Multiplication 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.

A multiplication table, by Adam Ries (1490s-1559)
18th century version of Napier's bones, a calculation device similar to an abacus

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 123456789101112131415161718192021222324252627282930
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 123456789101112131415161718192021222324252627282930
2 0 24681012141618202224262830323436384042444648505254565860
3 0 369121518212427303336394245485154576063666972757881848790
4 0 4812162024283236404448525660646872768084889296100104108112116120
5 0 5101520253035404550556065707580859095100105110115120125130135140145150
6 0 6121824303642485460667278849096102108114120126132138144150156162168174180
7 0 714212835424956637077849198105112119126133140147154161168175182189196203210
8 0 81624324048566472808896104112120128136144152160168176184192200208216224232240
9 0 918273645546372819099108117126135144153162171180189198207216225234243252261270
10 0 102030405060708090100110120130140150160170180190200210220230240250260270280290300
11 0 112233445566778899110121132143154165176187198209220231242253264275286297308319330
12 0 1224364860728496108120132144156168180192204216228240252264276288300312324336348360
13 0 13263952657891104117130143156169182195208221234247260273286299312325338351364377390
14 0 14284256708498112126140154168182196210224238252266280294308322336350364378392406420
15 0 153045607590105120135150165180195210225240255270285300315330345360375390405420435450
16 0 163248648096112128144160176192208224240256272288304320336352368384400416432448464480
17 0 1734516885102119136153170187204221238255272289306323340357374391408425442459476493510
18 0 1836547290108126144162180198216234252270288306324342360378396414432450468486504522540
19 0 1938577695114133152171190209228247266285304323342361380399418437456475494513532551570
20 0 20406080100120140160180200220240260280300320340360380400420440460480500520540560580600
21 0 21426384105126147168189210231252273294315336357378399420441462483504525546567588609630
22 0 22446688110132154176198220242264286308330352374396418440462484506528550572594616638660
23 0 23466992115138161184207230253276299322345368391414437460483506529552575598621644667690
24 0 24487296120144168192216240264288312336360384408432456480504528552576600624648672696720
25 0 255075100125150175200225250275300325350375400425450475500525550575600625650675700725750
26 0 265278104130156182208234260286312338364390416442468494520546572598624650676702728754780
27 0 275481108135162189216243270297324351378405432459486513540567594621648675702729756783810
28 0 285684112140168196224252280308336364392420448476504532560588616644672700728756784812840
29 0 295887116145174203232261290319348377406435464493522551580609638667696725754783812841870
30 0 306090120150180210240270300330360390420450480510540570600630660690720750780810840870900

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.

Matrix of multiples of 7
Matrix of multiples of 3
Multiplication table of 6 Illustration Multiplication table of 7 Illustration
6 x 1 = 6
6x1
 7 x 1  = 7
7x1
6 x 2  = 12
6x2
 7 x 2  = 14
7x2
6 x 3 = 18
6x3
 7 x 3 = 21
7x3
6 x 4 = 24
6x4
 7 x 4 = 28
7x4
6 x 5 = 30
6x5
 7 x 5 = 35
7x5
6 x 6 = 36
6x6
 7 x 6 = 42
7x6
6 x 7 = 42
6x7
 7 x 7 = 49
7x7
6 x 8 = 48
6x8
 7 x 8 = 56
7x8
6 x 9 = 54
6x9
 7 x 9 = 63
7x9
6 x 10= 60
6x10
 7 x 10= 70
7x10
6 x 11= 66
6x11
 7 x 11= 77
7x11
6 x 12= 72
6x12
 7 x 12= 84
7x12

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.

Addition Table
+ 0 1 2 3 4 5 6 7 8 9 10
0 01234 5 6 7 8 9 10
1 12345 6 7 8 9 10 11
2 23456 7 8 9 10 11 12
3 34567 8 9 10 11 12 13
4 45678 9 10 11 12 13 14
5 56789 10 11 12 13 14 15
6 678910 11 12 13 14 15 16
7 7891011 12 13 14 15 16 17
8 89101112 13 14 15 16 17 18
9 910111213 14 15 16 17 18 19
10 1011121314 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.

Division tables [8]
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. Nature article The 2,300-year-old matrix is the world's oldest decimal multiplication table
  5. "Multiplication & Division Table Charts 0-12 Printable PDF (FREE)" (PDF).
  6. "How to Complete a Division Table". study.com. Retrieved 2020-09-29.
  7. McIntosh, Janine; Ramagge, Jacqui. "The Improving Mathematics Education in Schools (TIMES) Project". Australian mathematical sciences institute.
  8. "Division Tables - From 1 to 12 For Easy Printing: With Customization Options". Helping With Math. 2020-02-28. Retrieved 2020-10-01.
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