2017:
5158:
43:
5127:
314:
140:
667:
200:
544:
589:
429:
817:
732:
3483:
2511:), one can continue the procedure past the ones place as far as desired. If the divisor has a fractional part, one can restate the problem by moving the decimal to the right in both numbers until the divisor has no fraction, which can make the problem easier to solve (e.g., 10/2.5 = 100/25 = 4).
1126:, which is the part of the first number that remains, when in the course of computing the quotient, no further full chunk of the size of the second number can be allocated. For example, if 21 apples are divided between 4 people, everyone receives 5 apples again, and 1 apple remains.
309:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,+\,{\text{term}}\\\scriptstyle {\text{summand}}\,+\,{\text{summand}}\\\scriptstyle {\text{addend}}\,+\,{\text{addend}}\\\scriptstyle {\text{augend}}\,+\,{\text{addend}}\end{matrix}}\right\}\,=\,}
1097:, the process of calculating the number of times one number is contained within another. For example, if 20 apples are divided evenly between 4 people, everyone receives 5 apples (see picture). However, this number of times or the number contained (divisor) need not be
472:
2535:
by aligning the divisor on the C scale with the dividend on the D scale. The quotient can be found on the D scale where it is aligned with the left index on the C scale. The user is responsible, however, for mentally keeping track of the decimal point.
2682:
under division. Apart from division by zero being undefined, the quotient is not an integer unless the dividend is an integer multiple of the divisor. For example, 26 cannot be divided by 11 to give an integer. Such a case uses one of five approaches:
2484:
Division is often introduced through the notion of "sharing out" a set of objects, for example a pile of lollies, into a number of equal portions. Distributing the objects several at a time in each round of sharing to each portion leads to the idea of
662:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\frac {\scriptstyle {\text{dividend}}}{\scriptstyle {\text{divisor}}}}\\\scriptstyle {\frac {\scriptstyle {\text{numerator}}}{\scriptstyle {\text{denominator}}}}\end{matrix}}\right\}\,=\,}
357:
3718:
3176:
1661:
751:
674:
4401:
1878:
984:
3113:
1521:
898:
1245:
means the size of each of 5 parts into which a set of size 20 is divided. For example, 20 apples divide into five groups of four apples, meaning that "twenty divided by five is equal to four". This is denoted as
2887:
1993:
2492:
By allowing one to subtract more multiples than what the partial remainder allows at a given stage, more flexible methods, such as the bidirectional variant of chunking, can be developed as well.
539:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{factor}}\,\times \,{\text{factor}}\\\scriptstyle {\text{multiplier}}\,\times \,{\text{multiplicand}}\end{matrix}}\right\}\,=\,}
2794:
1935:
1730:
454:
1009:
570:
3550:
842:
923:
2967:
return a rational number as the answer, as in case 3 above. These languages also provide functions to get the results of the other cases, either directly or from the result of case 3.
339:
2935:
2823:
2740:
2447:
424:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{term}}\,-\,{\text{term}}\\\scriptstyle {\text{minuend}}\,-\,{\text{subtrahend}}\end{matrix}}\right\}\,=\,}
2253:
2076:
2202:
1205:(which define multiplication and addition over single-variabled formulas). Those in which a division (with a single result) by all nonzero elements is defined are called
4167:. Division in this sense does not require ∗ to have any particular properties (such as commutativity, associativity, or an identity element). A magma for which both
2309:
2970:
Names and symbols used for integer division include div, /, \, and %. Definitions vary regarding integer division when the dividend or the divisor is negative:
1536:
3514:
812:{\displaystyle \scriptstyle \left.{\begin{matrix}\scriptstyle {\text{base}}^{\text{exponent}}\\\scriptstyle {\text{base}}^{\text{power}}\end{matrix}}\right\}\,=\,}
3019:
2412:
2364:
2138:
727:{\displaystyle \scriptstyle \left\{{\begin{matrix}\scriptstyle {\text{fraction}}\\\scriptstyle {\text{quotient}}\\\scriptstyle {\text{ratio}}\end{matrix}}\right.}
4317:
2318:-9.6 states it should not be used. This division sign is also used alone to represent the division operation itself, as for instance as a label on a key of a
3478:{\displaystyle {p+iq \over r+is}={(p+iq)(r-is) \over (r+is)(r-is)}={pr+qs+i(qr-ps) \over r^{2}+s^{2}}={pr+qs \over r^{2}+s^{2}}+i{qr-ps \over r^{2}+s^{2}}.}
2024:
used as a variant of the minus sign in an excerpt from an official
Norwegian trading statement form called «Næringsoppgave 1» for the taxation year 2010.
1745:
2598:
Euclidean division is the mathematical formulation of the outcome of the usual process of division of integers. It asserts that, given two integers,
941:
4956:
1036:
172:
1377:
5103:
863:
4920:
4728:
1364:(where the use of parentheses indicates that the operations inside parentheses are performed before the operations outside parentheses).
17:
1186:, which is not defined. In the 21-apples example, everyone would receive 5 apple and a quarter of an apple, thus avoiding any leftover.
5322:
107:
4929:
1129:
For division to always yield one number rather than an integer quotient plus a remainder, the natural numbers must be extended to
4895:
79:
60:
1193:, different ways of defining mathematical structure. Those in which a Euclidean division (with remainder) is defined are called
4747:
2847:
4875:
4708:
4857:
4813:
4646:
4203:. In a quasigroup, division in this sense is always possible, even without an identity element and hence without inverses.
1940:
86:
1325:. The set of all rational numbers is created by extending the integers with all possible results of divisions of integers.
2749:
1883:
4949:
2489:' – a form of division where one repeatedly subtracts multiples of the divisor from the dividend itself.
1029:
165:
4689:
3173:(when the divisor is nonzero) results in another complex number, which is found using the conjugate of the denominator:
2274:), and there is no implication that the division must be evaluated further. A second way to show division is to use the
5430:
5394:
5096:
1673:
435:
93:
3759:
990:
126:
4475:
1221:(for example, 1 and −1 in the ring of integers). Another generalization of division to algebraic structures is the
551:
1317:
division, where numbers have no fractional part, the remainder is kept separately (or exceptionally, discarded or
5507:
4562:
4499:
823:
75:
2889:
To make the distinction with the previous case, this division, with two integers as result, is sometimes called
904:
5512:
4942:
2568:
2495:
More systematically and more efficiently, two integers can be divided with pencil and paper with the method of
1022:
320:
158:
64:
1280:
leaves a remainder of 1, as 10 is not a multiple of 3. Sometimes this remainder is added to the quotient as a
5471:
5315:
5089:
3517:
4428:
produces an error message. However, in certain higher level mathematics division by zero is possible by the
4420:
in most mathematical systems is undefined, because zero multiplied by any finite number always results in a
2330:. The ÷ symbol is used to indicate subtraction in some European countries, so its use may be misunderstood.
1998:
This is unlike the case in multiplication, which is both left-distributive and right-distributive, and thus
5486:
5481:
2905:
3817:
2799:
2716:
2417:
1371:. That is, if there are multiple divisions in a row, the order of calculation goes from left to right:
1356:, meaning that when dividing multiple times, the order of division can change the result. For example,
5015:
2796:) Usually the resulting fraction should be simplified: the result of the division of 52 by 22 is also
143:
20 / 4 = 5, illustrated here with apples. This is said verbally, "Twenty divided by four equals five."
5269:
4994:
4495:
2975:
2619:
2370:
2217:
1368:
3713:{\displaystyle {pe^{iq} \over re^{is}}={pe^{iq}e^{-is} \over re^{is}e^{-is}}={p \over r}e^{i(q-s)}.}
2054:
1122:, which is the number of times the second number is completely contained in the first number, and a
5308:
4494:
Division by zero may be defined in some circumstances, either by extending the real numbers to the
3763:
2524:
can be used to divide two numbers, by subtracting the two numbers' logarithms, then looking up the
2181:
1202:
31:
5065:
100:
5257:
4266:
2964:
2959:
treat integer division as in case 5 above, so the answer is an integer. Other languages, such as
2826:
2377:. Leibniz disliked having separate symbols for ratio and division. However, in English usage the
1234:
1105:
1094:
53:
4783:
4725:
2706:
2504:
2259:
1656:{\displaystyle {\frac {a\pm b}{c}}=(a\pm b)/c=(a/c)\pm (b/c)={\frac {a}{c}}\pm {\frac {b}{c}}.}
5476:
4965:
4847:
4207:
4196:
3829:
2988:
can sometimes be used to quickly determine whether one integer divides exactly into another.
2699:
2695:
2679:
2288:
2208:
2160:
548:
151:
5252:
4243:
4211:
3813:
3547:
Division for complex numbers expressed in polar form is simpler than the definition above:
3000:
is another rational number when the divisor is not 0. The division of two rational numbers
2956:
2144:
4206:"Division" in the sense of "cancellation" can be done in any magma by an element with the
3490:
8:
5036:
4465:
4017:
3755:
2896:
2486:
2391:
2378:
2343:
2115:
2102:
1206:
1190:
4436:. In these algebras, the meaning of division is different from traditional definitions.
