456:
133:
36:
243:(including zero) is the next natural number and is the result of adding one to that number. The predecessor of a natural number (excluding zero) is the previous natural number and is the result of subtracting one from that number. For example, the successor of zero is one, and the predecessor of eleven is ten (
1090:
is a mathematical operation of repeated addition. When two numbers are multiplied, the resulting value is a product. The numbers being multiplied are multiplicands, multipliers, or factors. Multiplication can be expressed as "five times three equals fifteen", "five times three is fifteen" or "fifteen
702:
The
Austrian method, also known as the additions method, is taught in certain European countries. In contrast to the previous method, no borrowing is used, although there are crutches that vary according to certain countries. The method of addition involves augmenting the subtrahend. This transforms
2344:
Dividing 272 and 8, starting with the hundreds digit, 2 is not divisible by 8. Add 20 and 7 to get 27. The largest number that the divisor of 8 can be multiplied by without exceeding 27 is 3, so it is written under the tens column. Subtracting 24 (the product of 3 and 8) from 27 gives 3 as the
463:. The black numbers are the addends, the green number is the carry, and the blue number is the sum. In the rightmost digit, the addition of 9 and 7 is 16, carrying 1 into the next pair of the digit to the left, making its addition 1 + 5 + 2 = 8. Therefore, 59 + 27 = 86.
2586:
Elementary arithmetic is typically taught at the primary or secondary school levels and is governed by local educational standards. In the United States and Canada, there has been debate about the content and methods used to teach elementary arithmetic.
2814:
662:
is solved by borrowing a 10 from the tens place to add to the ones place in order to facilitate the subtraction. Subtracting 9 from 6 involves borrowing a 10 from the tens place, making the problem into
476:
is a mathematical operation that combines two or more numbers (called addends or summands) to produce a combined number (called the sum). The addition of two numbers is expressed with the plus sign (
1345:
1413:
540:
evaluates the difference between two numbers, where the minuend is the number being subtracted from, and the subtrahend is the number being subtracted. It is represented using the minus sign (
761:
Subtracting the numbers 792 and 308, starting with the ones column, 2 is smaller than 8. Using the borrowing method, 10 is borrowed from 90, reducing 90 to 80. This changes the problem to
1094:
Multiplication is represented using the multiplication sign (×), the asterisk (*), parentheses (), or a dot (⋅). The statement "five times three equals fifteen" can be written as "
2083:
1277:
511:
When the sum of a pair of digits results in a two-digit number, the "tens" digit is referred to as the "carry digit". In elementary arithmetic, students typically learn to add
1691:
1439:
Multiplying 729 and 3, starting on the ones column, the product of 9 and 3 is 27. 7 is written under the ones column and 2 is written above the tens column as a carry digit.
751:
2195:
2577:
2469:
1124:
2139:
1229:
1070:
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1197:
2392:
Going to the ones digit, the number is 2. Adding 30 (the remainder, 3, times 10) and 2 gets 32. The quotient of 32 and 8 is 4, which is written under the ones column.
693:
167:, broad range of application, and position as the foundation of all mathematics, elementary arithmetic is generally the first branch of mathematics taught in schools.
3109:
2168:
308:
408:
382:
3164:
660:
274:
785:
356:
336:
613:
587:
558:
494:
436:, starting with 1 for the first object and increasing by 1 for each subsequent object. The number of objects in the set is the count. This is also known as the
695:. This is indicated by crossing out the 8, writing a 7 above it, and writing a 1 above the 6. These markings are called "crutches", which were invented by
633:
is distinguished generally by the lack of preference for any specific technique, replaced by guiding students to invent their own methods of computation.
53:
2981:
2663:(sometimes including 0), multiplication under other number sets can satisfy more or less properties than those listed here, such as having an
2965:
2938:
2899:
2874:
2719:
3157:
136:
The symbols for elementary-level math operations. From top-left going clockwise: addition, division, multiplication, and subtraction.
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119:
100:
72:
3031:
190:
2784:
1584:
The product of 7 and 3 is 21, and since this is the last digit, 2 will not be written as a carry digit, but instead beside 1.
3304:
3299:
3150:
1288:
79:
57:
1431:
In the multiplication algorithm, the "tens" digit of the product of a pair of digits is referred to as the "carry digit".
3128:
1356:
86:
2930:
68:
46:
189:
are characters used to represent the value of numbers. An example of a numeral system is the predominantly used
3319:
626:
3068:
615:. In elementary arithmetic, the minuend is always larger than the subtrahend to produce a positive result.
3264:
3259:
2601:
2102:
2093:
2056:
160:
1244:
3006:
1664:
563:
Subtraction is not commutative, which means that the order of the numbers can change the final value;
3314:
3223:
3202:
2611:
1512:
The product of 2 and 3 is 6, and the carry digit adds 2 to 6, so 8 is written under the tens column.
