# Multiplication table

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Multiplication table

In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system.

The decimal multiplication table was traditionally taught as an essential part of elementary arithmetic around the world, as it lays the foundation for arithmetic operations with our base-ten numbers. Many educators believe it is necessary to memorize the table up to 9 × 9.

In his 1820 book The Philosophy of Arithmetic, mathematician John Leslie published a multiplication table up to 99 × 99, which allows numbers to be multiplied in pairs of digits at a time. Leslie also recommended that young pupils memorize the multiplication table up to 25 × 25.

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

The table is sometimes attributed to Pythagoras. It is also called the Table of Pythagoras in many languages (for example French, Italian and Russian), sometimes in English.

In 493 A.D., Victorius of Aquitaine wrote a 98-column multiplication table which gave (in Roman numerals) the product of every number from 2 to 50 times and the rows were "a list of numbers starting with one thousand, descending by hundreds to one hundred, then descending by tens to ten, then by ones to one, and then the fractions down to 1/144" (Maher & Makowski 2001, p.383)

The traditional rote learning of multiplication was based on memorization of columns in the table, in a form like

```  1 × 10 = 10
2 × 10 = 20
3 × 10 = 30
4 × 10 = 40
5 × 10 = 50
6 × 10 = 60
7 × 10 = 70
8 × 10 = 80
9 × 10 = 90
```

This form of writing the multiplication table in columns with complete number sentences is still used in some countries instead of the modern grid above.

## Patterns in the tables

There is a pattern in the multiplication table that can help people to memorize the table more easily. It uses the figures below:

```    →                 →
1 2 3             2   4
↑ 4 5 6 ↓         ↑       ↓
7 8 9             6   8
←                 ←
0                 0
Fig. 1             Fig. 2
```

For example, to memorize all the multiples of 7:

1. Look at the 7 in the first picture and follow the arrow.
2. The next number in the direction of the arrow is 4. So think of the next number after 7 that ends with 4, which is 14.
3. The next number in the direction of the arrow is 1. So think of the next number after 14 that ends with 1, which is 21.
4. After coming to the top of this column, start with the bottom of the next column, and travel in the same direction. The number is 8. So think of the next number after 21 that ends with 8, which is 28.
5. Proceed in the same way until the last number, 3, which corresponds to 63.
6. Next, use the 0 at the bottom. It corresponds to 70.
7. Then, start again with the 7. This time it will correspond to 77.
8. Continue like this.

Figure 1 is used for multiples of 1, 3, 7, and 9. Figure 2 is used for the multiples of 2, 4, 6, and 8. These patterns can be used to memorize the multiples of any number from 1 to 9, except 5.

## In abstract algebra

Multiplication tables can also define binary operations on groups, fields, rings, and other algebraic systems. In such contexts they can be called Cayley tables. For an example, see octonion.

## Standards-based mathematics reform in the USA

In 1989, the National Council of Teachers of Mathematics (NCTM) developed new standards which were based on the belief that all students should learn higher-order thinking skills, and which recommended reduced emphasis on the teaching of traditional methods that relied on rote memorization, such as multiplication tables. Widely adopted texts such as Investigations in Numbers, Data, and Space (widely known as TERC after its producer, Technical Education Research Centers) omitted aids such as multiplication tables in early editions. NCTM made it clear in their 2006 Focal Points that basic mathematics facts must be learned, though there is no consensus on whether rote memorization is the best method.

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### Look at other dictionaries:

• Multiplication table — Multiplication Mul ti*pli*ca tion, n. [L. multiplicatio: cf. F. multiplication. See {Multiply}.] 1. The act or process of multiplying, or of increasing in number; the state of being multiplied; as, the multiplication of the human species by… …   The Collaborative International Dictionary of English

• multiplication table — multiplication tables N COUNT A multiplication table is a list of the multiplications of numbers between one and twelve. Children often have to learn multiplication tables at school. Syn: table …   English dictionary

• multiplication table — n. a table for memorization showing the results of multiplying each number of a series, usually 1 to 12, by each of the numbers in succession …   English World dictionary

• multiplication table — ► NOUN ▪ a list of multiples of a particular number, typically from 1 to 12 …   English terms dictionary

• multiplication table — UK / US noun [countable] Word forms multiplication table : singular multiplication table plural multiplication tables maths a list that shows the results of multiplying all the combinations of two numbers between 1 and 12 together …   English dictionary

• multiplication table — noun : a table of the products of a set of numbers multiplied in some regular order; usually : a table of the products of the first 10 or 12 integers multiplied successively by 1, 2, 3, etc., up to 10 or 12 * * * noun, pl ⋯ tables [count]… …   Useful english dictionary

• multiplication table — table of the multiples of the numbers 1 10 …   English contemporary dictionary

• multiplication table — noun A table showing the products of each of the integers from 1 to 10 (or, especially formerly, 12) with each of the integers from 1 to 10 (or 12) …   Wiktionary

• multiplication table — multipli cation .table n a list, used especially by children in school, that shows the results when each number between one and twelve is multiplied by each number between one and twelve …   Dictionary of contemporary English

• multiplication table — multipli cation ,table noun count a list that shows the results of multiplying all the combinations of two numbers between 1 and 12 together …   Usage of the words and phrases in modern English