3778:
One can define a division operation for matrices. The usual way to do this is to define
5466:
5345:
5215:
5200:
4830:
4705:
4239:
3767:
2891:
2593:
2560:
2553:
2552:
compute division either by methods similar to long division, or by faster methods; see
2474:
2016:
1527:
1218:
1214:
1109:
4892:
3137:
results in another real number (when the divisor is nonzero). It is defined such that
1272:
Unlike the other basic operations, when dividing natural numbers there is sometimes a
5230:
4853:
4809:
4666:
4663:
4642:
4605:
4580:
4470:
2985:
1736:
4744:
4608:
4445:
2579:. This approach is often associated with the faster methods in computer arithmetic.
5385:
5274:
5220:
4872:
4411:
4259:
4255:
4242:
is finite and every nonzero element is cancellative, then by an application of the
4080:
2979:
2952:
2688:
2521:
2508:
2461:. The history of this notation is not entirely clear because it evolved over time.
2334:
2156:
1999:
1194:
1183:
4631:
Prime
Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics
5450:
5264:
5225:
4899:
4879:
4751:
4732:
4712:
4693:
4460:
4455:
4450:
4219:
4084:
2997:
2710:
2152:
1873:{\displaystyle {\frac {a}{b+c}}=a/(b+c)\;\neq \;(a/b)+(a/c)={\frac {ac+ab}{bc}}.}
1322:
1281:
1241:
means the number of 5s that must be added to get 20. In terms of partition,
1198:
1130:
1113:
2282:
though the term has additional meanings), common in arithmetic, in this manner:
5360:
5355:
5331:
5031:
5026:
4421:
4278:
3799:
3170:
3122:
2939:
2663:
2496:
1667:
1269:. In the example, 20 is the dividend, 5 is the divisor, and 4 is the quotient.
1222:
1090:
1065:
739:
460:
4583:
4502:
or when occurring as limit of divisions by numbers tending to 0. For example:
979:{\displaystyle \scriptstyle \log _{\text{base}}({\text{anti-logarithm}})\,=\,}
5501:
5289:
5210:
5061:
4924:
4767:
4686:
4638:
4634:
4311:
3837:
2525:
2500:
2470:
2458:
2275:
2011:
1210:
1138:
3121:
may be 0. This definition ensures that division is the inverse operation of
3108:{\displaystyle {p/q \over r/s}={p \over q}\times {s \over r}={ps \over qr}.}
5240:
5235:
5205:
4433:
2743:
2564:
2037:
4087:
with binary operation ∗ (which could nominally be termed multiplication),
2262:. A fraction is a division expression where both dividend and divisor are
1516:{\displaystyle a/b/c=(a/b)/c=a/(b\times c)\;\neq \;a/(b/c)=(a\times c)/b.}
5435:
5399:
5010:
5005:
4263:
4254:(in the technical sense) have a division operation, refer to the page on
3833:
3134:
2323:
2315:
1353:
1329:
1134:
1061:
893:{\displaystyle \scriptstyle {\sqrt{\scriptstyle {\text{radicand}}}}\,=\,}
345:
4934:
4396:{\displaystyle {\left({\frac {f}{g}}\right)}'={\frac {f'g-fg'}{g^{2}}}.}
5414:
5284:
5279:
5081:
5040:
4425:
4307:
4285:
4270:
4215:
4200:
3751:
2545:
2532:
2319:
2168:
1053:
2171:, allows the operands to be written in the reverse order by using the
5445:
5370:
5365:
4671:
4613:
4588:
4429:
2840:
2172:
1273:
1225:, in which the result of "division" is a group rather than a number.
929:
4016:
do not exist, division can also be defined as multiplication by the
2211:(fraction slash), but elevates the dividend and lowers the divisor:
42:
5409:
5375:
5350:
5147:
5112:
4989:
4984:
4808:. Brooks/Cole, Cengage Learning (Charles Van Wagner). p. 126.
4292:
2971:
2834:
2549:
1318:
1118:
1082:
1057:
848:
671:
188:
5300:
4915:
5247:
5143:
5126:
2263:
2207:
A typographical variation halfway between these two forms uses a
1314:
1217:
the elements by which division is always possible are called the
1098:
4806:
Mathematics for
Teachers: An Interactive Approach for Grades K–8
2575:
may be computed as the product by the multiplicative inverse of
4558:
3873:
does need to exist. To avoid confusion, division as defined by
2960:
2515:
2279:
2164:
2021:
1141:, division is the inverse operation to multiplication, that is
598:
2143:
This is the usual way of specifying division in most computer
2028:
Division is often shown in algebra and science by placing the
139:
5116:
4718:
3758:. Then, as in the case of integers, one has a remainder. See
2382:
2148:
2097:". A way to express division all on one line is to write the
4417:
1321:). When the remainder is kept as a fraction, it leads to a
757:
720:
595:
478:
363:
206:
4661:
4603:
2882:{\displaystyle {\tfrac {26}{11}}=2{\mbox{ remainder }}4.}
2337:-speaking countries, a colon is used to denote division:
2314:
This form is infrequent except in elementary arithmetic.
4791:
The
Unicode® Standard: Version 10.0 – Core Specification
4250:
by any nonzero element is possible. To learn about when
2687:
Say that 26 cannot be divided by 11; division becomes a
1276:
that will not go evenly into the dividend; for example,
4737:
4246:, every nonzero element of the ring is invertible, and
2825:. This simplification may be done by factoring out the
2147:, since it can easily be typed as a simple sequence of
2910:
2870:
2852:
2804:
2772:
2754:
2721:
2188:
1988:{\displaystyle {\frac {12}{2}}+{\frac {12}{4}}=6+3=9.}
994:
945:
908:
870:
867:
827:
781:
763:
760:
755:
709:
698:
687:
684:
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361:
324:
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233:
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204:
4793:. Unicode Consortium. June 2017. p. 280, Obelus.
4320:
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323:
203:
4706:
http://www.mathwords.com/a/associative_operation.htm
2974:
may be toward zero (so called T-division) or toward
2789:{\displaystyle {\tfrac {26}{11}}=2{\tfrac {4}{11}}.}
1930:{\displaystyle {\frac {12}{2+4}}={\frac {12}{6}}=2,}
1328:
Unlike multiplication and addition, division is not
1233:
The simplest way of viewing division is in terms of
4828:
4238:by left or right cancellation, respectively. If a
2381:is restricted to expressing the related concept of
2322:. The obelus was introduced by Swiss mathematician
2151:characters. (It is also the only notation used for
67:. Unsourced material may be challenged and removed.
4395:
3712:
3508:
3477:
3107:
2929:
2881:
2817:
2788:
2734:
2503:, if the divisor is larger. If the dividend has a
2441:
2406:
2358:
2303:
2247:
2196:
2132:
2070:
1987:
1929:
1872:
1724:
1655:
1530:over addition and subtraction, in the sense that
1515:
1003:
978:
917:
892:
836:
811:
726:
661:
564:
538:
448:
423:
333:
308:
4578:
4310:of the quotient of two functions is given by the
2582:
2230:
1725:{\displaystyle (a+b)\times c=a\times c+b\times c}
5499:
4873:http://mathworld.wolfram.com/DivisionbyZero.html
4803:
4222:, where not every element need have an inverse,
449:{\displaystyle \scriptstyle {\text{difference}}}
4930:Chinese Short Division Techniques on a Suan Pan
4765:
4424:of zero. Entry of such an expression into most
4230:can still be performed on elements of the form
3898:With left and right division defined this way,
2943:applied to case 2 or 3. It is sometimes called
2713:, so the result of the division of 26 by 11 is
2388:Since the 19th century, US textbooks have used
1004:{\displaystyle \scriptstyle {\text{logarithm}}}
30:"Divided" redirects here. For other uses, see
5316:
5097:
4950:
1030:
565:{\displaystyle \scriptstyle {\text{product}}}
166:
4127:, if this exists and is unique. Similarly,
3487:This process of multiplying and dividing by
2902:Give the integer quotient as the answer, so
2258:Any of these forms can be used to display a
3117:All four quantities are integers, and only
2978:(F-division); rarer styles can occur – see
2081:which can also be read out loud as "divide
1089:At an elementary level the division of two
837:{\displaystyle \scriptstyle {\text{power}}}
5323:
5309:
5104:
5090:
4957:
4943:
4687:http://www.mathwords.com/c/commutative.htm
4628:
3823:
3750:One can define the division operation for
1797:
1793:
1458:
1454:
1037:
1023:
918:{\displaystyle \scriptstyle {\text{root}}}
173:
159:
4964:
4761:
4759:
3806:, but it is far more common to write out
1189:Both forms of division appear in various
974:
970:
888:
884:
807:
803:
657:
653:
534:
530:
515:
511:
494:
490:
419:
415:
400:
396:
379:
375:
334:{\displaystyle \scriptstyle {\text{sum}}}
304:
300:
285:
281:
264:
260:
243:
239:
222:
218:
127:Learn how and when to remove this message
5111:
3516:is called 'realisation' or (by analogy)
2698:. This is the approach usually taken in
2015:
1367:Division is traditionally considered as
138:
4655:
4574:
4572:
4109:) is typically defined as the solution
1052:is one of the four basic operations of
14:
5500:
4845:
4756:
2991:
2036:with a horizontal line, also called a
1068:. What is being divided is called the
5304:
5085:
4938:
4822:
4662:
4604:
4579:
4552:
3762:, and, for hand-written computation,
3164:
2587:
4849:Advanced Abacus: Theory and Practice
4784:"6. Writing Systems and Punctuation"
4569:
3816:can also be defined in terms of the
2930:{\displaystyle {\tfrac {26}{11}}=2.}
1352:. Division is also not, in general,
65:adding citations to reliable sources
36:
5330:
4769:A History of Mathematical Notations
4405:
4074:
2514:Division can be calculated with an
24:
4597:
4218:algebras, and quasigroups. In an
3812:explicitly to avoid confusion. An
3128:
2531:Division can be calculated with a
1237:: from the quotition perspective,
25:
5524:
4909:
3760:Euclidean division of polynomials
3745:
2895:, because it is the basis of the
2818:{\displaystyle {\tfrac {26}{11}}}
2735:{\displaystyle {\tfrac {26}{11}}}
2479:
2442:{\displaystyle b{\overline {)a}}}
5156:
5125:
4476:Rule of division (combinatorics)
3999:
2694:Give an approximate answer as a
2571:. In these cases, a division by
2563:(modulo a prime number) and for
2369:This notation was introduced by
41:
4885:
4866:
4839:
4797:
4776:
4500:projectively extended real line
3861:. For this to be well defined,
2248:{\displaystyle {}^{a}\!/{}_{b}}
1228:
1080:, and the result is called the
52:needs additional citations for
4846:Kojima, Takashi (2012-07-09).