706:
501:
227:
system. Regardless of the numeral system used, the results of arithmetic operations are unaffected.
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2173:
3273:
3244:
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2333:
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224:
197:
93:
2550:
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1097:
2112:
1202:
1043:
630:
460:
236:
2907:
1129:
2479:
Another method of dividing taught in some schools is the bus stop method, sometimes notated as
2275:
1161:
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2961:
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2934:
2924:
2895:
2870:
2843:
2758:
2715:
2656:
1424:
666:
636:
American schools teach a method of subtraction using borrowing. A subtraction problem such as
311:). Every natural number has a successor, and every natural number except 0 has a predecessor.
201:
2864:
2147:
281:
2837:
2748:
2707:
2631:
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2315:
2231:
1418:
622:(e.g., find the size of a subset of a specific set), and find quantities in other contexts.
387:
361:
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247:
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2664:
2636:
1701:
Multiplying 789 and 345, starting with the ones column, the product of 789 and 5 is 3945.
764:
516:
341:
321:
186:
1820:
3 is in the hundreds digit. The multiplier is 300. The product of 789 and 300 is 236700.
1749:
4 is in the tens digit. The multiplier is 40, not 4. The product of 789 and 40 is 31560.
592:
566:
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3234:
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The order in which the addends are added does not affect the sum. This is known as the
479:
240:
182:
176:
156:
3293:
3269:
3042:
2807:"A Number System Invented by Inuit Schoolchildren Will Make Its Silicon Valley Debut"
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415:
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205:
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132:
2336:, involving subtracting more multiples from the partial remainder at each stage.
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2701:
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2626:
1285:. Rearranging the order of parentheses in a product does not change the result:
619:
537:
532:
444:
437:
315:
164:
152:
144:
35:
17:
3129:"Educators debate whether some math basics are 'a dead issue in the year 2016'"
2753:
2736:
3248:
507:
The sum of two numbers is unique; there is only one correct answer for a sums.
433:
411:
2762:
1234:
In elementary arithmetic, multiplication satisfies the following properties:
2394:
2351:
2346:
1921:
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1751:
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1514:
1441:
221:
217:
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520:
473:
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427:
148:
3142:
2737:"What is to be learnt? Critical aspects of elementary arithmetic skills"
512:
194:
2700:
Mitchelmore, Michael C.; White, Paul (2012). Seel, Norbert M. (ed.).
213:
209:
952:
In the hundreds column, the difference between 700 and 300 is 400.
2257:
454:
131:
2488:
The steps here are shown below, using the same example as above:
2863:
Musser, Gary L.; Peterson, Blake E.; Burger, William F. (2013).
2735:
Björklund, Camilla; Marton, Ference; Kullberg, Angelika (2021).
753:. A small 1 is marked below the subtrahend digit as a reminder.
3146:
1241:. Switching the order in a product does not change the result:
629:
method subtracts using methods suitable for hand calculation.
29:
3032:"Subtraction in the United States: An Historical Perspective"
2318:
is considered impossible at an elementary arithmetic level.
3069:"The Teaching of Arithmetic: A Manual for Teachers. pp. 177"
3007:"Subtraction Algorithms - Department of Mathematics at UTSA"
2866:
Mathematics for
Elementary Teachers: A Contemporary Approach
869:
In the tens column, the difference between 80 and 0 is 80.
560:). The minus sign is also used to notate negative numbers.
447:, the process of drawing a mark for each object in a set.
625:
There are several methods to accomplish subtraction. The
504:
of addition. (a + b) and (b + a) produce the same output.
1340:{\displaystyle a\times (b\times c)=(a\times b)\times c}
414:, meaning that any subset of the natural numbers has a
2655:
While elementary arithmetic mainly operates under the
432:
Counting assigns a natural number to each object in a
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2445:
2278:
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2150:
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1132:
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1046:
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709:
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569:
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482:
390:
364:
358:) the former. For example, three is less than eight (
344:
324:
284:
250:
1697:
Example of multiplication for multiple-digit factors
1435:
Example of multiplication for a single-digit factor
60:. Unsourced material may be challenged and removed.
2892:Discrete Maths and Its Applications Global Edition
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2332:A less systematic method involves the concept of
1408:{\displaystyle a\times (b+c)=a\times b+a\times c}
338:) another number, then the latter is less than (
2105:is an arithmetic operation, and the inverse of
2839:An Introduction to Proof through Real Analysis
699:, who used them in a study, in November 1937.
3158:
496:). It is performed according to these rules:
8:
2836:Madden, Daniel J.; Aubrey, Jason A. (2017).
1353:. Multiplication distributes over addition:
2777:"numeral system | mathematics | Britannica"
2706:. Boston, MA: Springer US. pp. 31–33.
