Multiplication tables

The multiplication table (also 1 × 1 or 1mal1 ) is a compilation of all products that result from the combination of two natural numbers from 1 to 10, mostly in tabular form. The multiplication table is the extension to natural numbers from 1 to 20. The multiplication table is part of the basic arithmetic knowledge of mathematics and is mostly learned by heart in elementary school .

As basics are metaphorically and basic knowledge of a field of knowledge or a skill called.

application

The ABCs is the written Multiply used both factors for finding the product of each item. For this, only the products from the digit combinations to are required, whereby the products with a factor of 0 are usually left out in the representation, instead the products with a factor of 10 are added based on the tradition of using Roman numerals . ${\ displaystyle \ textstyle 0 \ cdot 0}$ ${\ displaystyle \ textstyle 9 \ cdot 9}$

"But, to shorten the repeated summation of digits, it is expedient to construct a table, which must be engraved in the memory of the arithmetician."

"But in order to shorten the repeated adding of digits, it is useful to make a table that has to be impressed in the memory of the arithmeticist."

- John Leslie : The Philosophy of Arithmetic

This is also used in the written division .

The multiplication table is used to memorize products that are often needed.

presentation

After Adam Ries

Excerpt from Adam Ries' arithmetic book

In the Adam Risen arithmetic book of 1574, the following multiplication table is shown with the note "Above all, you have to know and memorize the once one as follows:" ( Adam Ries )

	times	is		times	is		times	is
1	1	1	2	8th	16	5	5	25th
1	2	2	2	9	18th	5	6th	30th
1	3	3	3	3	9	5	7th	35
1	4th	4th	3	4th	12	5	8th	40
1	5	5	3	5	15th	5	9	45
1	6th	6th	3	6th	18th	6th	6th	36
1	7th	7th	3	7th	21st	6th	7th	42
1	8th	8th	3	8th	24	6th	8th	48
1	9	9	3	9	27	6th	9	54
2	2	4th	4th	4th	16	7th	7th	49
2	3	6th	4th	5	20th	7th	8th	56
2	4th	8th	4th	6th	24	7th	9	63
2	5	10	4th	7th	28	8th	8th	64
2	6th	12	4th	8th	32	8th	9	72
2	7th	14th	4th	9	36	9	9	81

This compact representation dispenses with redundant information using the commutative law (2 · 3 = 3 · 2). It served as an aid when calculating on lines .

table

"Pythagoras board" as Napier ruler

The detailed tabular representation of the multiplication table is ascribed to Pythagoras and therefore also called the Pythagoras board or Pythagoras table in some languages , for example in French , English and Italian , but also in Montessori education .

The following table shows the multiplication table .

*	1	2	3	4th	5	6th	7th	8th	9	10
1	1	2	3	4th	5	6th	7th	8th	9	10
2	2	4th	6th	8th	10	12	14th	16	18th	20th
3	3	6th	9	12	15th	18th	21st	24	27	30th
4th	4th	8th	12	16	20th	24	28	32	36	40
5	5	10	15th	20th	25th	30th	35	40	45	50
6th	6th	12	18th	24	30th	36	42	48	54	60
7th	7th	14th	21st	28	35	42	49	56	63	70
8th	8th	16	24	32	40	48	56	64	72	80
9	9	18th	27	36	45	54	63	72	81	90
10	10	20th	30th	40	50	60	70	80	90	100

The multiplication table is subdivided according to the second factor into the 1 series , 2 series, 3 series etc. up to the 10 series. A table column therefore represents the corresponding row. The first factor is sought in the first column (left), the second factor is sought in the first row (top), and the product is at the intersection of the row and the column.

The following table shows the multiplication table with factors up to 20 (including the multiplication table ).