4726:Order of arithmetic operations
4699:
4680:
4622:
4546:
4488:
4446:400AD Sunzi division algorithm
3909:is in general not the same as
3773:
3702:
3690:
3340:
3322:
3289:
3274:
3271:
3256:
3251:
3236:
3233:
3218:
2832:Give the answer as an integer
2673:
2583:Division in different contexts
2539:
2499:, if the divisor is small, or
2427:
2398:
2109:(or denominator), as follows:
2071:{\displaystyle {\frac {a}{b}}}
2040:, between them. For example, "
1832:
1818:
1812:
1798:
1790:
1778:
1689:
1677:
1621:
1607:
1601:
1587:
1573:
1561:
1499:
1487:
1481:
1467:
1451:
1439:
1417:
1403:
967:
959:
13:
1:
5472:Conway chained arrow notation
4921:Division on a Japanese abacus
4832:History Of Mathematics Vol II
4539:
4262:can be used to show that any
4187:exist and are unique for all
4063:denote the pseudoinverses of
2457:, especially when discussing
2197:{\displaystyle b\backslash a}
4829:Smith, David Eugene (1925).
4563:Alexander Thom & Company
2955:requires special care. Some
2634:, the remainder, such that
2464:
2434:
7:
4925:Abacus: Mystery of the Bead
4439:
4301:
4273:to either the real numbers
2996:The result of dividing two
2005:
1056:. The other operations are
76:"Division" mathematics
18:Truncating integer division
10:
5529:
5424:Inverse for right argument
4902:Retrieved October 23, 2018
4882:Retrieved October 23, 2018
4715:Retrieved October 23, 2018
4696:Retrieved October 23, 2018
4416:Division of any number by
4409:
4226:by a cancellative element
3722:Again all four quantities
2591:
2468:
2175:as the division operator:
2009:
1074:, which is divided by the
29:
5482:Knuth's up-arrow notation
5459:
5423:
5384:
5338:
5196:
5165:
5154:
5132:
5123:
4972:
4804:Thomas Sonnabend (2010).
4629:Derbyshire, John (2004).
4496:extended real number line
3950:. However, it holds that
2567:, nonzero numbers have a
2371:Gottfried Wilhelm Leibniz
936:
928:
858:
847:
746:
738:
584:
576:
467:
459:
352:
344:
195:
187:
5487:Steinhaus–Moser notation
4766:Cajori, Florian (1929).
4481:
3867:need not exist, however
3836:, one can also define a
3764:polynomial long division
1313:, but in the context of
1095:possible interpretations
32:Divided (disambiguation)
4745:The Order of Operations
4267:normed division algebra
4004:To avoid problems when
3824:Left and right division
3754:in one variable over a
2965:computer algebra system
2951:Dividing integers in a
2827:greatest common divisor
2304:{\displaystyle a\div b}
2101:(or numerator), then a
1732:. However, division is
1342:is not always equal to
1235:quotition and partition
1106:division with remainder
5508:Division (mathematics)
4397:
3738:are real numbers, and
3714:
3536:are real numbers, and
3520:. All four quantities
3510:
3479:
3109:
2947:, and denoted by "//".
2931:
2883:
2819:
2790:
2736:
2569:multiplicative inverse
2443:
2408:
2360:
2305:
2266:(typically called the
2249:
2198:
2134:
2072:
2025:
1989:
1931:
1874:
1726:
1657:
1517:
1005:
980:
919:
894:
838:
813:
728:
663:
566:
540:
450:
425:
335:
310:
144:
5513:Elementary arithmetic
5477:Grzegorczyk hierarchy
4966:Elementary arithmetic
4893:"On Division by Zero"
4852:. Tuttle Publishing.
4772:. Open Court Pub. Co.
4724:George Mark Bergman:
4553:Blake, A. G. (1887).
4432:and algebras such as
4398:
4208:cancellation property
4197:Latin square property
3830:matrix multiplication
3715:
3511:
3480:
3110:
2957:programming languages
2932:
2884:
2872: remainder
2820:
2791:
2737:
2705:Give the answer as a
2700:numerical computation
2696:floating-point number
2507:part (expressed as a
2444:
2409:
2361:
2306:
2250:
2199:
2161:mathematical software
2145:programming languages
2135:
2073:
2048:" can be written as:
2020:Plus and minuses. An
2019:
2010:Further information:
1990:
1932:
1875:
1727:
1666:This is the same for
1658:
1518:
1006:
981:
920:
895:
839:
814:
729:
664:
567:
541:
451:
426:
336:
311:
152:Arithmetic operations
142:
4318:
4244:pigeonhole principle
4210:. Examples include
3887:is sometimes called
3814:elementwise division
3551:
3509:{\displaystyle r-is}
3491:
3177:
3020:
2906:
2848:
2800:
2750:
2717:
2656:|, where |
2418:
2392:
2344:
2289:
2218:
2182:
2116:
2055:
1941:
1884:
1880: For example
1746:
1674:
1537:
1378:
1191:algebraic structures
1137:. In these enlarged
1116:provides an integer
991:
942:
905:
864:
824:
752:
675:
590:
552:
473:
436:
358:
321:
201:
61:improve this article
27:Arithmetic operation
5451:Super-logarithm (4)
5410:Root extraction (3)
4916:Planetmath division
4835:. Ginn And Company.
4466:Order of operations
3544:may not both be 0.
3016:can be computed as
2992:Of rational numbers
2897:Euclidean algorithm
2662:| denotes the
2407:{\displaystyle b)a}
2359:{\displaystyle a:b}
2133:{\displaystyle a/b}
5467:Ackermann function
5361:Exponentiation (3)
5356:Multiplication (2)
5133:Division and ratio
4978:
4898:2019-08-17 at the
4891:Jesper Carlström.
4878:2018-10-23 at the
4750:2017-06-08 at the
4731:2017-03-05 at the
4711:2018-10-28 at the
4692:2018-10-28 at the
4667:"Integer Division"
4664:Weisstein, Eric W.
4609:"Division by Zero"
4606:Weisstein, Eric W.
4581:Weisstein, Eric W.
4393:
4149:) is the solution
3842:backslash-division
3768:synthetic division
3710:
3506:
3475:
3165:Of complex numbers
3105:
2986:Divisibility rules
2927:
2919:
2892:Euclidean division
2879:
2874:
2861:
2815:
2813:
2786:
2781:
2763:
2732:
2730:
2594:Euclidean division
2588:Euclidean division
2561:modular arithmetic
2554:Division algorithm
2475:Division algorithm
2439:
2404:
2356:
2301:
2278:(÷, also known as
2245:
2194:
2130:
2068:
2026:
1985:
1927:
1870:
1722:
1653:
1528:right-distributive
1513:
1110:Euclidean division
1001:
1000:
976:
975:
915:
914:
890:
889:
876:
834:
833:
809:
808:
797:
794:
776:
724:
723:
718:
715:
704:
693:
659:
658:
647:
644:
641:
634:
620:
617:
610:
562:
561:
536:
535:
524:
521:
500:
446:
445:
421:
420:
409:
406:
385:
331:
330:
306:
305:
294:
291:
270:
249:
228:
145:
5495:
5494:
5388:for left argument
5298:
5297:
5079:
5078:
5074:
5073:
4859:978-1-4629-0365-8
4815:978-0-495-56166-8
4743:Education Place:
4648:978-0-452-28525-5
4471:Repeating decimal
4388:
4335:
4256:division algebras
3895:in this context.
