# Multiplication table

mathematical table

A multiplication table is a tool used to learn how to multiply two numbers. The oldest known multiplication tables were written by the Babylonians about 4000 years ago.[1] Many people think it is important to know how to multiply two numbers by heart, usually up to 12 × 12, 30 × 30, 50 × 50, or 100 × 100.

Most children are introduced to the two, five and 10 times tables by year two - at the age of six and seven. Between the age of seven and eight, children start to learn the three, four and eight times tables.[2] The hardest multiplication is 6×8, which students got wrong 63% of the time. This was closely followed by 8×6, then 11×12, 12×8 and 8×12. The easiest multiplication, on the other hand, was 1×12, which students got wrong less than 5% of the time, followed by 1×6 and 9×1.[3]

In a multiplication table, a number on the first column is multiplied by a number on the first row. The number they corner up to is the answer. In the table below, 21 and 18 are multiplied to get 378, using the table. The numbers in bold are squares (numbers multiplied by themselves).

× 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60
3 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90
4 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 104 108 112 116 120
5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150
6 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 126 132 138 144 150 156 162 168 174 180
7 0 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 112 119 126 133 140 147 154 161 168 175 182 189 196 203 210
8 0 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160 168 176 184 192 200 208 216 224 232 240
9 0 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180 189 198 207 216 225 234 243 252 261 270
10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300
11 0 11 22 33 44 55 66 77 88 99 110 121 132 143 154 165 176 187 198 209 220 231 242 253 264 275 286 297 308 319 330
12 0 12 24 36 48 60 72 84 96 108 120 132 144 156 168 180 192 204 216 228 240 252 264 276 288 300 312 324 336 348 360
13 0 13 26 39 52 65 78 91 104 117 130 143 156 169 182 195 208 221 234 247 260 273 286 299 312 325 338 351 364 377 390
14 0 14 28 42 56 70 84 98 112 126 140 154 168 182 196 210 224 238 252 266 280 294 308 322 336 350 364 378 392 406 420
15 0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 360 375 390 405 420 435 450
16 0 16 32 48 64 80 96 112 128 144 160 176 192 208 224 240 256 272 288 304 320 336 352 368 384 400 416 432 448 464 480
17 0 17 34 51 68 85 102 119 136 153 170 187 204 221 238 255 272 289 306 323 340 357 374 391 408 425 442 459 476 493 510
18 0 18 36 54 72 90 108 126 144 162 180 198 216 234 252 270 288 306 324 342 360 378 396 414 432 450 468 486 504 522 540
19 0 19 38 57 76 95 114 133 152 171 190 209 228 247 266 285 304 323 342 361 380 399 418 437 456 475 494 513 532 551 570
20 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 500 520 540 560 580 600
21 0 21 42 63 84 105 126 147 168 189 210 231 252 273 294 315 336 357 378 399 420 441 462 483 504 525 546 567 588 609 630
22 0 22 44 66 88 110 132 154 176 198 220 242 264 286 308 330 352 374 396 418 440 462 484 506 528 550 572 594 616 638 660
23 0 23 46 69 92 115 138 161 184 207 230 253 276 299 322 345 368 391 414 437 460 483 506 529 552 575 598 621 644 667 690
24 0 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 384 408 432 456 480 504 528 552 576 600 624 648 672 696 720
25 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625 650 675 700 725 750
26 0 26 52 78 104 130 156 182 208 234 260 286 312 338 364 390 416 442 468 494 520 546 572 598 624 650 676 702 728 754 780
27 0 27 54 81 108 135 162 189 216 243 270 297 324 351 378 405 432 459 486 513 540 567 594 621 648 675 702 729 756 783 810
28 0 28 56 84 112 140 168 196 224 252 280 308 336 364 392 420 448 476 504 532 560 588 616 644 672 700 728 756 784 812 840
29 0 29 58 87 116 145 174 203 232 261 290 319 348 377 406 435 464 493 522 551 580 609 638 667 696 725 754 783 812 841 870
30 0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 480 510 540 570 600 630 660 690 720 750 780 810 840 870 900

## Warring States decimal multiplication table

A group of 21 strips of bamboo from 305 BC from the Warring States period is the world's oldest known decimal multiplication table.[4]

This is a Warring States decimal multiplication table used to find 12 × 34.5.