2325:. An abbreviated version of long division,
3182:
3165:
3151:
3143:
2321:Two numbers can be divided on paper using
2957:Psychology of Mathematics for Instruction
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120:Learn how and when to remove this message
515:and may also learn about topics such as
27:Numbers and the basic operations on them
2692:
2648:
1421:. Any number multiplied by 1 is itself.
428:Counting § Counting in mathematics
3110:"Debate about Teaching style of Maths"
2918:
2916:
618:Subtraction is also used to separate,
2869:. John Wiley & Sons. p. 87.
384:), thus eight is greater than three (
7:
3090:"The Teaching of Arithmetic. pp. 77"
2954:Resnick, L. B.; Ford, W. W. (2012).
2842:. John Wiley & Sons. p. 3.
2329:, can be used for smaller divisors.
2078:{\displaystyle 789\times 345=272205}
443:Counting can also be the process of
200:. Other numeral systems include the
58:adding citations to reliable sources
2926:An Introduction to Abstract Algebra
2703:Abstraction in Mathematics Learning
2542:= 32) 0 (32 - 32 = 0)
1272:{\displaystyle a\times b=b\times a}
1091:is the product of five and three".
2741:Educational Studies in Mathematics
1427:. Any number multiplied by 0 is 0.
25:
2817:from the original on 19 July 2023
318:. If one number is greater than (
2256:is restricted to the concept of
2211:. This can be read verbally as "
1686:{\displaystyle 3\times 729=2187}
410:). The natural numbers are also
34:
3127:Gollom, Mark (April 10, 2016).
3039:Microsoft Word - Issue 2 -9/23/
2982:"Everyday Mathematics4 at Home"
2787:from the original on 2023-08-10
746:{\displaystyle (80+16)-(39+10)}
231:Successor function and ordering
45:needs additional citations for
2483: result
2190:{\displaystyle {\frac {a}{b}}}
1378:
1366:
1328:
1316:
1310:
1298:
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1171:
1165:
740:
728:
722:
710:
235:In elementary arithmetic, the
1:
2712:10.1007/978-1-4419-1428-6_516
2679:and higher order number sets.
3088:Smith, David Eugene (1913).
2572:{\displaystyle 272\div 8=34}
2464:{\displaystyle 272\div 8=34}
1119:{\displaystyle 5\times 3=15}
2986:Everyday Mathematics Online
2783:. Paragraph 2, sentence 4.
2144:Division can be written as
2134:{\displaystyle c\times b=a}
1224:{\displaystyle 5\cdot 3=15}
1065:{\displaystyle 792-308=484}
314:The natural numbers have a
3336:
2960:. Routledge. p. 110.
2931:Cambridge University Press
2754:10.1007/s10649-021-10045-0
2091:
1151:{\displaystyle 5\ast 3=15}
1080:
703:the previous problem into
530:
466:
425:
191:Indo-Arabic numeral system
174:
163:. Due to its low level of
3180:
2805:Tillinghast-Raby, Amory.
2297:{\displaystyle a\div b=c}
1919:Adding all the products,
1192:{\displaystyle (5)(3)=15}
2252:. In English usage, the
688:{\displaystyle 70+16-39}
2890:Rosen, Kenneth (2013).
2671:and beyond, or lacking
2163:{\displaystyle a\div b}
627:traditional mathematics
303:{\displaystyle 11-1=10}
193:(0 to 9), which uses a
69:"Elementary arithmetic"
3067:Klapper, Paul (1916).
2602:Elementary mathematics
2573:
2465:
2298:
2191:
2164:
2135:
2094:Division (mathematics)
2079:
1687:
1409:
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1273:
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1193:
1152:
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1066:
781:
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689:
656:
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554:
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464:
404:
403:{\displaystyle 8>3}
378:
377:{\displaystyle 3<8}
352:
332:
304:
270:
137:
3305:Mathematics education
3300:Elementary arithmetic
3174:Elementary arithmetic
3011:mathresearch.utsa.edu
2582:Educational standards
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2466:
2299:
2234:-speaking cultures, "
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655:{\displaystyle 86-39}
610:
584:
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491:
458:
405:
379:
353:
333:
305:
271:
269:{\displaystyle 0+1=1}
141:Elementary arithmetic
135:
2923:Hall, F. M. (1972).