3680:
3667:
3592:
3470:
3418:
3369:
3293:
3210:
3100:
3077:
3064:
3051:
2982:for the details.
2918:
2873:
2860:
2812:
2780:
2762:
2729:
2678:Integers are not
2437:
2066:
1965:
1952:
1916:
1903:
1865:
1765:
1737:left-distributive
1648:
1635:
1556:
1195:Euclidean domains
1182:, then this is a
1047:
1046:
1014:
1013:
998:
965:
953:
912:
882:
880:
874:
831:
791:
786:
773:
768:
713:
702:
691:
642:
639:
632:
618:
615:
608:
559:
519:
509:
498:
488:
443:
404:
394:
383:
373:
328:
289:
279:
268:
258:
247:
237:
226:
216:
137:
136:
129:
111:
16:(Redirected from
5520:
5460:Related articles
5325:
5318:
5311:
5302:
5301:
5275:Musical interval
5188:
5187:
5185:
5184:
5181:
5178:
5160:
5159:
5129:
5106:
5099:
5092:
5083:
5082:
5054:
5029:
5008:
4987:
4975:
4974:
4959:
4952:
4945:
4936:
4935:
4903:
4889:
4883:
4870:
4864:
4863:
4843:
4837:
4836:
4826:
4820:
4819:
4801:
4795:
4794:
4788:
4780:
4774:
4773:
4763:
4754:
4741:
4735:
4722:
4716:
4703:
4697:
4684:
4678:
4677:
4676:
4659:
4653:
4652:
4626:
4620:
4619:
4618:
4601:
4595:
4594:
4593:
4576:
4567:
4566:
4550:
4533:
4532:
4530:
4528:
4527:
4522:
4519:
4492:
4412:Division by zero
4406:Division by zero
4402:
4400:
4399:
4394:
4389:
4387:
4386:
4377:
4376:
4359:
4350:
4345:
4341:
4340:
4336:
4328:
4260:Bott periodicity
4258:. In particular
4186:
4176:
4166:
4153:to the equation
4148:
4126:
4113:to the equation
4108:
4081:abstract algebra
4075:Abstract algebra
4062:
4056:
4050:
4033:
4015:
4009:
3995:
3971:
3949:
3934:
3923:
3908:
3886:
3872:
3866:
3860:
3818:Hadamard product
3811:
3797:
3791:
3719:
3717:
3716:
3711:
3706:
3705:
3681:
3673:
3668:
3666:
3665:
3664:
3649:
3648:
3632:
3631:
3630:
3615:
3614:
3598:
3593:
3591:
3590:
3589:
3573:
3572:
3571:
3555:
3515:
3513:
3512:
3507:
3484:
3482:
3481:
3476:
3471:
3469:
3468:
3467:
3455:
3454:
3444:
3427:
3419:
3417:
3416:
3415:
3403:
3402:
3392:
3375:
3370:
3368:
3367:
3366:
3354:
3353:
3343:
3299:
3294:
3292:
3254:
3216:
3211:
3209:
3195:
3181:
3133:Division of two
3114:
3112:
3111:
3106:
3101:
3099:
3091:
3083:
3078:
3070:
3065:
3057:
3052:
3050:
3046:
3037:
3033:
3024:
2998:rational numbers
2980:modulo operation
2953:computer program
2945:integer division
2936:
2934:
2933:
2928:
2920:
2911:
2888:
2886:
2885:
2880:
2875:
2871:
2862:
2853:
2824:
2822:
2821:
2816:
2814:
2805:
2795:
2793:
2792:
2787:
2782:
2773:
2764:
2755:
2741:
2739:
2738:
2733:
2731:
2722:
2689:partial function
2661:
2655:
2578:
2574:
2522:Logarithm tables
2509:decimal fraction
2448:
2446:
2445:
2440:
2438:
2433:
2425:
2413:
2411:
2410:
2405:
2365:
2363:
2362:
2357:
2328:Teutsche Algebra
2310:
2308:
2307:
2302:
2254:
2252:
2251:
2246:
2244:
2243:
2238:
2235:
2229:
2228:
2223:
2203:
2201:
2200:
2195:
2157:abstract algebra
2153:quotient objects
2139:
2137:
2136:
2131:
2126:
2077:
2075:
2074:
2069:
2067:
2059:
1994:
1992:
1991:
1986:
1966:
1958:
1953:
1945:
1936:
1934:
1933:
1928:
1917:
1909:
1904:
1902:
1888:
1879:
1877:
1876:
1871:
1866:
1864:
1856:
1839:
1828:
1808:
1777:
1766:
1764:
1750:
1731:
1729:
1728:
1723:
1662:
1660:
1659:
1654:
1649:
1641:
1636:
1628:
1617:
1597:
1580:
1557:
1552:
1541:
1522:
1520:
1519:
1514:
1506:
1477:
1466:
1438:
1424:
1413:
1396:
1388:
1369:left-associative
1363:
1362:24 / (6 / 2) = 8
1359:
1358:(24 / 6) / 2 = 2
1351:
1341:
1312:
1308:
1307:
1305:
1304:
1301:
1298:
1294:
1287:
1279:
1268:
1266:
1264:
1263:
1260:
1257:
1249:
1244:
1240:
1199:polynomial rings
1184:division by zero
1181:
1175:is not zero. If
1174:
1168:
1154:
1131:rational numbers
1093:is, among other
1039:
1032:
1025:
1010:
1008:
1007:
1002:
999:
996:
985:
983:
982:
977:
966:
963:
955:
954:
951:
924:
922:
921:
916:
913:
910:
899:
897:
896:
891:
883:
881:
878:
875:
872:
869:
843:
841:
840:
835:
832:
829:
818:
816:
815:
810:
802:
798:
793:
792:
789:
787:
784:
775:
774:
771:
769:
766:
733:
731:
730:
725:
722:
719:
714:
711:
703:
700:
692:
689:
668:
666:
665:
660:
652:
648:
643:
640:
637:
633:
630:
627:
619:
616:
613:
609:
606:
603:
571:
569:
568:
563:
560:
557:
545:
543:
542:
537:
529:
525:
520:
517:
510:
507:
499:
496:
489:
486:
455:
453:
452:
447:
444:
441:
430:
428:
427:
422:
414:
410:
405:
402:
395:
392:
384:
381:
374:
371:
340:
338:
337:
332:
329:
326:
315:
313:
312:
307:
299:
295:
290:
287:
280:
277:
269:
266:
259:
256:
248:
245:
238:
235:
227:
224:
217:
214:
185:
184:
175:
168:
161:
154:
147:
146:
132:
125:
121:
118:
112:
110:
69:
45:
37:
21:
5528:
5527:
5523:
5522:
5521:
5519:
5518:
5517:
5498:
5497:
5496:
5491:
5455:
5436:Subtraction (1)
5431:Predecessor (0)
5419:
5400:Subtraction (1)
5395:Predecessor (0)
5380:
5334:
5332:Hyperoperations
5329:
5299:
5294:
5265:Just intonation
5192:
5182:
5179:
5176:
5175:
5173:
5172:
5161:
5157:
5152:
5130:
5119:
5110:
5080:
5075:
5070:
5059:
5055:
5050:
5045:
5034:
5030:
5025:
5020:
5013:
5009:
5004:
4999:
4992:
4988:
4983:
4968:
4963:
4912:
4907:
4906:
4900:Wayback Machine
4890:
4886:
4880:Wayback Machine
4871:
4867:
4860:
4844:
4840:
4827:
4823:
4816:
4802:
4798:
4786:
4782:
4781:
4777:
4764:
4757:
4752:Wayback Machine
4742:
4738:
4733:Wayback Machine
4723:
4719:
4713:Wayback Machine
4704:
4700:
4694:Wayback Machine
4685:
4681:
4660:
4656:
4649:
4627:
4623:
4602:
4598:
4577:
4570:
4559:Dublin, Ireland
4551:
4547:
4542:
4537:
4536:
4523:
4520:
4514:
4513:
4511:
4510:
4503:
4493:
4489:
4484:
4461:Inverse element
4456:Galley division
4451:Division by two
4442:
4414:
4408:
4382:
4378:
4369:
4352:
4351:
4349:
4327:
4323:
4322:
4321:
4319:
4316:
4315:
4304:
4279:complex numbers
4220:integral domain
4178:
4168:
4154:
4140:
4114:
4100:
4077:
4058:
4052:
4035:
4021:
4011:
4005:
4002:
3973:
3951:
3936:
3925:
3910:
3899:
3874:
3868:
3862:
3845:
3826:
3807:
3793:
3779:
3776:
3748:
3686:
3682:
3672:
3654:
3650:
3641:
3637:
3633:
3620:
3616:
3607:
3603:
3599:
3597:
3582:
3578:
3574:
3564:
3560:
3556:
3554:
3552:
3549:
3548:
3518:rationalisation
3492:
3489:
3488:
3463:
3459:
3450:
3446:
3445:
3428:
3426:
3411:
3407:
3398:
3394:
3393:
3376:
3374:
3362:
3358:
3349:
3345:
3344:
3300:
3298:
3255:
3217:
3215:
3196:
3182:
3180:
3178:
3175:
3174:
3171:complex numbers
3167:
3149:if and only if
3131:
3129:Of real numbers
3092:
3084:
3082:
3069:
3056:
3042:
3038:
3029:
3025:
3023:
3021:
3018:
3017:
2994:
2909:
2907:
2904:
2903:
2869:
2851:
2849:
2846:
2845:
2803:
2801:
2798:
2797:
2771:
2753:
2751:
2748:
2747:
2720:
2718:
2715:
2714:
2711:rational number
2709:representing a
2676:
2657:
2651:
2618:≠ 0, there are
2596:
2590:
2585:
2576:
2572:
2542:
2528:of the result.
