:
1356:
1290:, so they have no intersection in the plane. Sometimes they are said to intersect at a
909:
835:
815:
795:
706:
686:
666:
476:
305:
168:
139:
5634:
5335:. Such a definition does not create contradictions, as further manipulations (such as
3921:, an infinite quantity that is neither positive nor negative. This quantity satisfies
6664:
6645:
6605:
6472:
5844:
5736:
5608:
5216:
4984:
4753:
2542:
2035:
1103:
774:
411:
6401:
6244:, APSYS '12, Seoul, 23–24 July 2012, New York: Association for Computing Machinery,
5280:
processors silently generate an incorrect result for the division and continue, and
6198:
6055:
5366:
5064:
4936:
4899:
4668:
4660:
4053:
3752:
read as "positive infinity" and "negative infinity" respectively, and representing
1872:
A compelling reason for not allowing division by zero is that allowing it leads to
762:
Now imagine instead that zero slices of bread are required per sandwich (perhaps a
354:
39:
6600:, New York: D. Van Nostrand, Ch. 5 "Thou Shalt Not Divide By Zero", pp. 77–96
6495:
Elementary
Mathematics from an Advanced Standpoint / Arithmetic, Algebra, Analysis
1486:
can be solved by rewriting it as an equivalent equation involving multiplication,
6427:
6177:
Cody, W. J. (Mar 1981), "Analysis of
Proposals for the Floating-Point Standard",
6072:
5257:, typically halt and display an error message after an attempt to divide by zero.
4803:). It does not, however, make sense to ask for a "value" of this distribution at
353:, a subtle mistake leading to absurd results. To prevent this, the arithmetic of
6324:
Tanter, Éric; Tabareau, Nicolas (2015), "Gradual certified programming in coq",
5479:
In China, Taiwan, and Japan, school textbooks typically distinguish between the
5204:
5188:
4836:
4644:
3679:
2779:
2382:
2296:
1163:
770:
88:
5579:
Cengiz, Nesrin; Rathouz, Margaret (2018), "Making Sense of
Equivalent Ratios",
4643:) in the whole number setting, this remains true as the setting expands to the
2330:(12th century) proposed that division by zero results in an infinite quantity,
894:
6568:
6407:
6149:
5981:
5764:
5715:
Wegman, Edward J.; Said, Yasmin H. (2010), "Natural homogeneous coordinates",
5445:
4980:
2300:
1347:
below. Vertical lines are sometimes said to have an "infinitely steep" slope.
6477:
Is Math(s) Real? How Simple
Questions Lead Us to Mathematics' Deepest Truths
6343:
6250:
6190:
6097:
5273:
5199:(+∞ or −∞), while numbers whose exponent is too small to represent instead "
5163:
2723:
2019:
The fallacy here arises from the assumption that it is legitimate to cancel
1873:
778:
6520:
Henkin, Leon; Smith, Norman; Varineau, Verne J.; Walsh, Michael J. (2012),
5660:
2023:
like any other number, whereas, in fact, doing so is a form of division by
6133:
5694:
5558:
769:
The quotitive concept of division lends itself to calculation by repeated
5932:
5196:
5192:
5184:
4799:
to a distribution on the whole space of real numbers (in effect by using
4229:
2394:
2377:
2316:
unsuccessfully tried to correct the mistake
Brahmagupta made in his book
585:
399:
160:
104:
6259:
5940:
5772:
5702:
5668:
5566:
5331:, define 1/0 = 0. This is due to the requirement that all functions are
1286:
dashed black in the figure. The vertical red and dashed black lines are
1098:
A geometrical appearance of the division-as-ratio interpretation is the
550:
can be defined to equal zero; it can be defined to equal a new explicit
5903:
5277:
4632:
6303:, Python Software Foundation, § "Concrete exceptions – exception
5728:
4898:. However, the localization of every commutative ring at zero is the
4266:
4265:, making the extended complex numbers topologically equivalent to a
6334:
5919:(1929), "Absurdities due to division by zero: An historical note",
5289:
4329:{\displaystyle {\hat {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}.}
3488:, but simplifying the quotient first shows that the limit exists:
17:
6652:
Introduction to Logic and to the
Methodology of Deductive Sciences
6497:, translated by Hedrick, E. R.; Noble, C. A. (3rd ed.), Dover
5254:
5248:
1099:
927:
893:
38:
4752:, division by zero is still impossible, but division by non-zero
6326:
DLS 2015: Proceedings of the 11th Symposium on Dynamic Languages
5447:
What happens when you divide by zero on a mechanical calculator?
4361:
ordinary complex arithmetic is extended by the additional rules
4338:
In the extended complex numbers, for any nonzero complex number
6559:
Zazkis, Rina; Liljedahl, Peter (2009), "Stories that explain",
6543:
Chapter Zero : Fundamental Notions of Abstract Mathematics
5237:/2 underflows to −0, and dividing 1 by this result produces 1/(
1259:
of a line through the origin is the vertical coordinate of the
6242:
APSYS '12: Proceedings of the Asia-Pacific Workshop on Systems
5269:
5208:
4115:, and should not be expected to behave like one. For example,
2034:, it is possible to disguise a division by zero to obtain an
1263:
between the line and a vertical line at horizontal coordinate
593:
366:
2808:
In some cases a function tends to two different values when
5717:
Wiley Interdisciplinary Reviews: Computational Statistics
4964:, the multiplication in the wheel no longer results in a
3950:, which is necessary in this context. In this structure,
3439:
the separate limits of the numerator and denominator are
100:
6102:
Mathematical Methods for Optical Physics and Engineering
6041:
5226:
For example, using single-precision IEEE arithmetic, if
5179:
In computing, most numerical calculations are done with
5647:
Cheng, Ivan (2010), "Fractions: A New Slant on Slope",
5519:
5121:
5087:
5001:
4778:
4575:
4546:
4406:
4369:
3467:
3430:{\displaystyle \lim _{x\to 1}{\dfrac {x^{2}-1}{x-1}},}
3339:
3307:
2644:
2411:
2347:
2345:
1828:
1747:
1648:
1611:
1457:
1238:
1209:
1174:
862:
733:
529:
449:
374:
255:
185:
115:
5517:
5488:
5183:, which since the 1980s has been standardized by the
5032:
4946:
4908:
4807: = 0; a sophisticated answer refers to the
4602:
4573:
4544:
4509:
4475:
4441:
4404:
4367:
4344:
4288:
4244:
4206:
4166:
4121:
4034:
3995:
3956:
3927:
3903:
3861:
3816:
3788:
3762:
3732:
3709:
3687:
3496:
3465:
3445:
3390:
3372:
3337:
3305:
3278:
3225:
3195:
3141:
3029:
3006:
2985:
2939:
2893:
2856:
2834:
2814:
2788:
2764:
2732:
2702:
2613:
2590:
2570:
2550:
2522:
2502:
2482:
2462:
2320:: "A number remains unchanged when divided by zero."
2248:
2209:
2140:
2095:
2058:
1995:
1953:
1921:
1889:
1826:
1786:
1745:
1720:
1686:
1646:
1609:
1574:
1551:
1529:
1492:
1455:
1409:
1365:
1327:
1300:
1269:
1236:
1207:
1172:
1142:
1116:
1072:
1043:
1016:
990:
964:
935:
912:
860:
838:
818:
798:
731:
709:
689:
669:
628:
564:
527:
499:
479:
432:
372:
328:
308:
282:
247:
212:
177:
142:
113:
2726:. While such a function is not formally defined for
1680:
requires determining an unknown quantity satisfying
1344:
5432:Kochenburger, Ralph J.; Turcio, Carolyn J. (1974),
5211:(not a number) value represents undefined results.
6400:
5843:
5535:
5500:
5134:
5100:
5047:
5014:
4952:
4920:
4791:
4614:
4588:
4559:
4530:
4496:
4462:
4428:
4391:
4353:
4328:
4253:
4212:
4189:
4133:
4040:
4014:
3975:
3942:
3909:
3881:
3839:
3802:
3774:
3744:
3718:
3695:
3654:
3480:
3451:
3429:
3355:
3323:
3287:
3262:
3123:
3012:
2991:
2971:
2929:A basic example of an infinite singularity is the
2912:
2875:
2840:
2820:
2800:
2770:
2747:
2714:
2684:
2622:
2599:
2576:
2556:
2531:
2508:
2488:
2468:
2448:
2360:
2260:
2227:
2188:
2126:
2077:
2007:
1977:
1939:
1907:
1844:
1806:
1771:
1728:
1706:
1672:
1630:
1595:
1560:
1537:
1515:
1478:
1441:
1396:
1333:
1312:
1278:
1251:
1222:
1193:
1154:
1128:
1087:
1058:
1028:
1002:
976:
950:
918:
881:
844:
824:
804:
752:
715:
695:
675:
648:
570:
542:
505:
485:
465:
387:
337:
314:
294:
268:
233:
198:
148:
128:
103:, is a unique and problematic special case. Using
6162:: CS1 maint: DOI inactive as of September 2024 (
4995:of integers mod 6. The meaning of the expression
4835:. Not all matrices have inverses. For example, a
4141:is undefined in this extension of the real line.
4048:. It is the natural way to view the range of the
1640:An analogous problem involving division by zero,
656:can be conceptually interpreted in several ways.
3616:
3550:
3498:
3374:
3227:
3197:
3143:
3077:
3031:
2979:which tends to positive or negative infinity as
2646:
2413:
107:notation, the general example can be written as
5966:, vol. 1, Houston: OpenStax, p. 454,
5800:
5788:
5467:
5419:
5407:
6104:, Cambridge University Press, pp. 88–93,
6060:, Prindle, Weber & Schmidt, pp. 29–30
6057:Elementary Calculus: An Infinitesimal Approach
5850:, New York: Oxford University Press, pp.
5846:The Nothing That Is: A Natural History of Zero
5339:) still require that the divisor is non-zero.