2612:Plus and minus signs
2551:
2514:7 ( 2 - 0 =
2443:
2276:
2174:
2148:
2113:
2057:
1665:
1357:
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1245:
1203:
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1044:
780:{\displaystyle 12-8}
765:
707:
667:
640:
593:
567:
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502:commutative property
480:
388:
362:
351:{\displaystyle <}
342:
331:{\displaystyle >}
322:
282:
248:
54:improve this article
3114:edmontonjournal.com
2811:Scientific American
2607:Chunking (division)
2530:2 (27 - 24 =
2485:(divisor) dividend
697:William A. Brownell
608:{\displaystyle 5-3}
589:is not the same as
582:{\displaystyle 3-5}
461:addition with carry
225:positional notation
204:(often used in the
198:positional notation
3186:
3048:on August 11, 2017
2781:www.britannica.com
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2461:
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2187:
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1116:
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777:
743:
685:
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631:Reform mathematics
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348:
328:
300:
266:
138:
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2967:978-1-136-55759-0
2940:978-0-521-08484-0
2901:978-0-07-131501-2
2876:978-1-118-48700-6
2721:978-1-4419-1428-6
2434:
2433:
2390:
2389:
2310:the divisor, and
2306:is the dividend,
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1034:
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553:{\displaystyle -}
489:{\displaystyle +}
130:
129:
122:
104:
16:(Redirected from
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3041:. Archived from
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3021:
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3018:
3017:
3003:
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2978:
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2802:
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2795:
2793:
2792:
2773:
2767:
2766:
2756:
2732:
2726:
2725:
2697:
2680:
2669:rational numbers
2653:
2632:Imaginary number
2622:Division by zero
2578:
2576:
2575:
2570:
2541:
2533:
2529:
2525:
2517:
2513:
2509:
2500:
2497:
2494:
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2468:
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2316:Division by zero
2303:
2301:
2300:
2295:
2251:
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2188:
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1412:
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1406:
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1227:
1222:
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1123:
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659:
658:
653:
614:
612:
611:
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588:
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557:
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517:negative numbers
495:
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409:
407:
406:
401:
383:
381:
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375:
357:
355:
354:
349:
337:
335:
334:
329:
309:
307:
306:
301:
275:
273:
272:
267:
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
18:Basic arithmetic
3335:
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3004:
3000:
2990:
2988:
2980:
2979:
2975:
2968:
2953:
2952:
2948:
2941:
2933:. p. 171.
2922:
2921:
2914:
2902:
2894:. McGraw Hill.
2889:
2888:
2884:
2877:
2862:
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2857:
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2835:
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2818:
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2799:
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2788:
2775:
2774:
2770:
2734:
2733:
2729:
2722:
2699:
2698:
2694:
2689:
2684:
2683:
2665:inverse element
2661:natural numbers
2654:
2650:
2645:
2637:Number sentence
2593:
2584:
2549:
2548:
2543:
2539:
2531:
2527:
2523:
2515:
2511:
2507:
2498:
2495:
2492:
2486:
2477:
2475:Bus stop method
2441:
2440:
2342:
2274:
2273:
2271:In an equation
2243:
2205:
2199:
2198:
2172:
2171:
2146:
2145:
2111:
2110:
2100:
2092:Main articles:
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1128:
1127:
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1041:
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762:
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705:
704:
665:
664:
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637:
591:
590:
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564:
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529:
478:
477:
471:
453:
430:
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386:
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359:
340:
339:
320:
319:
280:
279:
246:
245:
233:
202:Kaktovik system
183:numeral systems
179:
173:
171:Numeral systems
143:is a branch of
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
3333:
3331:
3323:
3322:
3320:Multiplication
3317:
3312:
3307:
3302:
3292:
3291:
3285:
3284:
3280:
3279:
3256:
3254:
3240:Multiplication
3231:
3229:
3210:
3208:
3189:
3187:
3181:
3178:
3177:
3172:
3170:
3169:
3162:
3155:
3147:
3141:
3140:
3135:
3134:
3119:
3101:
3080:
3059:
3022:
2998:
2973:
2966:
2946:
2939:
2912:
2900:
2882:
2875:
2855:
2848:
2828:
2797:
2768:
2747:(2): 261–284.
2727:
2720:
2691:
2690:
2688:
2685:
2682:
2681:
2647:
2646:
2644:
2641:
2640:
2639:
2634:
2629:
2624:
2619:
2614:
2609:
2604:
2599:
2597:Early numeracy
2592:
2589:
2583:
2580:
2568:
2565:
2562:
2559:
2556:
2490:
2481:
2476:
2473:
2472:
2471:
2460:
2457:
2454:
2451:
2448:
2432:
2431:
2428:
2425:
2423:
2420:
2419:
2416:
2414:
2412:
2408:
2407:
2404:
2401:
2398:
2388:
2387:
2385:
2382:
2380:
2377:
2376:
2373:
2371:
2369:
2365:
2364:
2361:
2358:
2355:
2341:
2338:
2327:short division
2314:the quotient.