*	1	2	3	4th	5	6th	7th	8th	9	10	11	12	13	14th	15th	16	17th	18th	19th	20th
1	001	002	003	004th	5	6th	7th	8th	9	10	11	12	13	14th	15th	16	17th	18th	19th	20th
2	2	4th	6th	8th	10	12	14th	16	18th	20th	22nd	24	26th	28	30th	32	34	36	38	40
3	3	6th	9	12	15th	18th	21st	24	27	30th	33	36	39	42	45	48	51	54	57	60
4th	4th	8th	12	16	20th	24	28	32	36	40	44	48	52	56	60	64	68	72	76	80
5	5	10	15th	20th	25th	30th	35	40	45	50	55	60	65	70	75	80	85	90	95	100
6th	6th	12	18th	24	30th	36	42	48	54	60	66	72	78	84	90	96	102	108	114	120
7th	7th	14th	21st	28	35	42	49	56	63	70	77	84	91	98	105	112	119	126	133	140
8th	8th	16	24	32	40	48	56	64	72	80	88	96	104	112	120	128	136	144	152	160
9	9	18th	27	36	45	54	63	72	81	90	99	108	117	126	135	144	153	162	171	180
10	10	20th	30th	40	50	60	70	80	90	100	110	120	130	140	150	160	170	180	190	200
11	11	22nd	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	26th	39	52	65	78	91	104	117	130	143	156	169	182	195	208	221	234	247	260
14th	14th	28	42	56	70	84	98	112	126	140	154	168	182	196	210	224	238	252	266	280
15th	15th	30th	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
17th	17th	34	51	68	85	102	119	136	153	170	187	204	221	238	255	272	289	306	323	340
18th	18th	36	54	72	90	108	126	144	162	180	198	216	234	252	270	288	306	324	342	360
19th	19th	38	57	76	95	114	133	152	171	190	209	228	247	266	285	304	323	342	361	380
20th	20th	40	60	80	100	120	140	160	180	200	220	240	260	280	300	320	340	360	380	400

Rows

Single rows of are square one as shown follows:

1-row 0 1 1 = 01 02 1 = 02 03 1 = 03 04 1 = 04 05 1 = 05 06 1 = 06 07 1 = 07 08 1 = 08 09 1 = 09 10 1 = 10	Row of 0 2 1 2 = 02 02 2 = 04 03 2 = 06 04 2 = 08 05 2 = 10 06 2 = 12 07 2 = 14 08 2 = 16 09 2 = 18 10 2 = 20	Row of 0 3 1 3 = 03 02 3 = 06 03 3 = 09 04 3 = 12 05 3 = 15 06 3 = 18 07 3 = 21 08 3 = 24 09 3 = 27 10 3 = 30	Row of 0 4 1 4 = 04 02 4 = 08 03 4 = 12 04 4 = 16 05 4 = 20 06 4 = 24 07 4 = 28 08 4 = 32 09 4 = 36 10 4 = 40	Series of 5 0 1 5 = 05 02 5 = 10 03 5 = 15 04 5 = 20 05 5 = 25 06 5 = 30 07 5 = 35 08 5 = 40 09 5 = 45 10 5 = 50
Row of 0 6 1 6 = 06 02 6 = 12 03 6 = 18 04 6 = 24 05 6 = 30 06 6 = 36 07 6 = 42 08 6 = 48 09 6 = 54 10 6 = 60	7th row 0 1 7 = 07 02 7 = 14 03 7 = 21 04 7 = 28 05 7 = 35 06 7 = 42 07 7 = 49 08 7 = 56 09 7 = 63 10 7 = 70	Row of 0 8 1 8 = 08 02 8 = 16 03 8 = 24 04 8 = 32 05 8 = 40 06 8 = 48 07 8 = 56 08 8 = 64 09 8 = 72 10 8 = 80	Row of 0 9 1 9 = 09 02 9 = 18 03 9 = 27 04 9 = 36 05 9 = 45 06 9 = 54 07 9 = 63 08 9 = 72 09 9 = 81 10 9 = 90	Row of 10s 0 1 10 = 010 02 10 = 020 03 10 = 030 04 10 = 040 05 10 = 050 06 10 = 060 07 10 = 070 08 10 = 080 09 10 = 090 10 10 = 100

Comparable in other number systems and number fonts

A multiplication table has been handed down in Greek numerals from the time around the birth of Christ . A student's record is considered evidence that multiplication tables were taught and learned at the time.