1162:However, if the slope is taken to be a single
683:is imagined to be split up into parts of size
4931:Nevertheless, any number system that forms a
4842:One can define a pseudo-division, by setting
4190:{\displaystyle \mathbb {C} \cup \{\infty \},}
4157:. Of major importance in this subject is the
2778:in this case does not represent any specific
357:and more general numerical structures called
349:while allowing division by zero can create a
167:is the number which yields the dividend when
8:
5142:is only shorthand for the formal expression
5067:. This equation has two distinct solutions,
4320:
4314:
4224:, which is defined to be contained in every
4181:
4175:
3882:{\displaystyle \mathbb {R} \cup \{\infty \}}
3876:
3870:
3356:{\displaystyle {\tfrac {\infty }{\infty }},}
6596:Riddles in Mathematics: A Book of Paradoxes
6138:Mathematical Structures in Computer Science
5162:for fields (or its equivalent) so that the
4589:{\displaystyle {\tfrac {\infty }{\infty }}}
4064:approaches the single point at infinity as
2299:(c. 598–668) is the earliest text to treat
1779:the equivalent multiplicative statement is
410:as their input tends to some value. When a
6017:
6333:
6249:
5837:
5835:
5833:
5649:Mathematics Teaching in the Middle School
5581:Mathematics Teaching in the Middle School
5547:Mathematics Teaching in the Middle School
5518:
5516:
5487:
5122:
5120:
5088:
5086:
5031:
5002:
5000:
4945:
4907:
4779:
4777:
4601:
4574:
4572:
4545:
4543:
4508:
4474:
4440:
4405:
4403:
4368:
4366:
4343:
4307:
4306:
4293:
4292:
4290:
4289:
4287:
4243:
4205:
4168:
4167:
4165:
4120:
4033:
3996:
3994:
3957:
3955:
3926:
3902:
3863:
3862:
3860:
3820:
3815:
3792:
3787:
3761:
3731:
3708:
3689:
3688:
3686:
3619:
3565:
3553:
3520:
3513:
3501:
3495:
3466:
3464:
3444:
3397:
3389:
3377:
3371:
3338:
3336:
3306:
3304:
3277:
3230:
3200:
3193:
3158:
3146:
3140:
3099:
3091:
3080:
3053:
3045:
3034:
3028:
3005:
2984:
2958:
2938:
2904:
2892:
2867:
2855:
2833:
2813:
2787:
2763:
2731:
2701:
2649:
2643:
2612:
2589:
2569:
2549:
2521:
2501:
2481:
2461:
2416:
2410:
2346:
2344:
2247:
2208:
2139:
2112:
2094:
2069:
2057:
1994:
1952:
1920:
1888:
1827:
1825:
1787:
1785:
1761:
1746:
1744:
1721:
1719:
1687:
1685:
1662:
1647:
1645:
1610:
1608:
1573:
1550:
1530:
1528:
1493:
1491:
1471:
1456:
1454:
1416:
1408:
1392:
1381:
1364:
1326:
1299:
1268:
1237:
1235:
1208:
1206:
1173:
1171:
1141:
1115:
1071:
1042:
1015:
989:
963:
934:
911:
861:
859:
837:
817:
797:
732:
730:
708:
688:
668:
632:
627:
563:
528:
526:
498:
478:
448:
431:
373:
371:
327:
307:
281:
254:
246:
211:
184:
176:
141:
114:
112:
6376:"Division by zero in type theory: a FAQ"
6029:
5295:consider the result of dividing by zero
4935:can be extended to a structure called a
4756:is possible. The same holds true in the
4429:{\displaystyle {\tfrac {z}{\infty }}=0,}
4392:{\displaystyle {\tfrac {z}{0}}=\infty ,}
4153:applies the concepts of calculus in the
2685:{\textstyle \lim _{x\to c}f(x)=\infty ,}
906:In another interpretation, the quotient
777:. Such an interminable division-by-zero
600:the program, among other possibilities.
6611:Zero: The Biography of a Dangerous Idea
6328:, Association for Computing Machinery,
6223:(Version D ed.), Arm Limited, 2010
6077:(2nd ed.), CRC Press, p. 20,
5388:
4531:{\displaystyle \infty \cdot z=\infty .}
1858:, and division by zero is not allowed.
6352:
6274:
6155:
4015:{\displaystyle {\frac {a}{\infty }}=0}
3976:{\displaystyle {\frac {a}{0}}=\infty }
1345:§ Projectively extended real line
6221:ARMv7-M Architecture Reference Manual
6005:
5993:
5873:
5824:
5812:
5784:
5395:
2456:means that the value of the function
466:{\displaystyle f(x)={\tfrac {1}{x}},}
7:
6563:, Sense Publishers, pp. 51–65,
6561:Teaching Mathematics as Storytelling
6453:Mathematical Fallacies and Paradoxes
5623:The Journal of Mathematical Behavior
4960:, and if the original system was an
3459:, so we have the indeterminate form
2634:. Instead, the function is said to "
1978:{\displaystyle 0\times 1=0\times 2.}
1772:{\displaystyle {\tfrac {0}{0}}={?},}
1673:{\displaystyle {\tfrac {6}{0}}={?},}
1230:is undefined. The real-valued slope
6457:, New York: Van Nostrand Reinhold,
5253:Handheld calculators, such as this
4983:(which for this reason is called a
2922:; such a function has two distinct
2541:In the case where the limit of the
2386:("ghosts of departed quantities").
1479:{\displaystyle {\tfrac {6}{3}}={?}}
99:where the divisor (denominator) is
6359:: CS1 maint: overridden setting (
6281:: CS1 maint: overridden setting (
5752:Educational Studies in Mathematics
5601:10.5951/mathteacmiddscho.24.3.0148
5593:10.5951/mathteacmiddscho.24.3.0148
5301:higher-level programming languages
4724:. This relation is shown to be an
4609:
4580:
4577:
4522:
4510:
4497:{\displaystyle \infty +z=\infty ,}
4488:
4476:
4463:{\displaystyle \infty +0=\infty ,}
4454:
4442:
4411:
4383:
4317:
4245:
4207:
4178:
4128:
4122:
4035:
4001:
3970:
3937:
3931:
3904:
3873:
3834:
3766:
3736:
3713:
3344:
3341:
3115:
3069:
2792:
2765:
2676:
1631:{\displaystyle {\tfrac {6}{3}}=2.}
1449:. Thus a division problem such as
1328:
882:{\displaystyle {\tfrac {10}{2}}=5}
753:{\displaystyle {\tfrac {10}{2}}=5}
565:
25:
5268:differ in behavior: for instance
5150:is the multiplicative inverse of
4663:and then this is expanded to the
2476:can be made arbitrarily close to
2449:{\textstyle \lim _{x\to c}f(x)=L}
1194:{\displaystyle {\tfrac {0}{1}}=0}
1166:then a horizontal line has slope
269:{\displaystyle q={\tfrac {a}{0}}}
241:By this definition, the quotient
199:{\displaystyle c={\tfrac {a}{b}}}
32:Division by zero (disambiguation)
6522:Retracing Elementary Mathematics
5536:{\displaystyle {\tfrac {N}{D}}.}
5443:For a video demonstration, see:
5115:In field theory, the expression
3943:{\displaystyle -\infty =\infty }
3324:{\displaystyle {\tfrac {0}{0}}.}
2134:The right side can be factored,
1845:{\displaystyle {\tfrac {0}{0}}.}
1397:{\displaystyle (5\times 3)/3={}}
781:is physically exhibited by some
703:(the divisor), and the quotient
345:Following the ordinary rules of
71:approaches zero from the right,
27:Class of mathematical expression
6415:from the original on 1999-04-29
4560:{\displaystyle {\tfrac {0}{0}}}
4134:{\displaystyle \infty +\infty }
3891:projectively extended real line
3851:Projectively extended real line
3481:{\displaystyle {\tfrac {0}{0}}}
3075:
2607:the function is not defined at
2189:{\displaystyle x-1=(x+1)(x-1).}
1739:When the problem is changed to
1442:{\displaystyle (5/3)\times 3=5}
1320:is represented by a new number
1252:{\displaystyle {\tfrac {y}{x}}}
1223:{\displaystyle {\tfrac {1}{0}}}
543:{\displaystyle {\tfrac {a}{0}}}
388:{\displaystyle {\tfrac {0}{0}}}
276:is nonsensical, as the product
129:{\displaystyle {\tfrac {a}{0}}}
79:approaches zero from the left,
75:tends to positive infinity. As
6506:, Cambridge University Press,
6399:Stutz, Michael (24 Jul 1998),
4615:{\displaystyle 0\cdot \infty }
4297:
3917:means an unsigned infinity or
3676:affinely extended real numbers
3643:
3631:
3623:
3595:
3583:
3580:
3568:
3557:
3505:
3381:
3251:
3245:
3234:
3221:
3215:
3204:
3184:
3178:
3170:
3164:
3150:
3084:
3038:
2949:
2943:
2897:
2860:
2670:
2664:
2653:
2437:
2431:
2420:
2180:
2168:
2165:
2153:
1707:{\displaystyle {?}\times 0=6.}
1516:{\displaystyle {?}\times 3=6,}
1424:
1410:
1378:
1366:
1136:and a vertical line has slope
442:
436:
322:rather than some other number
1:
6426:William Kahan (14 Oct 2011),
6374:Buzzard, Kevin (5 Jul 2020),
6134:"Wheels: On Division by Zero"
5960:"2.2 The Limit of a Function"
5635:10.1016/S0732-3123(03)00023-3
3331:(Another indeterminate form,
1807:{\displaystyle {?}\times 0=0}
902:, a kind of division by zero.
812:is imagined to be split into
590:positive or negative infinity
156:is the dividend (numerator).