2293:
2290:
2287:
2284:
2281:
2184:
2181:
2159:
2156:
2153:
2130:
2127:
2124:
2121:
2118:
2107:multiplication
2089:
2086:
2074:
2071:
2068:
2065:
2062:
2047:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
2002:
2001:
1998:
1995:
1992:
1989:
1986:
1984:
1981:
1980:
1977:
1974:
1971:
1968:
1966:
1964:
1961:
1960:
1957:
1954:
1951:
1949:
1947:
1945:
1941:
1940:
1937:
1934:
1931:
1929:
1927:
1925:
1915:
1914:
1911:
1908:
1905:
1902:
1899:
1895:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1875:
1872:
1869:
1866:
1863:
1861:
1858:
1857:
1854:
1851:
1848:
1845:
1843:
1840:
1839:
1836:
1833:
1830:
1828:
1826:
1816:
1815:
1812:
1809:
1806:
1803:
1799:
1798:
1795:
1792:
1789:
1786:
1783:
1782:
1779:
1776:
1773:
1770:
1767:
1766:
1763:
1760:
1757:
1755:
1745:
1744:
1741:
1738:
1735:
1731:
1730:
1727:
1724:
1721:
1717:
1716:
1713:
1710:
1707:
1698:
1695:
1694:
1693:
1682:
1679:
1676:
1673:
1670:
1654:
1653:
1650:
1647:
1644:
1640:
1639:
1636:
1634:
1632:
1628:
1627:
1624:
1621:
1618:
1615:
1614:
1612:
1609:
1607:
1604:
1603:
1600:
1597:
1590:
1580:
1579:
1576:
1573:
1571:
1568:
1567:
1564:
1562:
1560:
1556:
1555:
1552:
1549:
1546:
1543:
1542:
1540:
1537:
1535:
1532:
1531:
1528:
1521:
1518:
1508:
1507:
1504:
1502:
1500:
1497:
1496:
1493:
1491:
1489:
1485:
1484:
1481:
1478:
1475:
1472:
1471:
1469:
1464:
1462:
1459:
1458:
1451:
1448:
1445:
1436:
1433:
1429:
1428:
1422:
1416:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1351:Distributivity
1348:
1336:
1333:
1330:
1327:
1324:
1321:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1280:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1220:
1217:
1214:
1211:
1208:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1147:
1144:
1141:
1138:
1135:
1115:
1112:
1109:
1106:
1103:
1088:Multiplication
1083:Multiplication
1081:Main article:
1078:
1077:Multiplication
1075:
1074:
1073:
1061:
1058:
1055:
1052:
1049:
1032:
1031:
1028:
1025:
1022:
1019:
1018:
1015:
1012:
1009:
1005:
1004:
999:
994:
991:
988:
987:
982:
977:
975:
972:
971:
968:
965:
958:
948:
947:
944:
941:
939:
936:
935:
932:
929:
926:
922:
921:
916:
911:
908:
905:
904:
899:
894:
892:
889:
888:
885:
878:
875:
865:
864:
861:
859:
857:
854:
853:
850:
847:
844:
840:
839:
834:
829:
826:
823:
822:
817:
812:
810:
807:
806:
799:
796:
793:
776:
773:
770:
758:
755:
742:
739:
736:
733:
730:
727:
724:
721:
718:
715:
712:
684:
681:
678:
675:
672:
651:
648:
645:
604:
601:
598:
578:
575:
572:
549:
531:Main article:
528:
525:
509:
508:
505:
485:
467:Main article:
452:
449:
426:Main article:
423:
420:
399:
396:
393:
373:
370:
367:
347:
327:
316:total ordering
299:
296:
293:
290:
287:
265:
262:
259:
256:
253:
241:natural number
232:
229:
177:Numeral system
175:Main article:
172:
169:
157:multiplication
128:
127:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3332:
3321:
3318:
3316:
3313:
3311:
3308:
3306:
3303:
3301:
3298:
3297:
3295:
3277:
3275:
3271:
3266:
3261:
3255:
3252:
3250:
3246:
3241:
3236:
3230:
3227:
3225:
3220:
3215:
3209:
3206:
3204:
3199:
3194:
3188:
3185:
3184:
3179:
3175:
3168:
3163:
3161:
3156:
3154:
3149:
3148:
3145:
3139:
3138:
3130:
3123:
3120:
3115:
3111:
3105:
3102:
3091:
3084:
3081:
3070:
3063:
3060:
3044:
3040:
3033:
3030:Ross, Susan.