In 493 Victorius of Aquitaine put together a table with 98 columns to facilitate multiplication and division, in which he stated the products of the numbers from the fractions up to the value 1000 with the numbers from 2 to 50 in Roman numerals , the so-called Calculus Victorii.

For the sexagesimal was by Gaspar Schott , the Tabula Sexagenaria published 1,661th

Web links

Wiktionary: multiplication tables - explanations of meanings, word origins, synonyms, translations

Multiplication aids: historical tables

Individual evidence

^ Stephan Weiss: The Small Multiplication Table through the Centuries in Europe . (PDF) In: Journal of the Oughtred Society , 22, Fall 2013, p. 2.
↑ Stephan Weiss: The multiplication table through the centuries . (PDF) 2015.
^ ^A ^b John Leslie : The Philosophy of Arithmetic . Edinburgh 1820, p. 148 ( limited preview in Google Book search).
↑ Adam Risen arithmetic book on lines and Ziphren in all sorts of handling / business and purchase. Increased with new artificial rules and examples. 1574
↑ from M. Edouard Lucas: Calculating Machines . In: EL Youmans, WJ Youmans (Eds.): Popular Science Monthly . tape 26 . New York 1885, p. 451 (English).
↑ John Farrar: An Elementary Treatise on Arithmetic . Cambridge 1825, p. 17 ( limited preview in Google Book search).
^ Maria Montessori : Developmental materials in the child's school . Götz, Dörfles 2003, ISBN 3-9501011-7-9 (Italian: L'autoeducazione nelle scuole elementari . Translated by Karin Pellegrini).
↑ Stephan Weiss: The multiplier table, its design and use . (PDF) 2003
^ David W. Maher, John F. Makowski: Literary Evidence for Roman Arithmetic with Fractions . In: The University of Chicago (Ed.): Classical Philology . No. 96 , 2001, p. 376–399 (English, dmaher.org [PDF; 1,2 MB ; accessed on January 8, 2013]).
↑ Stephan Weiss: Reconstruction and Background of Gaspar Schott's Tabula Sexagenaria (1661) . (PDF)

[1] Stephan Weiss: The Small Multiplication Table through the Centuries in Europe . (PDF) In: Journal of the Oughtred Society , 22, Fall 2013, p. 2.

[2] Stephan Weiss: The multiplication table through the centuries . (PDF) 2015.

[leslie-3] A ^b John Leslie : The Philosophy of Arithmetic . Edinburgh 1820, p. 148 ( limited preview in Google Book search).

[4] Adam Risen arithmetic book on lines and Ziphren in all sorts of handling / business and purchase. Increased with new artificial rules and examples. 1574

[5] rom M. Edouard Lucas: Calculating Machines . In: EL Youmans, WJ Youmans (Eds.): Popular Science Monthly . tape 26 . New York 1885, p. 451 (English).

[6] John Farrar: An Elementary Treatise on Arithmetic . Cambridge 1825, p. 17 ( limited preview in Google Book search).

[7] Maria Montessori : Developmental materials in the child's school . Götz, Dörfles 2003, ISBN 3-9501011-7-9 (Italian: L'autoeducazione nelle scuole elementari . Translated by Karin Pellegrini).

[8] Stephan Weiss: The multiplier table, its design and use . (PDF) 2003

[MaherMakowski-9] David W. Maher, John F. Makowski: Literary Evidence for Roman Arithmetic with Fractions . In: The University of Chicago (Ed.): Classical Philology . No. 96 , 2001, p. 376–399 (English, dmaher.org [PDF; 1,2 MB ; accessed on January 8, 2013]).

[10] Stephan Weiss: Reconstruction and Background of Gaspar Schott's Tabula Sexagenaria (1661) . (PDF)