6592:Northrop, Eugene P. (1944),
6054:Keisler, H. Jerome (2023) ,
4837:matrix containing only zeros
4772:one can extend the function
3840:{\displaystyle 1/0=+\infty }
3775:{\displaystyle \pm \infty ,}
3696:{\displaystyle \mathbb {R} }
2361:{\textstyle {\tfrac {a}{0}}}
2127:{\displaystyle x-1=x^{2}-1.}
1596:{\displaystyle 2\times 3=6,}
159:The usual definition of the
5801:Zazkis & Liljedahl 2009
5789:Zazkis & Liljedahl 2009
5468:Zazkis & Liljedahl 2009
5438:, Santa Barbara: Hamilton,
5435:Computers in Modern Society
5420:Zazkis & Liljedahl 2009
5408:Zazkis & Liljedahl 2009
5135:{\textstyle {\frac {a}{b}}}
5101:{\textstyle {\frac {2}{2}}}
5015:{\textstyle {\frac {2}{2}}}
4866:. It can be proven that if
4792:{\textstyle {\frac {1}{x}}}
4052:and cotangent functions of
3983:can be defined for nonzero
2564:increases without bound as
1940:{\displaystyle 0\times 2=0}
1908:{\displaystyle 0\times 1=0}
1355:Division is the inverse of
554:, sometimes denoted by the
234:{\displaystyle c\cdot b=a.}
83:tends to negative infinity.
6722:
6524:, Literary Licensing LLC,
6301:Python 3 Library Reference
6132:Carlström, Jesper (2004),
4263:one-point compactification
3895:one-point compactification
3703:by adding two new numbers
3665:Alternative number systems
2913:{\displaystyle x\to c^{-}}
2876:{\displaystyle x\to c^{+}}
1865:
1736:to make a true statement.
612:
29:
6569:10.1163/9789087907358_008
6504:Numbers, Sets, and Axioms
6150:10.1017/S0960129503004110
6071:Conway, John H. (2000) ,
5765:10.1007/s10649-011-9330-5
5372:Zero to the power of zero
5205:positive or negative zero
5181:floating-point arithmetic
5175:Floating-point arithmetic
3745:{\displaystyle -\infty ,}
2972:{\displaystyle f(x)=1/x,}
2801:{\displaystyle -\infty .}
2048:. Multiply both sides by
1351:Inverse of multiplication
171:by the divisor. That is,
6502:Hamilton, A. G. (1982),
4271:stereographic projection
4254:{\displaystyle \infty ,}
4159:extended complex numbers
3719:{\displaystyle +\infty }
2632:mathematical singularity
2397:studies the behavior of
832:parts, and the quotient
420:mathematical singularity
402:studies the behavior of
361:leaves division by zero
295:{\displaystyle q\cdot 0}
43:The reciprocal function
6691:Fractions (mathematics)
6344:10.1145/2816707.2816710
6251:10.1145/2349896.2349905
6191:10.1109/C-M.1981.220379
5921:The Mathematics Teacher
5886:Roy, Rahul (Jan 2003),
5842:Kaplan, Robert (1999),
5683:The Mathematics Teacher
5022:should be the solution
4801:Cauchy principal values
4213:{\displaystyle \infty }
4108:from either direction.
4041:{\displaystyle \infty }
3910:{\displaystyle \infty }
3897:of the real line. Here
3756:. With the addition of
2771:{\displaystyle \infty }
2401:using the concept of a
2196:Dividing both sides by
2078:{\displaystyle x=x^{2}}
1989:from both sides yields
1334:{\displaystyle \infty }
615:Quotition and partition
609:The meaning of division
571:{\displaystyle \infty }
6706:Mathematical fallacies
6681:Division (mathematics)
5958:; et al. (2023),
5661:10.5951/MTMS.16.1.0034
5537:
5502:
5258:
5136:
5102:
5049:
5016:
4966:cancellative semigroup
4954:
4922:
4882:equals 0, then b = 0.
4793:
4616:
4590:
4561:
4532:
4498:
4464:
4430:
4393:
4355:
4330:
4255:
4214:
4191:
4135:
4042:
4016:
3977:
3944:
3911:
3883:
3841:
3804:
3776:
3746:
3720:
3697:
3678:are obtained from the
3656:
3482:
3453:
3431:
3357:
3325:
3289:
3264:
3125:
3014:
2993:
2973:
2914:
2877:
2842:
2822:
2802:
2772:
2749:
2716:
2686:
2624:
2601:
2578:
2558:
2533:
2516:sufficiently close to
2510:
2490:
2470:
2450:
2378:infinitesimal calculus
2362:
2337:
2310:
2262:
2229:
2228:{\displaystyle 1=x+1.}
2190:
2128:
2089:from each side to get
2079:
2009:
1979:
1941:
1909:
1846:
1808:
1773:
1730:
1708:
1674:
1632:
1597:
1562:
1539:
1517:
1480:
1443:
1398:
1335:
1314:
1280:
1253:
1224:
1195:
1156:
1130:
1089:
1060:
1030:
1004:
978:
952:
920:
903:
883:
846:
826:
806:
783:mechanical calculators
754:
717:
697:
677:
650:
572:
544:
507:
487:
467:
389:
339:
316:
296:
270:
235:
200:
150:
130:
84:
6701:Mathematical analysis
6641:(Dover reprint, 1999)
6635:Introduction to Logic
6614:, New York: Penguin,
6493:Klein, Felix (1925),
6449:Bunch, Bryan (1982),
6297:"Built-in Exceptions"
6217:"Divide instructions"
6152:(inactive 5 Sep 2024)
5695:10.5951/MT.104.2.0133
5559:10.5951/MTMS.9.7.0362
5538:
5503:
5293:programming languages
5272:processors trigger a
5252:
5137:
5103:
5050:
5017:
4955:
4923:
4811:of the distribution.
4794:
4744:Non-standard analysis
4617:
4591:
4562:
4533:
4499:
4465:
4431:
4394:
4356:
4331:
4273:, with the resulting
4256:
4215:
4192:
4136:
4043:
4017:
3978:
3945:
3912:
3884:
3842:
3805:
3777:
3747:
3721:
3698:
3657:
3483:
3454:
3432:
3358:
3326:
3299:, informally written
3290:
3265:
3126:
3015:
2994:
2974:
2915:
2878:
2843:
2823:
2803:
2773:
2750:
2717:
2687:
2625:
2602:
2579:
2559:
2534:
2511:
2491:
2471:
2451:
2363:
2332:
2305:
2292:Brāhmasphuṭasiddhānta
2263:
2230:
2191:
2129:
2080:
2010:
1980:
1942:
1910:
1866:Further information:
1847:
1809:
1774:
1731:
1709:
1675:
1633:
1598:
1563:
1540:
1518:
1481:
1444:
1399:
1336:
1315:
1281:
1254:
1225:
1196:
1157:
1131:
1090:
1061:
1059:{\displaystyle 10:0,}
1031:
1005:
979:
953:
921:
897:
884:
847:
827:
807:
755:
718:
698:
678:
651:
649:{\displaystyle N/D=Q}
604:Elementary arithmetic
573:
545:
508:
488:
473:tends to infinity as
468:
390:
340:
317:
297:
271:
236:
201:
165:elementary arithmetic
151:
131:
42:
6468:(Dover reprint 1997)
6402:"Sunk by Windows NT"
6074:On Numbers and Games
5933:10.5951/MT.22.6.0366
5515:
5486:
5343:Historical accidents
5221:arithmetic underflow
5119:
5085:
5081:, so the expression
5048:{\displaystyle 2x=2}
5030:
4999:
4944:
4906:
4776:
4726:equivalence relation
4661:Peano's axiom system
4622:are left undefined.
4600:
4571:
4542:
4507:
4473:
4439:
4402:
4365:
4342:
4286:
4242:
4204:
4164:
4119:
4032:
3993:
3954:
3925:
3901:
3859:
3814:
3786:
3760:
3730:
3707:
3685:
3494:
3463:
3443:
3370:
3335:
3303:
3276:
3139:
3027:
3004:
2983:
2937:
2891:
2854:
2832:
2812:
2786:
2762:
2748:{\displaystyle x=c,}
2730:
2700:
2642:
2611:
2588:
2568:
2548:
2520:
2500:
2480:
2460:
2409:
2343:
2318:Ganita Sara Samgraha
2261:{\displaystyle 1=2.}
2246:
2207:
2138:
2093:
2056:
2015:, a false statement.
1993:
1951:
1919:
1887:
1868:Mathematical fallacy
1824:
1784:
1743:
1718:
1684:
1644:
1607:
1572:
1549:
1527:
1490:
1453:
1407:
1363:
1325:
1298:
1267:
1234:
1205:
1170:
1155:{\displaystyle 1:0.}
1140:
1114:
1088:{\displaystyle 1:0,}
1070:
1041:
1014:
1003:{\displaystyle 5:1.}
988:
984:or, proportionally,
977:{\displaystyle 10:2}
962:
951:{\displaystyle N:D.}
933:
910:
858:
836:
816:
796:
729:
707:
687:
667:
626:
562:
525:
497:
477:
430:
370:
351:mathematical fallacy
326:
306:
280:
245:
210:
175:
140:
111:
30:For other uses, see
6676:Computer arithmetic
6110:2011mmop.book.....G
5501:{\displaystyle N:D}
5457:– via YouTube
5315:In proof assistants
5170:Computer arithmetic
4921:{\displaystyle 0=1}
4839:is not invertible.
4770:distribution theory
4764:Distribution theory
4730:equivalence classes
4228:, making those its
4220:and representing a
3803:{\displaystyle 1/0}
2931:reciprocal function
2715:{\displaystyle x=c}
2008:{\displaystyle 1=2}
1729:{\displaystyle {?}}
1538:{\displaystyle {?}}
1313:{\displaystyle 1:0}
1129:{\displaystyle 0:1}
1029:{\displaystyle 5:1}
592:, return a special
424:reciprocal function
422:. For example, the
396:is also undefined.