3026:
3023:
3012:
3008:
3002:
2999:
2987:
2983:
2977:
2974:
2969:
2963:
2959:
2958:
2950:
2947:
2942:
2936:
2932:
2928:
2927:
2919:
2917:
2913:
2909:
2903:
2897:
2893:
2886:
2883:
2878:
2872:
2868:
2867:
2859:
2856:
2851:
2849:9781119314721
2845:
2841:
2840:
2832:
2829:
2816:
2812:
2808:
2801:
2798:
2786:
2782:
2778:
2772:
2769:
2764:
2760:
2755:
2750:
2746:
2742:
2738:
2731:
2728:
2723:
2717:
2713:
2709:
2705:
2704:
2696:
2693:
2686:
2678:
2674:
2673:commutativity
2670:
2666:
2662:
2658:
2652:
2649:
2642:
2638:
2635:
2633:
2630:
2628:
2625:
2623:
2620:
2618:
2615:
2613:
2610:
2608:
2605:
2603:
2600:
2598:
2595:
2594:
2590:
2588:
2581:
2579:
2566:
2563:
2560:
2557:
2554:
2546:
2537:
2521:
2505:
2501:
2489:
2484:
2480:
2474:
2458:
2455:
2452:
2449:
2446:
2439:
2438:
2437:
2429:
2426:
2424:
2422:
2421:
2417:
2415:
2413:
2410:
2409:
2405:
2402:
2399:
2397:
2396:
2393:
2386:
2383:
2381:
2379:
2378:
2374:
2372:
2370:
2367:
2366:
2362:
2359:
2356:
2354:
2353:
2350:
2348:
2339:
2337:
2335:
2330:
2328:
2324:
2323:long division
2319:
2317:
2313:
2309:
2305:
2291:
2288:
2285:
2282:
2279:
2269:
2267:
2263:
2259:
2255:
2250:
2246:
2242:" is written
2241:
2237:
2233:
2228:
2226:
2222:
2218:
2214:
2208:
2202:
2182:
2179:
2157:
2154:
2151:
2142:
2128:
2125:
2122:
2119:
2116:
2109:. Given that
2108:
2104:
2099:
2098:Long division
2095:
2087:
2085:
2072:
2069:
2066:
2063:
2060:
2052:
2044:
2041:
2038:
2035:
2032:
2029:
2027:
2026:
2022:
2019:
2016:
2013:
2010:
2007:
2004:
2003:
1999:
1996:
1993:
1990:
1987:
1985:
1983:
1982:
1978:
1975:
1972:
1969:
1967:
1965:
1963:
1962:
1958:
1955:
1952:
1950:
1948:
1946:
1943:
1942:
1938:
1935:
1932:
1930:
1928:
1926:
1924:
1923:
1920:
1912:
1909:
1906:
1903:
1900:
1897:
1896:
1892:
1889:
1886:
1883:
1880:
1878:
1877:
1873:
1870:
1867:
1864:
1862:
1860:
1859:
1855:
1852:
1849:
1846:
1844:
1842:
1841:
1837:
1834:
1831:
1829:
1827:
1825:
1824:
1821:
1813:
1810:
1807:
1804:
1801:
1800:
1796:
1793:
1790:
1787:
1785:
1784:
1780:
1777:
1774:
1771:
1769:
1768:
1764:
1761:
1758:
1756:
1754:
1753:
1750:
1742:
1739:
1736:
1733:
1732:
1728:
1725:
1722:
1719:
1718:
1714:
1711:
1708:
1706:
1705:
1702:
1696:
1680:
1677:
1674:
1671:
1668:
1661:
1660:
1659:
1651:
1648:
1645:
1642:
1641:
1637:
1635:
1633:
1630:
1629:
1625:
1622:
1619:
1617:
1616:
1613:
1610:
1608:
1606:
1605:
1601:
1598:
1596:
1595:
1591:
1589:
1588:
1585:
1577:
1574:
1572:
1570:
1569:
1565:
1563:
1561:
1558:
1557:
1553:
1550:
1547:
1545:
1544:
1541:
1538:
1536:
1534:
1533:
1529:
1527:
1526:
1522:
1519:
1517:
1516:
1513:
1505:
1503:
1501:
1499:
1498:
1494:
1492:
1490:
1487:
1486:
1482:
1479:
1476:
1474:
1473:
1470:
1468:
1465:
1463:
1461:
1460:
1457:
1456:
1452:
1449:
1446:
1444:
1443:
1440:
1434:
1432:
1426:
1423:
1420:
1417:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1375:
1372:
1369:
1363:
1360:
1352:
1349:
1334:
1331:
1325:
1322:
1319:
1313:
1307:
1304:
1301:
1295:
1292:
1284:
1283:Associativity
1281:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1240:
1239:Commutativity
1237:
1236:
1235:
1232:
1218:
1215:
1212:
1209:
1206:
1186:
1183:
1177:
1168:
1145:
1142:
1139:
1136:
1133:
1113:
1110:
1107:
1104:
1101:
1092:
1089:
1084:
1076:
1072:
1059:
1056:
1053:
1050:
1047:
1039:
1038:
1037:
1029:
1026:
1023:
1021:
1020:
1016:
1013:
1010:
1007:
1006:
1003:
1000:
998:
995:
992:
990:
989:
986:
983:
981:
978:
976:
974:
973:
969:
966:
964:
963:
959:
957:
956:
953:
945:
942:
940:
938:
937:
933:
930:
927:
924:
923:
920:
917:
915:
912:
909:
907:
906:
903:
900:
898:
895:
893:
891:
890:
886:
884:
883:
879:
876:
874:
873:
870:
862:
860:
858:
856:
855:
851:
848:
845:
842:
841:
838:
835:
833:
830:
827:
825:
824:
821:
818:
816:
813:
811:
809:
808:
805:
804:
800:
797:
794:
792:
791:
788:
774:
771:
768:
756:
754:
737:
734:
731:
725:
719:
716:
713:
700:
698:
682:
679:
676:
673:
670:
649:
646:
643:
634:
632:
628:
623:
621:
616:
602:
599:
596:
576:
573:
570:
561:
547:
539:
534:
526:
524:
522:
518:
514:
513:whole numbers
506:
503:
499:
498:
497:
483:
475:
470:
462:
457:
450:
448:
446:
441:
439:
435:
429:
421:
419:
417:
416:least element
413:
397:
394:
391:
371:
368:
365:
345:
325:
317:
312:
310:
297:
294:
291:
288:
285:
276:
263:
260:
257:
254:
251:
242:
238:
230:
228:
226:
223:
219:
215:
211:
208:languages of
207:
203:
199:
196:
192:
188:
184:
178:
170:
168:
166:
162:
158:
154:
150:
146:
142:
134:
124:
121:
113:
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
3257:
3232:
3211:
3190:
3173:
3122:
3113:
3104:
3093:. Retrieved
3083:
3072:. Retrieved
3062:
3050:. Retrieved
3043:the original
3038:
3025:
3014:. Retrieved
3010:
3001:
2991:December 26,
2989:. Retrieved
2985:
2976:
2956:
2949:
2925:
2891:
2885:
2865:
2858:
2838:
2831:
2819:. Retrieved
2810:
2800:
2789:. Retrieved
2780:
2771:
2744:
2740:
2730:
2702:
2695:
2651:
2617:Peano axioms
2585:
2547:
2545:The result:
2544:
2538:( 8 ×
2535:
2526:= 24)
2522:( 8 ×
2519:
2506:( 8 ×
2503:
2491:
2487:
2482:
2478:
2436:The result:
2435:
2391:
2343:
2331:
2320:
2311:
2307:
2272:
2270:
2265:
2261:
2248:
2244:
2239:
2235:
2230:In some non-
2229:
2224:
2220:
2216:
2212:
2206:
2200:
2143:
2101:
2053:
2051:The result:
2050:
1918:
1819:
1748:
1700:
1658:The result:
1657:
1593:
1592:
1583:
1524:
1523:
1511:
1466:
1454:
1453:
1438:
1430:
1233:
1093:
1086:
1040:
1036:The result:
1035:
1001:
996:
984:
979:
961:
960:
951:
918:
913:
901:
896:
881:
880:
868:
836:
831:
819:
814:
802:
801:
760:
701:
635:
624:
617:
562:
536:
510:
472:
442:
440:of the set.
431:
412:well-ordered
313:
278:
244:
234:
220:), and is a
206:Eskimo-Aleut
180:
140:
139:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
3315:Subtraction
3219:Subtraction
2677:quaternions
2627:Real number
2510:= 0)
2502:8|272
2238:divided by
2215:divided by
538:Subtraction
533:Subtraction
527:Subtraction
459:Example of
438:cardinality
165:abstraction
153:subtraction
145:mathematics
3294:Categories
3095:2016-03-11
3074:2016-03-11
3016:2024-04-01
2908:Appendix I
2791:2022-11-24
2687:References
147:involving
80:newspapers
2763:0013-1954
2558:÷
2450:÷
2347:remainder
2283:÷
2155:÷
2120:×
2064:×
1672:×
1520:Hundreds
1447:Hundreds
1400:×
1388:×
1364:×
1332:×
1323:×
1305:×
1296:×
1264:×
1252:×
1210:⋅
1137:∗
1105:×
1051:−
877:Hundreds
795:Hundreds
772:−
726:−
680:−
647:−
600:−
574:−
548:−
521:fractions
289:−
237:successor
222:vigesimal
218:Greenland
3310:Addition
3265:Division
3198:Addition
3052:June 25,
2906:See the
2815:Archived
2785:Archived
2591:See also
2534:)
2334:chunking
2247: :
2103:Division
2088:Division
1594:Hundreds
1419:Identity
962:Hundreds
474:Addition
469:Addition
451:Addition
445:tallying
422:Counting
161:division
149:addition
110:May 2023
3274:∕
3224:−
3214:−
2821:24 July
2675:in the
2667:in the
2518:)
2340:Example
2232:English
2204:⁄
1199:", or "
757:Example
620:combine
195:decimal
94:scholar
3270:÷
3260:÷
3249:·
3245:×
3235:×
2964:
2937:
2898:
2873:
2846:
2761:
2718:
2264:is to
2258:ratios
2219:" or "
2073:272205
216:, and
214:Canada
210:Alaska
187:digits
159:, and
96:
89:
82:
75:
67:
3203:+
3193:+
3046:(PDF)
3035:(PDF)
2643:Notes
2254:colon
2223:over
2197:, or
2141:, ,
1602:Ones
1599:Tens
1530:Ones
1450:Tens
970:Ones
967:Tens
887:Ones
798:Tens
239:of a
101:JSTOR
87:books
3054:2019
2993:2022
2962:ISBN
2935:ISBN
2896:ISBN
2871:ISBN
2844:ISBN
2823:2023
2759:ISSN
2716:ISBN
2268:").