## Examples

The traditional form of multiplication tables are written in columns with complete number sentences, instead of the standard modern grid. This form is also taught in the schools. Some examples of traditional form of multiplication tables are multiplication tables of 6 and 7 given below.

Multiplication table of 6 Illustration Multiplication table of 7 Illustration
6 x 1 = 6 7 x 1  = 7
6 x 2  = 12 7 x 2  = 14
6 x 3 = 18 7 x 3 = 21
6 x 4 = 24 7 x 4 = 28
6 x 5 = 30 7 x 5 = 35
6 x 6 = 36 7 x 6 = 42
6 x 7 = 42 7 x 7 = 49
6 x 8 = 48 7 x 8 = 56
6 x 9 = 54 7 x 9 = 63
6 x 10= 60 7 x 10= 70
6 x 11= 66 7 x 11= 77
6 x 12= 72 7 x 12= 84

## Other operations

Addition and Division can also have their own tables. Similarly, Subtraction can also have its own table, although it is not commonly used.

+ 0 1 2 3 4 5 6 7 8 9 10
0 0 1 2 3 4 5 6 7 8 9 10
1 1 2 3 4 5 6 7 8 9 10 11
2 2 3 4 5 6 7 8 9 10 11 12
3 3 4 5 6 7 8 9 10 11 12 13
4 4 5 6 7 8 9 10 11 12 13 14
5 5 6 7 8 9 10 11 12 13 14 15
6 6 7 8 9 10 11 12 13 14 15 16
7 7 8 9 10 11 12 13 14 15 16 17
8 8 9 10 11 12 13 14 15 16 17 18
9 9 10 11 12 13 14 15 16 17 18 19
10 10 11 12 13 14 15 16 17 18 19 20

In mathematics, a division table, like multiplication table, is a mathematical table used to define a division operation for an algebraic system, or to obtain the solution to a certain equation.[5][6] The division symbol ÷ is used in the division table, known as the obelus. It was first used to signify division in 1659. Mathematicians however, almost never use the ÷ symbol for division. Instead they use fraction notation, called the vinculum.[7] Division tables are used for finding the Quotient in the Long division.