2482:
2477:
2469:Main articles:
2467:
2426:
2424:
2419:
2416:
2415:
2393:
2390:
2389:
2375:Acta eruditorum
2345:
2342:
2341:
2290:
2287:
2286:
2239:
2237:
2236:
2231:
2224:
2222:
2221:
2219:
2216:
2215:
2183:
2180:
2179:
2122:
2117:
2114:
2113:
2058:
2056:
2053:
2052:
2014:
2008:
1957:
1944:
1942:
1939:
1938:
1908:
1892:
1887:
1885:
1882:
1881:
1857:
1840:
1838:
1824:
1804:
1773:
1754:
1749:
1747:
1744:
1743:
1675:
1672:
1671:
1640:
1627:
1613:
1593:
1576:
1542:
1540:
1538:
1535:
1534:
1502:
1473:
1462:
1434:
1420:
1409:
1392:
1384:
1379:
1376:
1375:
1361:
1357:
1343:
1333:
1332:, meaning that
1323:rational number
1310:
1302:
1299:
1296:
1295:
1292:
1290:
1289:
1285:
1282:fractional part
1277:
1261:
1258:
1255:
1254:
1252:
1251:
1247:
1242:
1238:
1231:
1176:
1170:
1156:
1142:
1114:natural numbers
1091:natural numbers
1043:
995:
992:
989:
988:
962:
950:
946:
943:
940:
939:
909:
906:
903:
902:
877:
871:
868:
865:
862:
861:
828:
825:
822:
821:
796:
795:
788:
783:
782:
778:
777:
770:
765:
764:
759:
756:
753:
750:
749:
717:
716:
710:
706:
705:
699:
695:
694:
688:
683:
679:
676:
673:
672:
646:
645:
636:
629:
626:
622:
621:
612:
605:
602:
597:
594:
591:
588:
587:
556:
553:
550:
549:
523:
522:
516:
506:
502:
501:
495:
485:
480:
477:
474:
471:
470:
440:
437:
434:
433:
408:
407:
401:
391:
387:
386:
380:
370:
365:
362:
359:
356:
355:
325:
322:
319:
318:
293:
292:
286:
276:
272:
271:
265:
255:
251:
250:
244:
234:
230:
229:
223:
213:
208:
205:
202:
199:
198:
179:
150:
133:
122:
116:
113:
70:
68:
58:
46:
35:
28:
23:
22:
15:
12:
11:
5:
5526:
5516:
5515:
5510:
5493:
5492:
5490:
5489:
5484:
5479:
5474:
5469:
5463:
5461:
5457:
5456:
5454:
5453:
5448:
5443:
5438:
5433:
5427:
5425:
5421:
5420:
5418:
5417:
5415:Super-root (4)
5412:
5407:
5402:
5397:
5391:
5389:
5382:
5381:
5379:
5378:
5373:
5368:
5363:
5358:
5353:
5348:
5342:
5340:
5336:
5335:
5328:
5327:
5320:
5313:
5305:
5296:
5295:
5293:
5292:
5287:
5282:
5277:
5272:
5267:
5262:
5261:
5260:
5250:
5245:
5244:
5243:
5233:
5228:
5223:
5218:
5213:
5208:
5203:
5197:
5194:
5193:
5191:
5190:
5169:
5167:
5163:
5162:
5155:
5153:
5151:
5150:
5136:
5134:
5131:
5124:
5121:
5120:
5109:
5108:
5101:
5094:
5086:
5077:
5076:
5072:
5071:
5048:
5046:
5032:Multiplication
5023:
5021:
5002:
5000:
4981:
4979:
4973:
4970:
4969:
4962:
4961:
4954:
4947:
4939:
4933:
4932:
4927:
4923:selected from
4918:
4911:
4910:External links
4908:
4905:
4904:
4884:
4865:
4858:
4838:
4821:
4814:
4796:
4775:
4755:
4736:
4717:
4698:
4679:
4654:
4647:
4621:
4596:
4568:
4544:
4543:
4541:
4538:
4535:
4534:
4505:
4486:
4485:
4483:
4480:
4479:
4478:
4473:
4468:
4463:
4458:
4453:
4448:
4441:
4438:
4410:Main article:
4407:
4404:
4392:
4385:
4381:
4375:
4372:
4368:
4365:
4362:
4358:
4355:
4348:
4344:
4339:
4334:
4331:
4326:
4303:
4300:
4129:right division
4076:
4073:
4001:
3998:
3893:slash-division
3889:right division
3825:
3822:
3775:
3772:
3747:
3746:Of polynomials
3744:
3742:may not be 0.
3709:
3704:
3701:
3698:
3695:
3692:
3689:
3685:
3679:
3676:
3671:
3663:
3660:
3657:
3653:
3647:
3644:
3640:
3636:
3629:
3626:
3623:
3619:
3613:
3610:
3606:
3602:
3596:
3588:
3585:
3581:
3577:
3570:
3567:
3563:
3559:
3505:
3502:
3499:
3496:
3474:
3466:
3462:
3458:
3453:
3449:
3443:
3440:
3437:
3434:
3431:
3425:
3422:
3414:
3410:
3406:
3401:
3397:
3391:
3388:
3385:
3382:
3379:
3373:
3365:
3361:
3357:
3352:
3348:
3342:
3339:
3336:
3333:
3330:
3327:
3324:
3321:
3318:
3315:
3312:
3309:
3306:
3303:
3297:
3291:
3288:
3285:
3282:
3279:
3276:
3273:
3270:
3267:
3264:
3261:
3258:
3253:
3250:
3247:
3244:
3241:
3238:
3235:
3232:
3229:
3226:
3223:
3220:
3214:
3208:
3205:
3202:
3199:
3194:
3191:
3188:
3185:
3166:
3163:
3130:
3127:
3123:multiplication
3104:
3098:
3095:
3090:
3087:
3081:
3076:
3073:
3068:
3063:
3060:
3055:
3049:
3045:
3041:
3036:
3032:
3028:
2993:
2990:
2949:
2948:
2940:floor function
2926:
2923:
2917:
2914:
2900:
2878:
2868:
2865:
2859:
2856:
2830:
2811:
2808:
2785:
2779:
2776:
2770:
2767:
2761:
2758:
2728:
2725:
2703:
2692:
2675:
2672:
2664:absolute value
2592:Main article:
2589:
2586:
2584:
2581:
2541:
2538:
2497:short division
2481:
2480:Manual methods
2478:
2466:
2463:
2436:
2432:
2429:
2423:
2403:
2400:
2397:
2367:
2366:
2355:
2352:
2349:
2312:
2311:
2300:
2297:
2294:
2256:
2255:
2242:
2234:
2227:
2205:
2204:
2193:
2190:
2187:
2141:
2140:
2129:
2125:
2121:
2079:
2078:
2065:
2062:
2007:
2004:
1996:
1995:
1984:
1981:
1978:
1975:
1972:
1969:
1964:
1961:
1956:
1951:
1948:
1926:
1923:
1920:
1915:
1912:
1907:
1901:
1898:
1895:
1891:
1869:
1863:
1860:
1855:
1852:
1849:
1846:
1843:
1837:
1834:
1831:
1827:
1823:
1820:
1817:
1814:
1811:
1807:
1803:
1800:
1796:
1792:
1789:
1786:
1783:
1780:
1776:
1772:
1769:
1763:
1760:
1757:
1753:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1700:
1697:
1694:
1691:
1688:
1685:
1682:
1679:
1668:multiplication
1664:
1663:
1652:
1647:
1644:
1639:
1634:
1631:
1626:
1623:
1620:
1616:
1612:
1609:
1606:
1603:
1600:
1596:
1592:
1589:
1586:
1583:
1579:
1575:
1572:
1569:
1566:
1563:
1560:
1555:
1551:
1548:
1545:
1524:
1523:
1512:
1509:
1505:
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1480:
1476:
1472:
1469:
1465:
1461:
1457:
1453:
1450:
1447:
1444:
1441:
1437:
1433:
1430:
1427:
1423:
1419:
1416:
1412:
1408:
1405:
1402:
1399:
1395:
1391:
1387:
1383:
1230:
1227:
1223:quotient group
1211:division rings
1139:number systems
1066:multiplication
1045:
1044:
1042:
1041:
1034:
1027:
1019:
1016:
1015:
1012:
1011:
986:
973:
969:
964:anti-logarithm
961:
958:
949:
937:
934:
933:
926:
925:
900:
887:
859:
856:
855:
845:
844:
819:
806:
801:
780:
779:
762:
761:
758:
747:
744:
743:
740:Exponentiation
736:
735:
721:
708:
707:
697:
696:
686:
685:
682:
669:
656:
651:
624:
623:
600:
599:
596:
585:
582:
581:
574:
573:
546:
533:
528:
514:
504:
503:
493:
483:
482:
479:
468:
465:
464:
461:Multiplication
457:
456:
431:
418:
413:
399:
389:
388:
378:
368:
367:
364:
353:
350:
349:
342:
341:
316:
303:
298:
284:
274:
273:
263:
253:
252:
242:
232:
231:
221:
211:
210:
207:
196:
193:
192:
181:
180:
178:
177:
170:
163:
155:
135:
134:
49:
47:
40:
26:
9:
6:
4:
3:
2:
5525:
5514:
5511:
5509:
5506:
5505:
5503:
5488:
5485:
5483:
5480:
5478:
5475:
5473:
5470:
5468:
5465:
5464:
5462:
5458:
5452:
5449:
5447:
5446:Logarithm (3)
5444:
5442:
5439:
5437:
5434:
5432:
5429:
5428:
5426:
5422:
5416:
5413:
5411:
5408:
5406:
5403:
5401:
5398:
5396:
5393:
5392:
5390:
5387:
5383:
5377:
5374:
5372:
5371:Pentation (5)
5369:
5367:
5366:Tetration (4)
5364:
5362:
5359:
5357:
5354:
5352:
5349:
5347:
5346:Successor (0)
5344:
5343:
5341:
5337:
5333:
5326:
5321:
5319:
5314:
5312:
5307:
5306:
5303:
5291:
5288:
5286:
5283:
5281:
5278:
5276:
5273:
5271:
5268:
5266:
5263:
5259:
5256:
5255:
5254:
5251:
5249:
5246:
5242:
5239:
5238:
5237:
5234:
5232:
5229:
5227:
5224:
5222:
5219:
5217:
5214:
5212:
5209:
5207:
5204:
5202:
5199:
5198:
5195:
5171:
5170:
5168:
5164:
5149:
5145:
5141:
5138:
5137:
5135:
5128:
5122:
5118:
5114:
5107:
5102:
5100:
5095:
5093:
5088:
5087:
5084:
5069:
5067:
5063:
5058:
5053:
5047:
5044:
5042:
5038:
5033:
5028:
5022:
5019:
5017:
5012:
5007:
5001:
4998:
4996:
4991:
4986:
4980:
4977:
4976:
4971:
4967:
4960:
4955:
4953:
4948:
4946:
4941:
4940:
4937:
4931:
4928:
4926:
4922:
4919:
4917:
4914:
4913:
4901:
4897:
4894:
4888:
4881:
4877:
4874:
4869:
4861:
4855:
4851:
4850:
4842:
4834:
4833:
4825:
4817:
4811:
4807:
4800:
4792:
4785:
4779:
4771:
4770:
4762:
4760:
4753:
4749:
4746:
4740:
4734:
4730:
4727:
4721:
4714:
4710:
4707:
4702:
4695:
4691:
4688:
4683:
4674:
4673:
4668:
4665:
4658:
4650:
4644:
4640:
4639:Penguin Books
4636:
4635:New York City
4632:
4625:
4616:
4615:
4610:
4607:
4600:
4591:
4590:
4585:
4582:
4575:
4573:
4564:
4560:
4556:
4549:
4545:
4526:
4518:
4508:
4501:
4497:
4491:
4487:
4477:
4474:
4472:
4469:
4467:
4464:
4462:
4459:
4457:
4454:
4452:
4449:
4447:
4444:
4443:
4437:
4435:
4431:
4427:
4423:
4419:
4413:
4403:
4390:
4383:
4379:
4373:
4370:
4366:
4363:
4360:
4356:
4353:
4346:
4342:
4337:
4332:
4329:
4324:
4313:
4312:quotient rule
4309:
4299:
4297:
4294:
4290:
4287:
4283:
4280:
4276:
4272:
4268:
4265:
4261:
4257:
4253:
4249:
4245:
4241:
4237:
4233:
4229:
4225:
4221:
4217:
4213:
4209:
4204:
4202:
4198:
4194:
4190:
4185:
4181:
4175:
4171:
4165:
4161:
4157:
4152:
4147:
4143:
4138:
4134:
4130:
4125:
4121:
4117:
4112:
4107:
4103:
4098:
4094:
4090:
4089:left division
4086:
4082:
4072:
4070:
4066:
4061:
4055:
4049:
4046:
4042:
4038:
4032:
4028:
4024:
4019:
4018:pseudoinverse
4014:
4008:
4000:Pseudoinverse
3997:
3993:
3989:
3985:
3981:
3977:
3970:
3966:
3962:
3958:
3954:
3947:
3943:
3939:
3933:
3929:
3922:
3918:
3914:
3906:
3902:
3896:
3894:
3890:
3885:
3881:
3877:
3871:
3865:
3859:
3856:
3852:
3848:
3843:
3840:or so-called
3839:
3838:left division
3835:
3831:
3821:
3819:
3815:
3810:
3805:
3801:
3796:
3790:
3786:
3782:
3771:
3769:
3765:
3761:
3757:
3753:
3743:
3741:
3737:
3733:
3729:
3725:
3720:
3707:
3699:
3696:
3693:
3687:
3683:
3677:
3674:
3669:
3661:
3658:
3655:
3651:
3645:
3642:
3638:
3634:
3627:
3624:
3621:
3617:
3611:
3608:
3604:
3600:
3594:
3586:
3583:
3579:
3575:
3568:
3565:
3561:
3557:
3545:
3543:
3539:
3535:
3531:
3527:
3523:
3519:
3503:
3500:
3497:
3494:
3485:
3472:
3464:
3460:
3456:
3451:
3447:
3441:
3438:
3435:
3432:
3429:
3423:
3420:
3412:
3408:
3404:
3399:
3395:
3389:
3386:
3383:
3380:
3377:
3371:
3363:
3359:
3355:
3350:
3346:
3337:
3334:
3331:
3328:
3325:
3319:
3316:
3313:
3310:
3307:
3304:
3301:
3295:
3286:
3283:
3280:
3277:
3268:
3265:
3262:
3259:
3248:
3245:
3242:
3239:
3230:
3227:
3224:
3221:
3212:
3206:
3203:
3200:
3197:
3192:
3189:
3186:
3183:
3172:
3169:Dividing two
3162:
3160:
3156:
3152:
3148:
3144:
3140:
3136:
3126:
3124:
3120:
3115:
3102:
3096:
3093:
3088:
3085:
3079:
3074:
3071:
3066:
3061:
3058:
3053:
3047:
3043:
3039:
3034:
3030:
3026:
3015:
3011:
3007:
3003:
2999:
2989:
2987:
2983:
2981:
2977:
2973:
2968:
2966:
2962:
2958:
2954:
2946:
2942:
2941:
2924:
2921:
2915:
2912:
2901:
2898:
2894:
2893:
2876:
2866:
2863:
2857:
2854:
2843:
2842:
2837:
2836:
2831:
2828:
2809:
2806:
2783:
2777:
2774:
2768:
2765:
2759:
2756:
2745:
2726:
2723:
2712:
2708:
2704:
2701:
2697:
2693:
2690:
2686:
2685:
2684:
2681:
2671:
2669:
2665:
2660:
2654:
2649:
2645:
2641:
2637:
2633:
2629:
2625:
2621:
2617:
2613:
2609:
2605:
2601:
2595:
2580:
2570:
2566:
2562:
2557:
2555:
2551:
2547:
2537:
2534:
2529:
2527:
2526:antilogarithm
2523:
2519:
2517:
2512:
2510:
2506:
2502:
2501:long division
2498:
2493:
2490:
2488:
2476:
2472:
2471:Long division
2462:
2460:
2459:long division
2456:
2452:
2430:
2421:
2401:
2395:
2386:
2384:
2380:
2376:
2372:
2353:
2350:
2347:
2340:
2339:
2338:
2336:
2331:
2329:
2325:
2321:
2317:
2298:
2295:
2292:
2285:
2284:
2283:
2281:
2277:
2276:division sign
2273:
2269:
2265:
2261:
2240:
2232:
2225:
2214:
2213:
2212:
2210:
2191:
2185:
2178:
2177:
2176:
2174:
2170:
2166:
2162:
2158:
2154:
2150:
2146:
2127:
2123:
2119:
2112:
2111:
2110:
2108:
2104:
2100:
2096:
2092:
2088:
2084:
2063:
2060:
2051:
2050:
2049:
2047:
2043:
2039:
2035:
2031:
2023:
2018:
2013:
2012:Division sign
2003:
2001:
1982:
1979:
1976:
1973:
1970:
1967:
1962:
1959:
1954:
1949:
1946:
1924:
1921:
1918:
1913:
1910:
1905:
1899:
1896:
1893:
1889:
1867:
1861:
1858:
1853:
1850:
1847:
1844:
1841:
1835:
1829:
1825:
1821:
1815:
1809:
1805:
1801:
1794:
1787:
1784:
1781:
1774:
1770:
1767:
1761:
1758:
1755:
1751:
1742:
1741:
1740:
1738:
1735:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1695:
1692:
1686:
1683:
1680:
1669:
1650:
1645:
1642:
1637:
1632:
1629:
1624:
1618:
1614:
1610:
1604:
1598:
1594:
1590:
1584:
1581:
1577:
1570:
1567:
1564:
1558:
1553:
1549:
1546:
1543:
1533:
1532:
1531:
1529:
1510:
1507:
1503:
1496:
1493:
1490:
1484:
1478:
1474:
1470:
1463:
1459:
1455:
1448:
1445:
1442:
1435:
1431:
1428:
1425:
1421:
1414:
1410:
1406:
1400:
1397:
1393:
1389:
1385:
1381:
1374:
1373:
1372:
1370:
1365:
1355:
1350:
1346:
1340:
1336:
1331:
1326:
1324:
1320:
1316:
1283:
1275:
1270:
1236:
1226:
1224:
1220:
1216:
1212:
1208:
1204:
1203:indeterminate
1200:
1196:
1192:
1187:
1185:
1179:
1173:
1169:, as long as
1167:
1163:
1159:
1153:
1149:
1145:
1140:
1136:
1132:
1127:
1125:
1121:
1120:
1115:
1111:
1107:
1102:
1100:
1096:
1092:
1087:
1085:
1084:
1079:
1078:
1073:
1072:
1067:
1063:
1059:
1055:
1051:
1040:
1035:
1033:
1028:
1026:
1021:
1020:
1018:
1017:
987:
971:
956:
947:
938:
935:
931:
927:
901:
885:
860:
857:
853:
851:
846:
820:
804:
799:
748:
745:
741:
737:
734:
680:
670:
654:
649:
586:
583:
579:
575:
572:
547:
531:
526:
512:
491:
469:
466:
462:
458:
432:
416:
411:
397:
376:
354:
351:
347:
343:
317:
301:
296:
282:
261:
240:
219:
197:
194:
190:
186:
183:
182:
176:
171:
169:
164:
162:
157:
156:
153:
149:
148:
141:
131:
128:
120:
109:
106:
102:
99:
95:
92:
88:
85:
81:
78: –
77:
73:
72:Find sources:
66:
62:
56:
55:
50:This article
48:
44:
39:
38:
33:
19:
5441:Division (2)
5440:
5405:Division (2)
5404:
5376:Hexation (6)
5351:Addition (1)
5139:
5056:
5051:
5049:
5024:
5003:
4982:
4887:
4868:
4848:
4841:
4831:
4824:
4805:
4799:
4790:
4778:
4768:
4739:
4720:
4701:
4682:
4670:
4657:
4630:
4624:
4612:
4599:
4587:
4554:
4548:
4524:
4516:
4506:
4490:
4415:
4305:
4295:
4288:
4281:
4274:
4251:
4247:
4235:
4231:
4227:
4223:
4205:
4192:
4188:
4183:
4179:
4173:
4169:
4163:
4159:
4155:
4150:
4145:
4141:
4136:
4132:
4128:
4123:
4119:
4115:
4110:
4105:
4101:
4096:
4092:
4088:
4078:
4068:
4064:
4059:
4053:
4047:
4044:
4040:
4036:
4030:
4026:
4022:
4012:
4006:
4003:
3991:
3987:
3983:
3979:
3975:
3968:
3964:
3960:
3956:
3952:
3945:
3941:
3937:
3935:the same as
3931:
3927:
3920:
3916:
3912:
3904:
3900:
3897:
3892:
3888:
3883:
3879:
3875:
3869:
3863:
3857:
3854:
3850:
3846:
3841:
3827:
3808:
3803:
3798:denotes the
3794:
3788:
3784:
3780:
3777:
3749:
3739:
3735:
3731:
3727:
3723:
3721:
3546:
3541:
3537:
3533:
3529:
3525:
3521:
3486:
3168:
3158:
3154:
3150:
3146:
3142:
3138:
3135:real numbers
3132:
3118:
3116:
3013:
3009:
3005:
3001:
2995:
2984:
2969:
2950:
2944:
2938:
2937:This is the
2890:
2839:
2833:
2744:mixed number
2677:
2667:
2658:
2652:
2647:
2643:
2639:
2635:
2631:
2627:
2623:
2615:
2614:, such that
2611:
2607:
2603:
2599:
2597:
2565:real numbers
2558:
2543:
2530:
2520:
2513:
2494:
2491:
2483:
2454:
2450:
2387:
2374:
2373:in his 1684
2368:
2333:In some non-
2332:
2327:
2313:
2271:
2267:
2257:
2206:
2142:
2106:
2098:
2094:
2090:
2086:
2082:
2080:
2045:
2041:
2038:fraction bar
2033:
2029:
2027:
2000:distributive
1997:
1733:
1665:
1526:Division is
1525:
1366:
1348:
1344:
1338:
1334:
1327:
1288:is equal to
1271:
1232:
1229:Introduction
1197:and include
1188:
1177:
1171:
1165:
1161:
1157:
1151:
1147:
1143:
1135:real numbers
1128:
1123:
1117:
1103:
1088:
1081:
1076:
1075:
1070:
1069:
1049:
1048:
849:
577:
518:multiplicand
123:
117:October 2014
114:
104:
97:
90:
83:
71:
59:Please help
54:verification
51:
5253:Irreducible
5183:Denominator
5011:Subtraction
4426:calculators
4286:quaternions
4020:. That is,
3834:commutative
3774:Of matrices
3752:polynomials
2674:Of integers
2650:< |
2546:calculators
2540:By computer
2453:divided by
2326:in 1659 in
2324:Johann Rahn
2316:ISO 80000-2
2272:denominator
2105:, then the
2044:divided by
1354:associative
1330:commutative
1062:subtraction
638:denominator
346:Subtraction
5502:Categories
5285:Percentage
5280:Paper size
5189:= Quotient
4584:"Division"
4555:Arithmetic
4540:References
4498:or to the
4308:derivative
4271:isomorphic
4216:quaternion
4214:algebras,
4201:quasigroup
4083:, given a
2963:and every
2533:slide rule
2505:fractional
2449:to denote
2320:calculator
2169:GNU Octave
2163:, such as
1248:20 / 5 = 4
1054:arithmetic
508:multiplier
442:difference
403:subtrahend
87:newspapers
5258:Reduction
5216:Continued
5201:Algebraic
5177:Numerator
5113:Fractions
4672:MathWorld
4614:MathWorld
4589:MathWorld
4430:zero ring
4364:−
4293:octonions
4291:, or the
4139:(written
4099:(written
3924:, nor is
3697:−
3656:−
3622:−
3498:−
3436:−
3332:−
3281:−
3243:−
3067:×
2841:remainder
2742:(or as a
2622:integers
2550:computers
2465:Computing
2435:¯
2296:÷
2268:numerator
2189:∖
2173:backslash
2032:over the
1795:≠
1717:×
1705:×
1693:×
1638:±
1605:±
1568:±
1547:±
1494:×
1456:≠
1446:×
1274:remainder
1124:remainder
997:logarithm
957:
930:Logarithm
631:numerator
513:×
492:×
398:−
377:−
5231:Egyptian
5166:Fraction
5148:Quotient
5140:Dividend
5057:Division
4990:Addition
4896:Archived
4876:Archived
4748:Archived
4729:Archived
4709:Archived
4690:Archived
4440:See also
4374:′
4357:′
4343:′
4302:Calculus
4269:must be
4252:algebras
4248:division
4224:division
4191:and all
4051:, where
3828:Because
3792:, where
2972:rounding
2835:quotient
2707:fraction
2646:and 0 ≤
2628:quotient
2604:dividend
2487:chunking
2264:integers
2260:fraction
2159:.) Some
2099:dividend
2030:dividend
2006:Notation
1291:3
1119:quotient
1099:integers
1083:quotient
1071:dividend
1058:addition
1050:Division
873:radicand
772:exponent
701:quotient
690:fraction
607:dividend
578:Division
189:Addition
5386:Inverse
5339:Primary
5248:Integer
5221:Decimal
5186:
5174:
5144:Divisor
5066:∕
5016:−
5006:−
4529:
4512:
4422:product
4199:) is a
4010:and/or
3832:is not
3800:inverse
2612:divisor
2544:Modern
2335:English
2209:solidus
2107:divisor
2034:divisor
1319:rounded
1315:integer
1311:3.33...
1306:
1265:
1253:
1213:. In a
1201:in one
1112:of two
1077:divisor
852:th root
614:divisor
558:product
393:minuend
246:summand
236:summand
101:scholar
5241:Silver
5236:Golden
5226:Dyadic
5211:Binary
5206:Aspect
5117:ratios
5062:÷
5052:÷
5041:·
5037:×
5027:×
4856:
4812:
4645:
4434:wheels
4284:, the
4277:, the
4212:matrix
2961:MATLAB
2838:and a
2680:closed
2630:, and
2626:, the
2620:unique
2610:, the
2606:, and
2602:, the
2516:abacus
2383:ratios
2280:obelus
2165:MATLAB
2089:" or "
2022:obelus
1360:, but
1286:10 / 3
1278:10 / 3
1243:20 / 5
1239:20 / 5
1207:fields
1155:means
1064:, and
879:degree
497:factor
487:factor
288:addend
278:augend
267:addend
257:addend
103:
96:
89:
82:
74:
4995:+
4985:+
4787:(PDF)
4482:Notes
4195:(the
4085:magma
3959:) = (
3756:field
3161:≠ 0.