6545:, Addison-Wesley,
6042:Henkin et al. 2012
5904:10.1007/BF02834448
5533:
5528:
5510:value of the ratio
5498:
5297:undefined behavior
5274:hardware exception
5259:
5245:Integer arithmetic
5132:
5098:
5055:. But in the ring
5045:
5012:
4950:
4918:
4789:
4626:Higher mathematics
4612:
4586:
4584:
4557:
4555:
4528:
4494:
4460:
4426:
4415:
4389:
4378:
4354:{\displaystyle z,}
4351:
4326:
4275:spherical distance
4251:
4210:
4187:
4131:
4070:approaches either
4038:
4012:
3973:
3940:
3907:
3879:
3837:
3800:
3772:
3754:points at infinity
3742:
3716:
3693:
3670:Extended real line
3652:
3630:
3564:
3512:
3478:
3476:
3449:
3427:
3422:
3388:
3353:
3348:
3321:
3316:
3297:indeterminate form
3288:{\displaystyle 0,}
3285:
3260:
3254:
3241:
3224:
3211:
3157:
3121:
3098:
3052:
3010:
2989:
2969:
2910:
2873:
2838:
2818:
2798:
2768:
2745:
2712:
2682:
2660:
2623:{\displaystyle x,}
2620:
2600:{\displaystyle c,}
2597:
2574:
2554:
2532:{\displaystyle c.}
2529:
2506:
2486:
2466:
2446:
2427:
2358:
2356:
2258:
2225:
2186:
2124:
2075:
2005:
1975:
1937:
1905:
1842:
1837:
1804:
1769:
1756:
1726:
1704:
1670:
1657:
1628:
1620:
1593:
1561:{\displaystyle 2,}
1558:
1535:
1513:
1476:
1466:
1439:
1394:
1331:
1310:
1279:{\displaystyle 1,}
1276:
1249:
1247:
1220:
1218:
1191:
1183:
1152:
1126:
1085:
1066:or proportionally
1056:
1026:
1000:
974:
948:
916:
904:
879:
871:
842:
822:
802:
790:partitive division
750:
742:
713:
693:
673:
661:quotitive division
646:
568:
540:
538:
515:indeterminate form
506:{\displaystyle 0.}
503:
483:
463:
458:
385:
383:
347:elementary algebra
338:{\displaystyle a.}
335:
312:
292:
266:
264:
231:
196:
194:
146:
126:
124:
85:
6578:978-90-8790-734-1
6552:978-0-201-82653-1
6539:Schumacher, Carol
6486:978-1-541-60182-6
6305:ZeroDivisionError
6269:978-1-4503-1669-9
6119:978-0-521-51610-5
5973:978-1-947172-13-5
5861:978-0-19-514237-2
5815:, pp. 82–83.
5527:
5398:, pp. 75–83.
5130:
5096:
5010:
4953:{\displaystyle 1}
4787:
4750:hyperreal numbers
4583:
4554:
4414:
4377:
4300:
4222:point at infinity
4004:
3965:
3919:point at infinity
3615:
3610:
3549:
3544:
3497:
3475:
3452:{\displaystyle 0}
3421:
3373:
3347:
3315:
3255:
3226:
3196:
3188:
3142:
3107:
3076:
3061:
3030:
3013:{\displaystyle 0}
2992:{\displaystyle x}
2841:{\displaystyle c}
2821:{\displaystyle x}
2645:
2577:{\displaystyle x}
2557:{\displaystyle f}
2509:{\displaystyle x}
2489:{\displaystyle L}
2469:{\displaystyle f}
2412:
2355:
1836:
1755:
1656:
1619:
1465:
1292:point at infinity
1246:
1217:
1182:
919:{\displaystyle Q}
900:1 : 0
870:
845:{\displaystyle Q}
825:{\displaystyle D}
805:{\displaystyle N}
741:
716:{\displaystyle Q}
696:{\displaystyle D}
676:{\displaystyle N}
552:point at infinity
537:
486:{\displaystyle x}
457:
382:
315:{\displaystyle 0}
263:
206:is equivalent to
193:
149:{\displaystyle a}
123:
16:(Redirected from
6713:
6657:
6655:
6640:
6638:
6624:
6601:
6599:
6581:
6555:
6534:
6516:
6498:
6489:
6467:
6456:
6436:
6435:
6434:
6423:
6417:
6416:
6404:
6396:
6390:
6389:
6388:
6387:
6371:
6365:
6364:
6358:
6350:
6337:
6321:
6315:
6314:
6313:
6312:
6306:
6293:
6287:
6286:
6280:
6272:
6253:
6237:
6231:
6230:
6229:
6228:
6213:
6207:
6206:
6174:
6168:
6167:
6161:
6153:
6129:
6123:
6122:
6094:
6088:
6087:
6068:
6062:
6061:
6051:
6045:
6039:
6033:
6027:
6021:
6015:
6009:
6003:
5997:
5991:
5985:
5984:
5951:
5945:
5943:
5913:
5907:
5906:
5883:
5877:
5871:
5865:
5864:
5849:
5839:
5828:
5822:
5816:
5810:
5804:
5798:
5792:
5782:
5776:
5775:
5746:
5740:
5739:
5729:10.1002/wics.122
5712:
5706:
5705:
5678:
5672:
5671:
5644:
5638:
5637:
5618:
5612:
5611:
5576:
5570:
5569:
5542:
5540:
5539:
5534:
5529:
5520:
5507:
5505:
5504:
5499:
5477:
5471:
5465:
5459:
5458:
5456:
5455:
5442:
5429:
5423:
5422:, p. 55–56.
5417:
5411:
5410:, p. 52–53.
5405:
5399:
5393:
5377:L'Hôpital's rule
5321:proof assistants
5232:
5161:
5141:
5139:
5138:
5133:
5131:
5123:
5107:
5105:
5104:
5099:
5097:
5089:
5080:
5073:
5054:
5052:
5051:
5046:
5026:of the equation
5021:
5019:
5018:
5013:
5011:
5003:
4959:
4957:
4956:
4951:
4933:commutative ring
4927:
4925:
4924:
4919:
4892:commutative ring
4886:Abstract algebra
4809:singular support
4798:
4796:
4795:
4790:
4788:
4780:
4723:
4713:
4689:
4682:
4665:ring of integers
4637:rational numbers
4621:
4619:
4618:
4613:
4595:
4593:
4592:
4587:
4585:
4576:
4566:
4564:
4563:
4558:
4556:
4547:
4537:
4535:
4534:
4529:
4503:
4501:
4500:
4495:
4469:
4467:
4466:
4461:
4435:
4433:
4432:
4427:
4416:
4407:
4398:
4396:
4395:
4390:
4379:
4370:
4360:
4358:
4357:
4352:
4335:
4333:
4332:
4327:
4310:
4302:
4301:
4296:
4291:
4260:
4258:
4257:
4252:
4219:
4217:
4216:
4211:
4196:
4194:
4193:
4188:
4171:
4151:complex analysis
4140:
4138:
4137:
4132:
4107:
4106:
4104:
4103:
4100:
4097:
4088:
4087:
4085:
4084:
4081:
4078:
4069:
4063:
4050:tangent function
4047:
4045:
4044:
4039:
4027:
4021:
4019:
4018:
4013:
4005:
3997:
3988:
3982:
3980:
3979:
3974:
3966:
3958:
3949:
3947:
3946:
3941:
3916:
3914:
3913:
3908:
3888:
3886:
3885:
3880:
3866:
3846:
3844:
3843:
3838:
3824:
3809:
3807:
3806:
3801:
3796:
3781:
3779:
3778:
3773:
3751:
3749:
3748:
3743:
3725:
3723:
3722:
3717:
3702:
3700:
3699:
3694:
3692:
3661:
3659:
3658:
3653:
3629:
3611:
3609:
3598:
3566:
3563:
3545:
3543:
3532:
3525:
3524:
3514:
3511:
3487:
3485:
3484:
3479:
3477:
3468:
3458:
3456:
3455:
3450:
3436:
3434:
3433:
3428:
3423:
3420:
3409:
3402:
3401:
3391:
3387:
3362:
3360:
3359:
3354:
3349:
3340:
3330:
3328:
3327:
3322:
3317:
3308:
3294:
3292:
3291:
3286:
3269:
3267:
3266:
3261:
3256:
3240:
3210:
3194:
3189:
3187:
3173:
3159:
3156:
3130:
3128:
3127:
3122:
3108:
3100:
3097:
3096:
3095:
3062:
3054:
3051:
3050:
3049:
3021:
3019:
3017:
3016:
3011:
2998:
2996:
2995:
2990:
2978:
2976:
2975:
2970:
2962:
2924:one-sided limits
2921:
2919:
2917:
2916:
2911:
2909:
2908:
2884:
2882:
2880:
2879:
2874:
2872:
2871:
2847:
2845:
2844:
2839:
2827:
2825:
2824:
2819:
2807:
2805:
2804:
2799:
2777:
2775:
2774:
2769:
2754:
2752:
2751:
2746:
2721:
2719:
2718:
2713:
2691:
2689:
2688:
2683:
2659:
2636:tend to infinity
2629:
2627:
2626:
2621:
2606:
2604:
2603:
2598:
2583:
2581:
2580:
2575:
2563:
2561:
2560:
2555:
2538:
2536:
2535:
2530:
2515:
2513:
2512:
2507:
2495:
2493:
2492:
2487:
2475:
2473:
2472:
2467:
2455:
2453:
2452:
2447:
2426:
2376:'s criticism of
2368:is contained in
2367:
2365:
2364:
2359:
2357:
2348:
2280:
2273:
2267:
2265:
2264:
2259:
2241:
2234:
2232:
2231:
2226:
2202:
2195:
2193:
2192:
2187:
2133:
2131:
2130:
2125:
2117:
2116:
2088:
2084:
2082:
2081:
2076:
2074:
2073:
2051:
2047:
2026:
2022:
2014:
2012:
2011:
2006:
1988:
1984:
1982:
1981:
1976:
1946:
1944:
1943:
1938:
1914:
1912:
1911:
1906:
1851:
1849:
1848:
1843:
1838:
1829:
1815:
1813:
1811:
1810:
1805:
1791:
1778:
1776:
1775:
1770:
1765:
1757:
1748:
1735:
1733:
1732:
1727:
1725:
1713:
1711:
1710:
1705:
1691:
1679:
1677:
1676:
1671:
1666:
1658:
1649:
1637:
1635:
1634:
1629:
1621:
1612:
1602:
1600:
1599:
1594:
1567:
1565:
1564:
1559:
1544:
1542:
1541:
1536:
1534:
1522:
1520:
1519:
1514:
1497:
1485:
1483:
1482:
1477:
1475:
1467:
1458:
1448:
1446:
1445:
1440:
1420:
1403:
1401:
1400:
1395:
1393:
1385:
1342:
1340:
1338:
1337:
1332:
1319:
1317:
1316:
1311:
1294:, and the ratio
1285:
1283:
1282:
1277:
1258:
1256:
1255:
1250:
1248:
1239:
1229:
1227:
1226:
1221:
1219:
1210:
1200:
1198:
1197:
1192:
1184:
1175:
1161:
1159:
1158:
1153:
1135:
1133:
1132:
1127:
1094:
1092:
1091:
1086:
1065:
1063:
1062:
1057:
1035:
1033:
1032:
1027:
1009:
1007:
1006:
1001:
983:
981:
980:
975:
957:
955:
954:
949:
925:
923:
922:
917:
901:
890:
888:
886:
885:
880:
872:
863:
851:
849:
848:
843:
831:
829:
828:
823:
811:
809:
808:
803:
775:never terminates
761:
759:
757:
756:
751:
743:
734:
722:
720:
719:
714:
702:
700:
699:
694:
682:
680:
679:
674:
655:
653:
652:
647:
636:
579:
577:
575:
574:
569:
549:
547:
546:
541:
539:
530:
512:
510:
509:
504:
492:
490:
489:
484:
472:
470:
469:
464:
459:
450:
416:tend to infinity
394:
392:
391:
386:
384:
375:
344:
342:
341:
336:
321:
319:
318:
313:
301:
299:
298:
293:
275:
273:
272:
267:
265:
256:
240:
238:
237:
232:
205:
203:
202:
197:
195:
186:
155:
153:
152:
147:
135:
133:
132:
127:
125:
116:
93:division by zero
82:
78:
74:
70:
66:
65:
63:
62:
57:
54:
21:
6721:
6720:
6716:
6715:
6714:
6712:
6711:
6710:
6686:Computer errors
6661:
6660:
6644:
6629:Suppes, Patrick
6627:
6622:
6604:
6591:
6588:
6586:Further reading
6579:
6558:
6553:
6537:
6532:
6519:
6514:
6501:
6492:
6487:
6479:, Basic Books,
6471:
6465:
6448:
6445:
6440:
6439:
6432:
6425:
6424:
6420:
6398:
6397:
6393:
6385:
6383:
6373:
6372:
6368:
6355:cite conference
6351:
6323:
6322:
6318:
6310:
6308:
6304:
6295:
6294:
6290:
6277:cite conference
6273:
6270:
6239:
6238:
6234:
6226:
6224:
6215:
6214:
6210:
6176:
6175:
6171:
6154:
6131:
6130:
6126:
6120:
6096:
6095:
6091:
6085:
6070:
6069:
6065:
6053:
6052:
6048:
6040:
6036:
6028:
6024:
6018:Schumacher 1996
6016:
6012:
6004:
6000:
5992:
5988:
5974:
5956:Strang, Gilbert
5954:Herman, Edwin;
5953:
5952:
5948:
5917:Cajori, Florian
5915:
5914:
5910:
5885:
5884:
5880:
5872:
5868:
5862:
5841:
5840:
5831:
5823:
5819:
5811:
5807:
5799:
5795:
5783:
5779:
5748:
5747:
5743:
5714:
5713:
5709:
5680:
5679:
5675:
5646:
5645:
5641:
5620:
5619:
5615:
5578:
5577:
5573:
5544:
5513:
5512:
5484:
5483:
5478:
5474:
5466:
5462:
5453:
5451:
5444:
5431:
5430:
5426:
5418:
5414:
5406:
5402:
5394:
5390:
5385:
5363:
5345:
5317:
5247:
5227:
5191:and an integer
5177:
5172:
5159:
5117:
5116:
5083:
5082:
5075:
5068:
5028:
5027:
4997:
4996:
4962:integral domain
4942:
4941:
4904:
4903:
4888:
4858:represents the
4817:
4774:
4773:
4766:
4758:surreal numbers
4746:
4715:
4714:if and only if
4695:
4694:on this set by
4692:binary relation
4684:
4672:
4649:complex numbers
4628:
4598:
4597:
4569:
4568:
4540:
4539:
4505:
4504:
4471:
4470:
4437:
4436:
4400:
4399:
4363:
4362:
4340:
4339:
4284:
4283:
4240:
4239:
4226:exterior domain
4202:
4201:
4199:infinity symbol
4162:
4161:
4155:complex numbers
4149:The subject of
4147:
4117:
4116:
4101:
4098:
4095:
4094:
4092:
4090:
4082:
4079:
4076:
4075:
4073:
4071:
4065:
4057:
4030:
4029:
4023:
3991:
3990:
3984:
3952:
3951:
3923:
3922:
3899:
3898:
3857:
3856:
3853:
3812:
3811:
3784:
3783:
3758:
3757:
3728:
3727:
3705:
3704:
3683:
3682:
3672:
3667:
3599:
3567:
3533:
3516:
3515:
3492:
3491:
3461:
3460:
3441:
3440:
3410:
3393:
3392:
3368:
3367:
3333:
3332:
3301:
3300:
3274:
3273:
3174:
3160:
3137:
3136:
3087:
3041:
3025:
3024:
3002:
3001:
3000:
2981:
2980:
2935:
2934:
2900:
2889:
2888:
2886:
2863:
2852:
2851:
2849:
2830:
2829:
2810:
2809:
2784:
2783:
2760:
2759:
2757:infinity symbol
2728:
2727:
2698:
2697:
2640:
2639:
2609:
2608:
2586:
2585:
2566:
2565:
2546:
2545:
2518:
2517:
2498:
2497:
2478:
2477:
2458:
2457:
2407:
2406:
2392:
2374:George Berkeley
2341:
2340:
2287:
2275:
2271:
2268:
2244:
2243:
2236:
2205:
2204:
2197:
2136:
2135:
2108:
2091:
2090:
2086:
2065:
2054:
2053:
2049:
2042:
2038:. For example:
2024:
2020:
1991:
1990:
1986:
1949:
1948:
1917:
1916:
1885:
1884:
1870:
1864:
1822:
1821:
1782:
1781:
1780:
1741:
1740:
1716:
1715:
1682:
1681:
1642:
1641:
1605:
1604:
1570:
1569:
1547:
1546:
1525:
1524:
1488:
1487:
1451:
1450:
1405:
1404:
1361:
1360:
1353:
1323:
1322:
1321:
1296:
1295:
1265:
1264:
1232:
1231:
1203:
1202:
1168:
1167:
1138:
1137:
1112:
1111:
1108:Cartesian plane
1068:
1067:
1039:
1038:
1012:
1011:
986:
985:
960:
959:
931:
930:
926:represents the
908:
907:
899:
856:
855:
853:
834:
833:
814:
813:
794:
793:
792:, the dividend
727:
726:
724:
705:
704:
685:
684:
665:
664:
663:, the dividend
624:
623:
617:
611:
606:
560:
559:
558:
556:infinity symbol
523:
522:
495:
494:
475:
474:
428:
427:
368:
367:
324:
323:
304:
303:
278:
277:
243:
242:
208:
207:
173:
172:
138:
137:
109:
108:
80:
76:
72:
68:
58:
55:
52:
51:
49:
44:
35:
28:
23:
22:
15:
12:
11:
5:
6719:
6717:
6709:
6708:
6703:
6698:
6693:
6688:
6683:
6678:
6673:
6663:
6662:
6659:
6658:
6646:Tarski, Alfred
6642:
6625:
6620:
6606:Seife, Charles
6602:
6587:
6584:
6583:
6582:
6577:
6556:
6551:
6535:
6531:978-1258291488
6530:
6517:
6513:978-0521287616
6512:
6499:
6490:
6485:
6473:Cheng, Eugenia
6469:
6463:
6444:
6441:
6438:
6437:
6418:
6391:
6366:
6316:
6288:
6268:
6232:
6208:
6169:
6144:(1): 143–184,
6124:
6118:
6089:
6083:
6063:
6046:
6034:
6022:
6010:
5998:
5986:
5972:
5946:
5927:(6): 366–368,
5908:
5878:
5866:
5860:
5829:
5817:
5805:
5793:
5787:, p. 78;
5777:
5759:(3): 409–428,
5741:
5723:(6): 678–685,
5707:
5689:(2): 133–137,
5673:
5639:
5629:(3): 297–317,
5613:
5587:(3): 148–155,
5571:
5553:(7): 362–367,
5532:
5526:
5523:
5497:
5494:
5491:
5472:
5460:
5424:
5412:
5400:
5387:
5386:
5384:
5381:
5380:
5379:
5374:
5369:
5362:
5359:
5358:
5357:
5344:
5341:
5337:cancelling out
5316:
5313:
5246:
5243:
5207:(+0 or −0). A
5176:
5173:
5171:
5168:
5129:
5126:
5095:
5092:
5044:
5041:
5038:
5035:
5009:
5006:
4949:
4917:
4914:
4911:
4887:
4884:
4816:
4815:Linear algebra
4813:
4786:
4783:
4765:
4762:
4754:infinitesimals
4745:
4742:
4627:
4624:
4611:
4608:
4605:
4582:
4579:
4553:
4550:
4527:
4524:
4521:
4518:
4515:
4512:
4493:
4490:
4487:
4484:
4481:
4478:
4459:
4456:
4453:
4450:
4447:
4444:
4425:
4422:
4419:
4413:
4410:
4388:
4385:
4382:
4376:
4373:
4350:
4347:
4325:
4322:
4319:
4316:
4313:
4309:
4305:
4299:
4295:
4280:Riemann sphere
4250:
4247:
4209:
4186:
4183:
4180:
4177:
4174:
4170:
4146:
4145:Riemann sphere
4143:
4130:
4127:
4124:
4037:
4011:
4008:
4003:
4000:
3972:
3969:
3964:
3961:
3939:
3936:
3933:
3930:
3906:
3878:
3875:
3872:
3869:
3865:
3852:
3849:
3836:
3833:
3830:
3827:
3823:
3819:
3799:
3795:
3791:
3771:
3768:
3765:
3741:
3738:
3735:
3715:
3712:
3691:
3671:
3668:
3666:
3663:
3651:
3648:
3645:
3642:
3639:
3636:
3633:
3628:
3625:
3622:
3618:
3614:
3608:
3605:
3602:
3597:
3594:
3591:
3588:
3585:
3582:
3579:
3576:
3573:
3570:
3562:
3559:
3556:
3552:
3548:
3542:
3539:
3536:
3531:
3528:
3523:
3519:
3510:
3507:
3504:
3500:
3474:
3471:
3448:
3426:
3419:
3416:
3413:
3408:
3405:
3400:
3396:
3386:
3383:
3380:
3376:
3352:
3346:
3343:
3320:
3314:
3311:
3284:
3281:
3259:
3253:
3250:
3247:
3244:
3239:
3236:
3233:
3229:
3223:
3220:
3217:
3214:
3209:
3206:
3203:
3199:
3192:
3186:
3183:
3180:
3177:
3172:
3169:
3166:
3163:
3155:
3152:
3149:
3145:
3120:
3117:
3114:
3111:
3106:
3103:
3094:
3090:
3086:
3083:
3079:
3074:
3071:
3068:
3065:
3060:
3057:
3048:
3044:
3040:
3037:
3033:
3009:
2988:
2968:
2965:
2961:
2957:
2954:
2951:
2948:
2945:
2942:
2907:
2903:
2899:
2896:
2870:
2866:
2862:
2859:
2837:
2817:
2797:
2794:
2791:
2767:
2744:
2741:
2738:
2735:
2722:as a vertical
2711:
2708:
2705:
2681:
2678:
2675:
2672:
2669:
2666:
2663:
2658:
2655:
2652:
2648:
2619:
2616:
2596:
2593:
2573:
2553:
2528:
2525:
2505:
2485:
2465:
2445:
2442:
2439:
2436:
2433:
2430:
2425:
2422:
2419:
2415:
2391:
2388:
2354:
2351:
2286:
2285:Early attempts
2283:
2257:
2254:
2251:
2224:
2221:
2218:
2215:
2212:
2185:
2182:
2179:
2176:
2173:
2170:
2167:
2164:
2161:
2158:
2155:
2152:
2149:
2146:
2143:
2123:
2120:
2115:
2111:
2107:
2104:
2101:
2098:
2072:
2068:
2064:
2061:
2040:
2017:
2016:
2004:
2001:
1998:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1936:
1933:
1930:
1927:
1924:
1904:
1901:
1898:
1895:
1892:
1863:
1860:
1841:
1835:
1832:
1803:
1800:
1797:
1794:
1790:
1768:
1764:
1760:
1754:
1751:
1724:
1703:
1700:
1697:
1694:
1690:
1669:
1665:
1661:
1655:
1652:
1627:
1624:
1618:
1615:
1592:
1589:
1586:
1583:
1580:
1577:
1557:
1554:
1533:
1512:
1509:
1506:
1503:
1500:
1496:
1474:
1470:
1464:
1461:
1438:
1435:
1432:
1429:
1426:
1423:
1419:
1415:
1412:
1391:
1388:
1384:
1380:
1377:
1374:
1371:
1368:
1357:multiplication
1352:
1349:
1330:
1309:
1306:
1303:
1275:
1272:
1245:
1242:
1216:
1213:
1190:
1187:
1181:
1178:
1151:
1148:
1145:
1125:
1122:
1119:
1084:
1081:
1078:
1075:
1055:
1052:
1049:
1046:
1025:
1022:
1019:
999:
996:
993:
973:
970:
967:
947:
944:
941:
938:
915:
878:
875:
869:
866:
841:
821:
801:
749:
746:
740:
737:
712:
692:
672:
645:
642:
639:
635:
631:
610:
607:
605:
602:
567:
536:
533:
502:
482:
462:
456:
453:
447:
444:
441:
438:
435:
381:
378:
334:
331:
311:
291:
288:
285:
262:
259:
253:
250:
230:
227:
224:
221:
218:
215:
192:
189:
183:
180:
145:
122:
119:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6718:
6707:
6704:
6702:
6699:
6697:
6694:
6692:
6689:
6687:
6684:
6682:
6679:
6677:
6674:
6672:
6669:
6668:
6666:
6654:
6653:
6647:
6643:
6637:
6636:
6630:
6626:
6623:
6621:0-14-029647-6
6617:
6613:
6612:
6607:
6603:
6598:
6597:
6590:
6589:
6585:
6580:
6574:
6570:
6566:
6562:
6557:
6554:
6548:
6544:
6540:
6536:
6533:
6527:
6523:
6518:
6515:
6509:
6505:
6500:
6496:
6491:
6488:
6482:
6478:
6474:
6470:
6466:
6464:0-442-24905-5
6460:
6455:
6454:
6447:
6446:
6442:
6431:
6430:
6422:
6419:
6414:
6410:
6409:
6403:
6395:
6392:
6381:
6377:
6370:
6367:
6362:
6356:
6349:
6345:
6341:
6336:
6331:
6327:
6320:
6317:
6302:
6298:
6292:
6289:
6284:
6278:
6271:
6265:
6261:
6257:
6252:
6247:
6243:
6236:
6233:
6222:
6218:
6212:
6209:
6205:
6200:
6196:
6192:
6188:
6184:
6180:
6173:
6170:
6165:
6159:
6151:
6147:
6143:
6139:
6135:
6128:
6125:
6121:
6115:
6111:
6107:
6103:
6099:
6093:
6090:
6086:
6084:9781568811277
6080:
6076:
6075:
6067:
6064:
6059:
6058:
6050:
6047:
6044:, p. 292
6043:
6038:
6035:
6031:
6030:Hamilton 1982
6026:
6023:
6020:, p. 149
6019:
6014:
6011:
6007:
6002:
5999:
5995:
5990:
5987:
5983:
5979:
5975:
5969:
5965:
5961:
5957:
5950:
5947:
5942:
5938:
5934:
5930:
5926:
5922:
5918:
5912:
5909:
5905:
5901:
5897:
5893:
5889:
5882:
5879:
5875:
5870:
5867:
5863:
5857:
5853:
5848:
5847:
5838:
5836:
5834:
5830:
5826:
5821:
5818:
5814:
5809:
5806:
5803:, p. 55.
5802:
5797:
5794:
5790:
5786:
5781:
5778:
5774:
5770:
5766:
5762:
5758:
5754:
5753:
5745:
5742:
5738:
5734:
5730:
5726:
5722:
5718:
5711:
5708:
5704:
5700:
5696:
5692:
5688:
5684:
5677:
5674:
5670:
5666:
5662:
5658:
5654:
5650:
5643:
5640:
5636:
5632:
5628:
5624:
5617:
5614:
5610:
5606:
5602:
5598:
5594:
5590:
5586:
5582:
5575:
5572:
5568:
5564:
5560:
5556:
5552:
5548:
5530:
5524:
5521:
5511:
5495:
5492:
5489:
5482:
5476:
5473:
5469:
5464:
5461:
5449:
5448:
5441:
5437:
5436:
5428:
5425:
5421:
5416:
5413:
5409:
5404:
5401:
5397:
5392:
5389:
5382:
5378:
5375:
5373:
5370:
5368:
5365:
5364:
5360:
5355:
5353:
5347:
5346:
5342:
5340:
5338:
5334:
5330:
5326:
5322:
5314:
5312:
5310:
5306:
5302:
5299:. In typical
5298:
5294:
5291:
5287:
5283:
5279:
5275:
5271:
5267:
5263:
5256:
5251:
5244:
5242:
5240:
5236:
5230:
5224:
5222:
5218:
5217:negative zero
5212:
5210:
5206:
5202:
5198:
5194:
5190:
5186:
5182:
5174:
5169:
5167:
5165:
5157:
5153:
5149:
5145:
5127:
5124:
5113:
5111:
5093:
5090:
5078:
5071:
5066:
5062:
5058:
5042:
5039:
5036:
5033:
5025:
5007:
5004:
4994:
4990:
4986:
4985:division ring
4982:
4978:
4974:
4969:
4967:
4963:
4947:
4938:
4934:
4929:
4915:
4912:
4909:
4901:
4897:
4893:
4885:
4883:
4881:
4877:
4873:
4870:exists, then
4869:
4865:
4861:
4860:pseudoinverse
4857:
4853:
4850: =
4849:
4845:
4840:
4838:
4834:
4830:
4826:
4822:
4814:
4812:
4810:
4806:
4802:
4784:
4781:
4771:
4763:
4761:
4759:
4755:
4751:
4743:
4741:
4738:
4736:
4731:
4727:
4722:
4718:
4711:
4707:
4703:
4699:
4693:
4687:
4680:
4676:
4671:of integers,
4670:
4669:ordered pairs
4666:
4662:
4656:
4652:
4650:
4646:
4642:
4638:
4634:
4625:
4623:
4606:
4603:
4551:
4548:
4525:
4519:
4516:
4513:
4491:
4485:
4482:
4479:
4457:
4451:
4448:
4445:
4423:
4420:
4417:
4408:
4386:
4380:
4374:
4371:
4348:
4345:
4336:
4323:
4311:
4303:
4281:
4276:
4272:
4268:
4264:
4248:
4236:
4234:
4233:neighborhoods
4231:
4227:
4223:
4200:
4184:
4172:
4160:
4156:
4152:
4144:
4142:
4125:
4114:
4109:
4068:
4061:
4055:
4051:
4026:
4009:
4006:
3998:
3987:
3967:
3962:
3959:
3934:
3928:
3920:
3896:
3893:, which is a
3892:
3867:
3850:
3848:
3831:
3828:
3825:
3821:
3817:
3797:
3793:
3789:
3769:
3763:
3755:
3739:
3733:
3710:
3681:
3677:
3669:
3664:
3662:
3649:
3646:
3640:
3637:
3634:
3626:
3620:
3612:
3606:
3603:
3600:
3592:
3589:
3586:
3577:
3574:
3571:
3560:
3554:
3546:
3540:
3537:
3534:
3529:
3526:
3521:
3517:
3508:
3502:
3489:
3472:
3469:
3446:
3437:
3424:
3417:
3414:
3411:
3406:
3403:
3398:
3394:
3384:
3378:
3365:
3350:
3318:
3312:
3309:
3298:
3282:
3279:
3270:
3257:
3248:
3242:
3237:
3231:
3218:
3212:
3207:
3201:
3190:
3181:
3175:
3167:
3161:
3153:
3147:
3134:
3131:
3118:
3112:
3109:
3104:
3101:
3092:
3088:
3081:
3072:
3066:
3063:
3058:
3055:
3046:
3042:
3035:
3022:
3007:
2986:
2966:
2963:
2959:
2955:
2952:
2946:
2940:
2932:
2927:
2925:
2905:
2901:
2894:
2868:
2864:
2857:
2835:
2815:
2795:
2789:
2781:
2758:
2742:
2739:
2736:
2733:
2725:
2709:
2706:
2703:
2696:has the line
2695:
2679:
2673:
2667:
2661:
2656:
2650:
2637:
2633:
2617:
2614:
2594:
2591:
2571:
2551:
2544:
2543:real function
2539:
2526:
2523:
2503:
2483:
2463:
2443:
2440:
2434:
2428:
2423:
2417:
2404:
2400:
2396:
2389:
2387:
2385:
2384:
2379:
2375:
2371:
2352:
2349:
2336:
2331:
2329:
2325:
2321:
2319:
2315:
2309:
2304:
2302:
2298:
2294:
2293:
2284:
2282:
2278:
2255:
2252:
2249:
2239:
2235:Substituting
2222:
2219:
2216:
2213:
2210:
2200:
2183:
2177:
2174:
2171:
2162:
2159:
2156:
2150:
2147:
2144:
2141:
2121:
2118:
2113:
2109:
2105:
2102:
2099:
2096:
2070:
2066:
2062:
2059:
2045:
2039:
2037:
2036:invalid proof
2033:
2028:
2002:
1999:
1996:
1972:
1969:
1966:
1963:
1960:
1957:
1954:
1934:
1931:
1928:
1925:
1922:
1902:
1899:
1896:
1893:
1890:
1882:
1881:
1880:
1877:
1875:
1869:
1861:
1859:
1857:
1852:
1839:
1833:
1830:
1819:
1816:in this case
1801:
1798:
1795:
1792:
1788:
1766:
1762:
1758:
1752:
1749:
1737:
1722:
1701:
1698:
1695:
1692:
1688:
1667:
1663:
1659:
1653:
1650:
1638:
1625:
1622:
1616:
1613:
1603:so therefore
1590:
1587:
1584:
1581:
1578:
1575:
1555:
1552:
1531:
1510:
1507:
1504:
1501:
1498:
1494:
1472:
1468:
1462:
1459:
1436:
1433:
1430:
1427:
1421:
1417:
1413:
1389:
1386:
1382:
1375:
1372:
1369:
1358:
1350:
1348:
1346:
1307:
1304:
1301:
1293:
1289:
1273:
1270:
1262:
1243:
1240:
1214:
1211:
1188:
1185:
1179:
1176:
1165:
1149:
1146:
1143:
1123:
1120:
1117:
1109:
1105:
1104:straight line
1101:
1096:
1082:
1079:
1076:
1073:
1053:
1050:
1047:
1044:
1023:
1020:
1017:
997:
994:
991:
971:
968:
965:
945:
942:
939:
936:
929:
913:
896:
892:
876:
873:
867:
864:
839:
819:
799:
791:
786:
784:
780:
776:
772:
767:
765:
747:
744:
738:
735:
710:
690:
670:
662:
657:
643:
640:
637:
633:
629:
622:
616:
608:
603:
601:
599:
595:
591:
587:
582:
557:
553:
534:
531:
518:
516:
500:
480:
460:
454:
451:
445:
439:
433:
425:
421:
418:", a type of
417:
413:
412:real function
409:
405:
401:
397:
395:
379:
376:
364:
360:
356:
352:
348:
332:
329:
309:
289:
286:
283:
260:
257:
251:
248:
228:
225:
222:
219:
216:
213:
190:
187:
181:
178:
170:
166:
162:
157:
143:
120:
117:
106:
102:
98:
94:
90:
61:
47:
41:
37:
33:
19:
6651:
6634:
6609:
6595:
6560:
6542:
6521:
6503:
6494:
6476:
6452:
6428:
6421:
6406:
6394:
6384:, retrieved
6380:Xena Project
6379:
6369:
6347:
6325:
6319:
6309:, retrieved
6300:
6291:
6260:1721.1/86949
6241:
6235:
6225:, retrieved
6220:
6211:
6202:
6182:
6178:
6172:
6141:
6137:
6127:
6101:
6092:
6073:
6066:
6056:
6049:
6037:
6032:, p. 19
6025:
6013:
6008:, p. 26
6001:
5996:, p. 63
5989:
5963:
5949:
5924:
5920:
5911:
5898:(1): 30–40,
5895:
5891:
5881:
5876:, p. 15
5869:
5845:
5827:, p. 14
5820:
5808:
5796:
5791:, p. 55
5780:
5756:
5750:
5744:
5720:
5716:
5710:
5686:
5682:
5676:
5655:(1): 34–41,
5652:
5648:
5642:
5626:
5622:
5616:
5584:
5580:
5574:
5550:
5546:
5509:
5480:
5475:
5463:
5452:, retrieved
5450:, 7 Mar 2021
5446:
5439:
5434:
5427:
5415:
5403:
5391:
5367:Zero divisor
5351:
5318:
5260:
5238:
5234:
5228:
5225:
5213:
5178:
5155:
5151:
5147:
5143:
5114:
5076:
5069:
5065:zero divisor
5060:
5056:
5023:
4992:
4988:
4970:
4930:
4900:trivial ring
4896:localization
4889:
4879:
4875:
4871:
4867:
4863:
4855:
4851:
4847:
4843:
4841:
4818:
4804:
4767:
4747:
4739:
4735:transitivity
4720:
4716:
4709:
4705:
4701:
4697:
4685:
4678:
4674:
4657:
4653:
4640:
4629:
4337:
4237:
4148:
4110:
4066:
4059:
4054:trigonometry
4024:
3985:
3854:
3680:real numbers
3673:
3490:
3438:
3366:
3271:
3135:
3132:
3023:
2928:
2540:
2496:by choosing
2393:
2381:
2372:philosopher
2338:
2333:
2327:
2322:
2317:
2311:
2306:
2290:
2288:
2276:
2269:
2237:
2198:
2043:
2029:
2018:
1878:
1871:
1855:
1853:
1817:
1738:
1639:
1354:
1261:intersection
1097:
905:
789:
787:
768:
764:lettuce wrap
660:
658:
618:
594:not-a-number
583:
519:
398:
355:real numbers
158:
92:
86:
59:
45:
36:
5189:significand
4854:, in which
4690:, define a
4278:called the
4230:topological
2848:from above
2780:real number
2638:", denoted
2383:The Analyst
2380:in 1734 in
2370:Anglo-Irish
2324:Bhāskara II
2297:Brahmagupta
2085:. Subtract
1985:Cancelling
1164:real number
771:subtraction
89:mathematics
6671:0 (number)
6665:Categories
6408:Wired News
6386:2024-01-21
6335:1506.04205
6311:2024-01-22
6227:2024-06-12
6098:Gbur, Greg
6006:Klein 1925
5994:Klein 1925
5982:1022848630
5874:Bunch 1982
5825:Bunch 1982
5813:Cheng 2023
5785:Cheng 2023
5454:2024-01-06
5396:Cheng 2023
5323:, such as
5303:, such as
4981:skew field
4829:multiplied
2885:and below
2630:a type of
613:See also:
596:value, or
302:is always
169:multiplied
6204:negative.
6185:(3): 65,
5892:Resonance
5737:121947341
5609:188092067
5309:exception
5201:underflow
5164:zero ring
5110:undefined
5063:, 2 is a
4641:undefined
4610:∞
4607:⋅
4581:∞
4578:∞
4538:However,
4523:∞
4514:⋅
4511:∞
4489:∞
4477:∞
4455:∞
4443:∞
4412:∞
4384:∞
4318:∞
4312:∪
4298:^
4246:∞
4208:∞
4179:∞
4173:∪
4129:∞
4123:∞
4036:∞
4002:∞
3971:∞
3938:∞
3932:∞
3929:−
3905:∞
3874:∞
3868:∪
3835:∞
3767:∞
3764:±
3737:∞
3734:−
3714:∞
3624:→
3604:−
3575:−
3558:→
3538:−
3527:−
3506:→
3415:−
3404:−
3382:→
3345:∞
3342:∞
3235:→
3205:→
3151:→
3116:∞
3113:−
3093:−
3085:→
3070:∞
3039:→
2999:tends to
2906:−
2898:→
2861:→
2828:tends to
2793:∞
2790:−
2766:∞
2724:asymptote
2677:∞
2654:→
2584:tends to
2421:→
2399:functions
2175:−
2145:−
2119:−
2100:−
1970:×
1958:×
1947:one gets
1926:×
1894:×
1874:fallacies
1862:Fallacies
1856:undefined
1793:×
1693:×
1579:×
1499:×
1428:×
1373:×
1329:∞
779:algorithm
586:computing
566:∞
493:tends to
404:functions
363:undefined
287:⋅
217:⋅
6696:Infinity
6648:(1941),
6631:(1957),
6608:(2000),
6541:(1996),
6475:(2023),
6413:archived
6179:Computer
6158:citation
6100:(2011),
5964:Calculus
5941:27951153
5773:41413121
5703:20876802
5669:41183440
5567:41181943
5508:and the
5361:See also
5352:Yorktown
5276:, while
5197:infinity
5193:exponent
5185:IEEE 754
5146:, where
4902:, where
4728:and its
4647:or even
4633:integers
3855:The set
2755:and the
2692:and its
2395:Calculus
2390:Calculus
2328:Līlāvatī
2314:Mahāvīra
2312:In 830,
1568:because
1288:parallel
621:division
400:Calculus
161:quotient
136:, where
105:fraction
97:division
6443:Sources
6199:9923085
6106:Bibcode
5354:(CG-48)
5278:PowerPC
5262:Integer
5233:, then
4833:inverse
4748:In the
4105:
4093:
4086:
4074:
4028:is not
3889:is the
2242:yields
2203:yields
2052:to get
2032:algebra
1106:in the
406:in the
64:
50:
6618:
6575:
6549:
6528:
6510:
6483:
6461:
6382:(Blog)
6266:
6197:
6116:
6081:
5980:
5970:
5939:
5858:
5771:
5735:
5701:
5667:
5607:
5599:
5565:
5305:Python
4977:fields
4821:matrix
4596:, and
4267:sphere
3989:, and
2030:Using
1523:where
359:fields
6433:(PDF)
6330:arXiv
6195:S2CID
5937:JSTOR
5852:68–75
5769:JSTOR
5733:S2CID
5699:JSTOR
5665:JSTOR
5605:S2CID
5597:JSTOR
5563:JSTOR
5481:ratio
5383:Notes
5333:total
5319:Many
5307:, an
5255:TI-86
5203:" to
5160:0 ≠ 1
4973:rings
4937:wheel
4878:. If
4825:added
4704:) ≃ (
4683:with
4113:field
4022:when
2694:graph
2403:limit
2308:zero.