 1 ÷ 1 = 1 2 ÷ 1 = 2 3 ÷ 1 = 3 4 ÷ 1 = 4 5 ÷ 1 = 5 6 ÷ 1 = 6 7 ÷ 1 = 7 8 ÷ 1 = 8 9 ÷ 1 = 9 10 ÷ 1 = 10 11 ÷ 1 = 11 12 ÷ 1 = 12 2 ÷ 2 = 1 4 ÷ 2 = 2 6 ÷ 2 = 3 8 ÷ 2 = 4 10 ÷ 2 = 5 12 ÷ 2 = 6 14 ÷ 2 = 7 16 ÷ 2 = 8 18 ÷ 2 = 9 20 ÷ 2 = 10 22 ÷ 2 = 11 24 ÷ 2 = 12 3 ÷ 3 = 1 6 ÷ 3 = 2 9 ÷ 3 = 3 12 ÷ 3 = 4 15 ÷ 3 = 5 18 ÷ 3 = 6 21 ÷ 3 = 7 24 ÷ 3 = 8 27 ÷ 3 = 9 30 ÷ 3 = 10 33 ÷ 3 = 11 36 ÷ 3 = 12 4 ÷ 4 = 1 8 ÷ 4 = 2 12 ÷ 4 = 3 16 ÷ 4 = 4 20 ÷ 4 = 5 24 ÷ 4 = 6 28 ÷ 4 = 7 32 ÷ 4 = 8 36 ÷ 4 = 9 40 ÷ 4 = 10 44 ÷ 4 = 11 48 ÷ 4 = 12 5 ÷ 5 = 1 10 ÷ 5 = 2 15 ÷ 5 = 3 20 ÷ 5 = 4 25 ÷ 5 = 5 30 ÷ 5 = 6 35 ÷ 5 = 7 40 ÷ 5 = 8 45 ÷ 5 = 9 50 ÷ 5 = 10 55 ÷ 5 = 11 60 ÷ 5 = 12 6 ÷ 6 = 1 12 ÷ 6 = 2 18 ÷ 6 = 3 24 ÷ 6 = 4 30 ÷ 6 = 5 36 ÷ 6 = 6 42 ÷ 6 = 7 48 ÷ 6 = 8 54 ÷ 6 = 9 60 ÷ 6 = 10 66 ÷ 6 = 11 72 ÷ 6 = 12 7 ÷ 7 = 1 14 ÷ 7 = 2 21 ÷ 7 = 3 28 ÷ 7 = 4 35 ÷ 7 = 5 42 ÷ 7 = 6 49 ÷ 7 = 7 56 ÷ 7 = 8 63 ÷ 7 = 9 70 ÷ 7 = 10 77 ÷ 7 = 11 84 ÷ 7 = 12 8 ÷ 8 = 1 16 ÷ 8 = 2 24 ÷ 8 = 3 32 ÷ 8 = 4 40 ÷ 8 = 5 48 ÷ 8 = 6 56 ÷ 8 = 7 64 ÷ 8 = 8 72 ÷ 8 = 9 80 ÷ 8 = 10 88 ÷ 8 = 11 96 ÷ 8 = 12 9 ÷ 9 = 1 18 ÷ 9 = 2 27 ÷ 9 = 3 36 ÷ 9 = 4 45 ÷ 9 = 5 54 ÷ 9 = 6 63 ÷ 9 = 7 72 ÷ 9 = 8 81 ÷ 9 = 9 90 ÷ 9 = 10 99 ÷ 9 = 11 108 ÷ 9 = 12 10 ÷ 10 = 1 20 ÷ 10 = 2 30 ÷ 10 = 3 40 ÷ 10 = 4 50 ÷ 10 = 5 60 ÷ 10 = 6 70 ÷ 10 = 7 80 ÷ 10 = 8 90 ÷ 10 = 9 100 ÷ 10 = 10 110 ÷ 10 = 11 120 ÷ 10 = 12 11 ÷ 11 = 1 22 ÷ 11 = 2 33 ÷ 11 = 3 44 ÷ 11 = 4 55 ÷ 11 = 5 66 ÷ 11 = 6 77 ÷ 11 = 7 88 ÷ 11 = 8 99 ÷ 11 = 9 110 ÷ 11 = 10 121 ÷ 11 = 11 132 ÷ 11 = 12 12 ÷ 12 = 1 24 ÷ 12 = 2 36 ÷ 12 = 3 48 ÷ 12 = 4 60 ÷ 12 = 5 72 ÷ 12 = 6 84 ÷ 12 = 7 96 ÷ 12 = 8 108 ÷ 12 = 9 120 ÷ 12 = 10 132 ÷ 12 = 11 144 ÷ 12 = 12

## References

1. Jane Qiu (January 7, 2014). "Ancient times table hidden in Chinese bamboo strips". Nature News. doi:10.1038/nature.2014.14482. S2CID 130132289.
2. "Nine-year-olds should recite times tables by heart, says Schools Minister". The Telegraph. Retrieved 2020-09-29.
3. "Which times table do you find the hardest?". 3P Learning. 2014-02-21. Archived from the original on 2020-09-23. Retrieved 2020-09-29.
4. "How to Complete a Division Table". study.com. Retrieved 2020-09-29.
5. McIntosh, Janine; Ramagge, Jacqui. "The Improving Mathematics Education in Schools (TIMES) Project". Australian mathematical sciences institute.
6. "Division Tables - From 1 to 12 For Easy Printing: With Customization Options". Helping With Math. 2020-02-28. Retrieved 2020-10-01.