2844:, so
2746:, so
2379:colon
2149:ASCII
2103:slash
2093:over
1739:, as
1670:, as
1284:, so
1250:, or
1219:units
932:(log)
830:power
790:power
712:ratio
108:JSTOR
94:books
5290:Unit
5115:and
4854:ISBN
4810:ISBN
4643:ISBN
4531:= 1.
4515:sin
4418:zero
4306:The
4264:real
4240:ring
4177:and
4067:and
4057:and
4034:and
3978:) \
3972:and
3967:) /
3930:) \
3919:) /
3540:and
3157:and
3008:and
2548:and
2473:and
2270:and
2167:and
1937:but
1215:ring
1209:and
1104:The
952:base
911:root
785:base
767:base
382:term
372:term
225:term
215:term
80:news
5270:LCD
5064:or
5039:or
4504:lim
4234:or
4135:by
4131:of
4095:by
4091:of
4079:In
3986:\ (
3955:/ (
3940:\ (
3903:/ (
3891:or
3844:as
3802:of
3766:or
2666:of
2559:In
2414:or
2155:in
2085:by
1734:not
1309:or
1267:= 4
1180:= 0
1133:or
1108:or
1086:.
948:log
854:(√)
742:(^)
580:(÷)
463:(×)
348:(−)
327:sum
191:(+)
63:by
5504::
5146:=
5142:÷
5068:)
5043:)
5018:)
4997:)
4789:.
4758:^
4669:.
4641:.
4637::
4633:.
4611:.
4586:.
4571:^
4561::
4557:.
4509:→0
4314::
4298:.
4236:ca
4232:ab
4182:/
4172:\
4162:=
4158:∗
4144:/
4122:=
4118:∗
4104:\
4071:.
4043:=
4039:\
4031:AB
4029:=
4025:/
3996:.
3990:\
3982:=
3976:AB
3963:/
3957:BC
3944:\
3928:AB
3915:/
3905:BC
3884:AB
3882:=
3878:/
3853:=
3849:\
3820:.
3809:AB
3789:AB
3787:=
3783:/
3770:.
3734:,
3730:,
3726:,
3532:,
3528:,
3524:,
3155:cb
3153:=
3145:=
3125:.
2976:−∞
2925:2.
2916:11
2913:26
2877:4.
2858:11
2855:26
2810:11
2807:26
2778:11
2760:11
2757:26
2727:11
2724:26
2670:.
2642:+
2640:bq
2638:=
2556:.
2518:.
2385:.
2002:.
1983:9.
1960:12
1947:12
1911:12
1890:12
1347:/
1337:/
1256:20
1164:=
1160:×
1150:/
1146:=
1101:.
1060:,
5324:e
5317:t
5310:v
5180:/
5105:e
5098:t
5091:v
5060:(
5035:(
5014:(
4993:(
4958:e
4951:t
4944:v
4862:.
4818:.
4675:.
4651:.
4617:.
4592:.
4565:.
4525:x
4521:/
4517:x
4507:x
4391:.
4384:2
4380:g
4371:g
4367:f
4361:g
4354:f
4347:=
4338:)
4333:g
4330:f
4325:(
4296:O
4289:H
4282:C
4275:R
4228:a
4193:b
4189:a
4184:a
4180:b
4174:b
4170:a
4164:b
4160:a
4156:y
4151:y
4146:a
4142:b
4137:a
4133:b
4124:b
4120:x
4116:a
4111:x
4106:b
4102:a
4097:a
4093:b
4069:B
4065:A
4060:B
4054:A
4048:B
4045:A
4041:B
4037:A
4027:B
4023:A
4013:B
4007:A
3994:)
3992:C
3988:A
3984:B
3980:C
3974:(
3969:B
3965:C
3961:A
3953:A
3948:)
3946:C
3942:B
3938:A
3932:C
3926:(
3921:C
3917:B
3913:A
3911:(
3907:)
3901:A
3880:B
3876:A
3870:A
3864:B
3858:B
3855:A
3851:B
3847:A
3804:B
3795:B
3785:B
3781:A
3740:r
3736:s
3732:r
3728:q
3724:p
3708:.
3703:)
3700:s
3694:q
3691:(
3688:i
3684:e
3678:r
3675:p
3670:=
3662:s
3659:i
3652:e
3646:s
3643:i
3639:e
3635:r
3628:s
3625:i
3618:e
3612:q
3609:i
3605:e
3601:p
3595:=
3587:s
3584:i
3580:e
3576:r
3569:q
3566:i
3562:e
3558:p
3542:s
3538:r
3534:s
3530:r
3526:q
3522:p
3504:s
3501:i
3495:r
3473:.
3465:2
3461:s
3457:+
3452:2
3448:r
3442:s
3439:p
3433:r
3430:q
3424:i
3421:+
3413:2
3409:s
3405:+
3400:2
3396:r
3390:s
3387:q
3384:+
3381:r
3378:p
3372:=
3364:2
3360:s
3356:+
3351:2
3347:r
3341:)
3338:s
3335:p
3329:r
3326:q
3323:(
3320:i
3317:+
3314:s
3311:q
3308:+
3305:r
3302:p
3296:=
3290:)
3287:s
3284:i
3278:r
3275:(
3272:)
3269:s
3266:i
3263:+
3260:r
3257:(
3252:)
3249:s
3246:i
3240:r
3237:(
3234:)
3231:q
3228:i
3225:+
3222:p
3219:(
3213:=
3207:s
3204:i
3201:+
3198:r
3193:q
3190:i
3187:+
3184:p
3159:b
3151:a
3147:c
3143:b
3141:/
3139:a
3119:p
3103:.
3097:r
3094:q
3089:s
3086:p
3080:=
3075:r
3072:s
3062:q
3059:p
3054:=
3048:s
3044:/
3040:r
3035:q
3031:/
3027:p
3014:s
3012:/
3010:r
3006:q
3004:/
3002:p
2922:=
2899:.
2867:2
2864:=
2829:.
2784:.
2775:4
2769:2
2766:=
2702:.
2691:.
2668:b
2659:b
2653:b
2648:r
2644:r
2636:a
2632:r
2624:q
2616:b
2608:b
2600:a
2577:x
2573:x
2485:'
2455:b
2451:a
2431:a
2428:)
2422:b
2402:a
2399:)
2396:b
2354:b
2351::
2348:a
2299:b
2293:a
2241:b
2233:/
2226:a
2192:a
2186:b
2128:b
2124:/
2120:a
2095:b
2091:a
2087:b
2083:a
2064:b
2061:a
2046:b
2042:a
1980:=
1977:3
1974:+
1971:6
1968:=
1963:4
1955:+
1950:2
1925:,
1922:2
1919:=
1914:6
1906:=
1900:4
1897:+
1894:2
1868:.
1862:c
1859:b
1854:b
1851:a
1848:+
1845:c
1842:a
1836:=
1833:)
1830:c
1826:/
1822:a
1819:(
1816:+
1813:)
1810:b
1806:/
1802:a
1799:(
1791:)
1788:c
1785:+
1782:b
1779:(
1775:/
1771:a
1768:=
1762:c
1759:+
1756:b
1752:a
1720:c
1714:b
1711:+
1708:c
1702:a
1699:=
1696:c
1690:)
1687:b
1684:+
1681:a
1678:(
1651:.
1646:c
1643:b
1633:c
1630:a
1625:=
1622:)
1619:c
1615:/
1611:b
1608:(
1602:)
1599:c
1595:/
1591:a
1588:(
1585:=
1582:c
1578:/
1574:)
1571:b
1565:a
1562:(
1559:=
1554:c
1550:b
1544:a
1511:.
1508:b
1504:/
1500:)
1497:c
1491:a
1488:(
1485:=
1482:)
1479:c
1475:/
1471:b
1468:(
1464:/
1460:a
1452:)
1449:c
1443:b
1440:(
1436:/
1432:a
1429:=
1426:c
1422:/
1418:)
1415:b
1411:/
1407:a
1404:(
1401:=
1398:c
1394:/
1390:b
1386:/
1382:a
1349:a
1345:b
1339:b
1335:a
1303:3
1300:/
1297:1
1293:+
1262:5
1259:/
1178:b
1172:b
1166:c
1162:b
1158:a
1152:b
1148:c
1144:a
1038:e
1031:t
1024:v
972:=
968:)
960:(
886:=
850:n
805:=
800:}
681:{
655:=
650:}
532:=
527:}
417:=
412:}
302:=
297:}
283:+
262:+
241:+
220:+
174:e
167:t
160:v
130:)
124:(
119:)
115:(
105:·
98:·
91:·
84:·
57:.
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