2096:and
1681:2187
1525:Tens
1455:Ones
1425:Zero
1158:", "
1126:", "
882:Tens
803:Ones
519:and
395:>
369:<
346:<
326:>
277:and
73:news
3272:or
3247:or
2749:doi
2745:107
2708:doi
2659:of
2657:set
2555:272
2447:272
2304:, a
2227:".
2067:345
2061:789
1675:729
1231:".
1060:484
1054:308
1048:792
434:set
181:In
56:by
3296::
3276:)
3251:)
3226:)
3205:)
3112:.
3037:.
3009:.
2984:.
2929:.
2915:^
2910:.
2813:.
2809:.
2779:.
2757:.
2743:.
2739:.
2714:.
2567:34
2536:32
2520:24
2459:34
2430:4
2418:8
2406:2
2375:8
2363:2
2349:.
2260:("
2170:,
2045:5
2023:0
2000:0
1979:5
1959:5
1939:9
1913:0
1901:3
1898:2
1893:0
1881:3
1874:5
1856:5
1838:9
1814:0
1802:3
1797:5
1781:5
1765:9
1743:5
1729:5
1715:9
1652:7
1638:3
1626:9
1611:2
1578:7
1566:3
1554:9
1539:2
1506:7
1495:3
1483:9
1219:15
1187:15
1146:15
1114:15
1030:4
1017:8
946:4
934:8
863:4
852:8
787:.
769:12
738:10
732:39
720:16
714:80
683:39
677:16
671:70
650:39
644:86
523:.
418:.
298:10
286:11
212:,
185:,
155:,
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3201:(
3166:e
3159:t
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2943:.
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2852:.
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2794:.
2765:.
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2724:.
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2564:=
2561:8
2540:4
2532:3
2528:3
2524:3
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2512:2
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2504:0
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2496:3
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2427:3
2411:÷
2403:7
2400:2
2384:3
2368:÷
2360:7
2357:2
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2308:b
2292:c
2289:=
2286:b
2280:a
2266:b
2262:a
2249:b
2245:a
2240:b
2236:a
2225:b
2221:a
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2213:a
2207:b
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2183:b
2180:a
2158:b
2152:a
2129:a
2126:=
2123:b
2117:c
2070:=
2042:0
2039:2
2036:2
2033:7
2030:2
2020:0
2017:7
2014:6
2011:3
2008:2
2005:+
1997:6
1994:5
1991:1
1988:3
1976:4
1973:9
1970:3
1956:4
1953:3
1944:×
1936:8
1933:7
1910:0
1907:7
1904:6
1890:6
1887:5
1884:1
1871:4
1868:9
1865:3
1853:4
1850:3
1847:×
1835:8
1832:7
1811:6
1808:5
1805:1
1794:4
1791:9
1788:3
1778:4
1775:3
1772:×
1762:8
1759:7
1740:4
1737:9
1734:3
1726:4
1723:3
1720:×
1712:8
1709:7
1678:=
1669:3
1649:8
1646:1
1643:2
1631:×
1623:2
1620:7
1575:8
1559:×
1551:2
1548:7
1488:×
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1477:7
1467:2
1415:.
1403:c
1397:a
1394:+
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1373:+
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1367:(
1361:a
1347:.
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1326:b
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1267:a
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1249:a
1216:=
1213:3
1207:5
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1181:)
1178:3
1175:(
1172:)
1169:5
1166:(
1143:=
1140:3
1134:5
1111:=
1108:3
1102:5
1057:=
1027:8
1024:4
1014:0
1011:3
1008:−
1002:2
997:9
993:7
985:2
980:8
943:8
931:0
928:3
925:−
919:2
914:9
910:7
902:2
897:8
849:0
846:3
843:−
837:2
832:9
828:7
820:2
815:8
775:8
741:)
735:+
729:(
723:)
717:+
711:(
674:+
603:3
597:5
577:5
571:3
484:+
398:3
392:8
372:8
366:3
295:=
292:1
264:1
261:=
258:1
255:+
252:0
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117:(
112:)
108:(
98:·
91:·
84:·
77:·
50:.
20:)
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