1883:From
1102:of a
1100:slope
928:ratio
598:crash
408:limit
67:. As
6616:ISBN
6573:ISBN
6547:ISBN
6526:ISBN
6508:ISBN
6481:ISBN
6459:ISBN
6361:link
6283:link
6264:ISBN
6164:link
6114:ISBN
6079:ISBN
5978:OCLC
5968:ISBN
5856:ISBN
5350:USS
5329:Lean
5327:and
5288:and
5266:CPUs
5231:= −2
5074:and
4975:and
4827:and
4645:real
4058:tan(
3726:and
3674:The
2301:zero
2289:The
2041:Let
1915:and
1343:see
619:The
101:zero
6565:doi
6340:doi
6256:hdl
6246:doi
6187:doi
6146:doi
5929:doi
5900:doi
5761:doi
5725:doi
5691:doi
5687:104
5657:doi
5631:doi
5589:doi
5555:doi
5325:Coq
5290:C++
5282:ARM
5270:x86
5209:NaN
5108:is
5079:= 4
5072:= 1
4862:of
4819:In
4768:In
4737:).
4688:≠ 0
4089:or
3617:lim
3551:lim
3499:lim
3375:lim
3228:lim
3198:lim
3144:lim
3078:lim
3032:lim
2647:lim
2414:lim
2326:'s
2295:of
2279:− 1
2274:as
2240:= 1
2201:− 1
2046:= 1
1876:.
1818:any
788:In
659:In
584:In
163:in
87:In
18:1/0
6667::
6571:,
6411:,
6405:,
6378:,
6357:}}
6353:{{
6346:,
6338:,
6299:,
6279:}}
6275:{{
6262:,
6254:,
6219:,
6201:,
6193:,
6183:14
6181:,
6160:}}
6156:{{
6142:14
6140:,
6136:,
6112:,
5976:,
5962:,
5935:,
5925:22
5923:,
5894:,
5890:,
5854:,
5832:^
5767:,
5757:79
5755:,
5731:,
5719:,
5697:,
5685:,
5663:,
5653:16
5651:,
5627:22
5625:,
5603:,
5595:,
5585:24
5583:,
5561:,
5549:,
5223:.
5144:ab
5112:.
5059:/6
4991:/6
4968:.
4874:=
4852:ab
4760:.
4721:bc
4719:=
4717:ad
4708:,
4700:,
4681:)}
4677:,
4673:{(
4651:.
4567:,
4261:a
4235:.
4056::
3847:.
3650:2.
2933:,
2926:.
2281:.
2256:2.
2223:1.
2122:1.
2027:.
1973:2.
1702:6.
1626:2.
1150:0.
1045:10
998:1.
966:10
889:).
865:10
785:.
760:).
736:10
501:0.
426:,
95:,
91:,
48:=
6567::
6363:)
6342::
6332::
6307:"
6285:)
6258::
6248::
6189::
6166:)
6148::
6108::
5944:.
5931::
5902::
5896:8
5763::
5727::
5721:2
5693::
5659::
5633::
5591::
5557::
5551:9
5531:.
5525:D
5522:N
5496:D
5493::
5490:N
5286:C
5239:x
5235:x
5229:x
5156:b
5152:b
5148:b
5128:b
5125:a
5094:2
5091:2
5077:x
5070:x
5061:Z
5057:Z
5043:2
5040:=
5037:x
5034:2
5024:x
5008:2
5005:2
4993:Z
4989:Z
4948:1
4916:1
4913:=
4910:0
4880:b
4876:b
4872:b
4868:b
4864:b
4856:b
4848:b
4846:/
4844:a
4805:x
4785:x
4782:1
4712:)
4710:d
4706:c
4702:b
4698:a
4696:(
4686:b
4679:b
4675:a
4604:0
4552:0
4549:0
4526:.
4520:=
4517:z
4492:,
4486:=
4483:z
4480:+
4458:,
4452:=
4449:0
4446:+
4424:,
4421:0
4418:=
4409:z
4387:,
4381:=
4375:0
4372:z
4349:,
4346:z
4324:.
4321:}
4315:{
4308:C
4304:=
4294:C
4249:,
4185:,
4182:}
4176:{
4169:C
4126:+
4102:2
4099:/
4096:π
4091:−
4083:2
4080:/
4077:π
4072:+
4067:x
4062:)
4060:x
4025:a
4010:0
4007:=
3999:a
3986:a
3968:=
3963:0
3960:a
3935:=
3877:}
3871:{
3864:R
3832:+
3829:=
3826:0
3822:/
3818:1
3798:0
3794:/
3790:1
3770:,
3740:,
3711:+
3690:R
3647:=
3644:)
3641:1
3638:+
3635:x
3632:(
3627:1
3621:x
3613:=
3607:1
3601:x
3596:)
3593:1
3590:+
3587:x
3584:(
3581:)
3578:1
3572:x
3569:(
3561:1
3555:x
3547:=
3541:1
3535:x
3530:1
3522:2
3518:x
3509:1
3503:x
3473:0
3470:0
3447:0
3425:,
3418:1
3412:x
3407:1
3399:2
3395:x
3385:1
3379:x
3351:,
3319:.
3313:0
3310:0
3283:,
3280:0
3258:.
3252:)
3249:x
3246:(
3243:g
3238:c
3232:x
3222:)
3219:x
3216:(
3213:f
3208:c
3202:x
3191:=
3185:)
3182:x
3179:(
3176:g
3171:)
3168:x
3165:(
3162:f
3154:c
3148:x
3119:.
3110:=
3105:x
3102:1
3089:0
3082:x
3073:,
3067:+
3064:=
3059:x
3056:1
3047:+
3043:0
3036:x
3020::
3008:0
2987:x
2967:,
2964:x
2960:/
2956:1
2953:=
2950:)
2947:x
2944:(
2941:f
2920:)
2902:c
2895:x
2887:(
2883:)
2869:+
2865:c
2858:x
2850:(
2836:c
2816:x
2796:.
2743:,
2740:c
2737:=
2734:x
2710:c
2707:=
2704:x
2680:,
2674:=
2671:)
2668:x
2665:(
2662:f
2657:c
2651:x
2618:,
2615:x
2595:,
2592:c
2572:x
2552:f
2527:.
2524:c
2504:x
2484:L
2464:f
2444:L
2441:=
2438:)
2435:x
2432:(
2429:f
2424:c
2418:x
2353:0
2350:a
2277:x
2272:0
2253:=
2250:1
2238:x
2220:+
2217:x
2214:=
2211:1
2199:x
2184:.
2181:)
2178:1
2172:x
2169:(
2166:)
2163:1
2160:+
2157:x
2154:(
2151:=
2148:1
2142:x
2114:2
2110:x
2106:=
2103:1
2097:x
2087:1
2071:2
2067:x
2063:=
2060:x
2050:x
2044:x
2025:0
2021:0
2003:2
2000:=
1997:1
1987:0
1967:0
1964:=
1961:1
1955:0
1935:0
1932:=
1929:2
1923:0
1903:0
1900:=
1897:1
1891:0
1840:.
1834:0
1831:0
1814:;
1802:0
1799:=
1796:0
1789:?
1767:,
1763:?
1759:=
1753:0
1750:0
1723:?
1699:=
1696:0
1689:?
1668:,
1664:?
1660:=
1654:0
1651:6
1623:=
1617:3
1614:6
1591:,
1588:6
1585:=
1582:3
1576:2
1556:,
1553:2
1532:?
1511:,
1508:6
1505:=
1502:3
1495:?
1473:?
1469:=
1463:3
1460:6
1437:5
1434:=
1431:3
1425:)
1422:3
1418:/
1414:5
1411:(
1390:=
1387:3
1383:/
1379:)
1376:3
1370:5
1367:(
1341:;
1308:0
1305::
1302:1
1274:,
1271:1
1244:x
1241:y
1215:0
1212:1
1189:0
1186:=
1180:1
1177:0
1147::
1144:1
1124:1
1121::
1118:0
1083:,
1080:0
1077::
1074:1
1054:,
1051:0
1048::
1024:1
1021::
1018:5
995::
992:5
972:2
969::
946:.
943:D
940::
937:N
914:Q
877:5
874:=
868:2
854:(
840:Q
820:D
800:N
748:5
745:=
739:2
725:(
711:Q
691:D
671:N
644:Q
641:=
638:D
634:/
630:N
578:;
535:0
532:a
481:x
461:,
455:x
452:1
446:=
443:)
440:x
437:(
434:f
380:0
377:0
333:.
330:a
310:0
290:0
284:q
261:0
258:a
252:=
249:q
229:.
226:a
223:=
220:b
214:c
191:b
188:a
182:=
179:c
144:a
121:0
118:a
81:y
77:x
73:y
69:x
60:x
56:/
53:1
46:y
34:.
20:)
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