Wikipedia

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.

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, 12 and 9 are multiplied to get 108, 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
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
2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
3 0 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60
4 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80
5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
6 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120
7 0 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 112 119 126 133 140
8 0 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160
9 0 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180
10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
11 0 11 22 33 44 55 66 77 88 99 110 121 132 143 154 165 176 187 198 209 220
12 0 12 24 36 48 60 72 84 96 108 120 132 144 156 168 180 192 204 216 228 240
13 0 13 26 39 52 65 78 91 104 117 130 143 156 169 182 195 208 221 234 247 260
14 0 14 28 42 56 70 84 98 112 126 140 154 168 182 196 210 224 238 252 266 280
15 0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300
16 0 16 32 48 64 80 96 112 128 144 160 176 192 208 224 240 256 272 288 304 320
17 0 17 34 51 68 85 102 119 136 153 170 187 204 221 238 255 272 289 306 323 340
18 0 18 36 54 72 90 108 126 144 162 180 198 216 234 252 270 288 306 324 342 360
19 0 19 38 57 76 95 114 133 152 171 190 209 228 247 266 285 304 323 342 361 380
20 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

Warring States decimal multiplication table

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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

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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

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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 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.

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

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  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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