Multiplication table

Amultiplication tableis a tool used to learn how tomultiplytwo numbers. The oldest known multiplication tables were written by theBabyloniansabout 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, 12 and 9 are multiplied to get 108, using the table. The numbers in bold aresquares(numbers multiplied by themselves).

×	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
1	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
2	2	4	6	8	10	12	14	16	18	20	22	24	26	28	30	32	34	36	38	40
3	3	6	9	12	15	18	21	24	27	30	33	36	39	42	45	48	51	54	57	60
4	4	8	12	16	20	24	28	32	36	40	44	48	52	56	60	64	68	72	76	80
5	5	10	15	20	25	30	35	40	45	50	55	60	65	70	75	80	85	90	95	100
6	6	12	18	24	30	36	42	48	54	60	66	72	78	84	90	96	102	108	114	120
7	7	14	21	28	35	42	49	56	63	70	77	84	91	98	105	112	119	126	133	140
8	8	16	24	32	40	48	56	64	72	80	88	96	104	112	120	128	136	144	152	160
9	9	18	27	36	45	54	63	72	81	90	99	108	117	126	135	144	153	162	171	180
10	10	20	30	40	50	60	70	80	90	100	110	120	130	140	150	160	170	180	190	200
11	11	22	33	44	55	66	77	88	99	110	121	132	143	154	165	176	187	198	209	220
12	12	24	36	48	60	72	84	96	108	120	132	144	156	168	180	192	204	216	228	240
13	13	26	39	52	65	78	91	104	117	130	143	156	169	182	195	208	221	234	247	260
14	14	28	42	56	70	84	98	112	126	140	154	168	182	196	210	224	238	252	266	280
15	15	30	45	60	75	90	105	120	135	150	165	180	195	210	225	240	255	270	285	300
16	16	32	48	64	80	96	112	128	144	160	176	192	208	224	240	256	272	288	304	320
17	17	34	51	68	85	102	119	136	153	170	187	204	221	238	255	272	289	306	323	340
18	18	36	54	72	90	108	126	144	162	180	198	216	234	252	270	288	306	324	342	360
19	19	38	57	76	95	114	133	152	171	190	209	228	247	266	285	304	323	342	361	380
20	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

A group of 21 strips ofbamboofrom 305BCfrom the Warring States period is the world's oldest knowndecimalmultiplication 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 theschools.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	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

AdditionandDivisioncan also have their own tables. Similarly,Subtractioncan 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

Inmathematics,adivision table,like multiplication table, is a mathematical table used to define adivision operationfor 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 theobelus.It was first used to signify division in 1659. Mathematicians however, almost never use the ÷ symbol for division. Instead they usefraction notation,called thevinculum.^[7]Division tables are used for finding theQuotientin theLong 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

↑Jane Qiu (January 7, 2014)."Ancient times table hidden in Chinese bamboo strips".Nature News.doi:10.1038/nature.2014.14482.S2CID 130132289.
↑"Nine-year-olds should recite times tables by heart, says Schools Minister".The Telegraph.Retrieved2020-09-29.
↑"Which times table do you find the hardest?".3P Learning.2014-02-21. Archived fromthe originalon 2020-09-23.Retrieved2020-09-29.
↑NaturearticleThe 2,300-year-old matrix is the world's oldest decimal multiplication table
↑"Multiplication & Division Table Charts 0-12 Printable PDF (FREE)"(PDF).
↑"How to Complete a Division Table".study.Retrieved2020-09-29.
↑McIntosh, Janine; Ramagge, Jacqui."The Improving Mathematics Education in Schools (TIMES) Project".Australian mathematical sciences institute.
↑"Division Tables - From 1 to 12 For Easy Printing: With Customization Options".Helping With Math.2020-02-28.Retrieved2020-10-01.

[Qiu-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.Retrieved2020-09-29.

[3] "Which times table do you find the hardest?".3P Learning.2014-02-21. Archived fromthe originalon 2020-09-23.Retrieved2020-09-29.

[4] NaturearticleThe 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.Retrieved2020-09-29.

[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.Retrieved2020-10-01.

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