# 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 base-ten numbers. Many educators believe it is necessary to memorize the table up to 9 × 9.

## History

### In pre-modern time

The oldest known multiplication tables were used by the Babylonians about 4000 years ago. However, they used a base of 60. The oldest known tables using a base of 10 are the Chinese decimal multiplication table on bamboo strips dating to about 305 BC, during China's Warring States period.

The multiplication table is sometimes attributed to the ancient Greek mathematician Pythagoras (570–495 BC). It is also called the Table of Pythagoras in many languages (for example French, Italian and Russian), sometimes in English. The Greco-Roman mathematician Nichomachus (60–120 AD), a follower of Neopythagoreanism, included a multiplication table in his Introduction to Arithmetic, whereas the oldest surviving Greek multiplication table is on a wax tablet dated to the 1st century AD and currently housed in the British Museum.

In 493 AD, 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."

### In modern time

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 50 × 50.

The illustration below shows a table up to 20 × 20, which is a size commonly used nowadays in English-world schools.

× 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 91 104 117 130 145 156 169 182 195 208 221 234 247 260
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 85 102 119 126 153 170 187 204 221 238 255 272 289 306 323 340
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

In China, however, because multiplication of integers is commutative, many schools use a smaller table as below. Some schools even remove the first column since 1 is the multiplicative identity.

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

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, such as Bosnia and Herzegovina,[citation needed] instead of the modern grids 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 → → ↑ ↓ ↑ ↓ ← ← Figure 1: Odd Figure 2: Even Cycles of the unit digit of multiples of integers ending in 1, 3, 7 and 9 (upper row), and 2, 4, 6 and 8 (lower row) on a telephone keypad

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 0 to 10, except 5. As you would start on the number you are multiplying, when you multiply by 0, you stay on 0 (0 is external and so the arrows have no effect on 0, otherwise 0 is used as a link to create a perpetual cycle). The pattern also works with multiples of 10, by starting at 1 and simply adding 0, giving you 10, then just apply every number in the pattern to the "tens" unit as you would normally do as usual to the "ones" unit.

For example, to recall 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, corresponding 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.

### Multiplication by 6 to 10

Multiplying two whole numbers, each from 6 to 10 can be achieved using fingers and thumbs as follows:

1. Number the fingers and thumbs from 10 to 6, then 6 to 10 from left to right, as in the figure.
2. Bend the finger or thumb on each hand corresponding to each number, and all the fingers between them.
3. The number of bent fingers or thumbs gives the tens digit.
4. To the above is added the product of the unbent fingers or thumbs on the left and right sides.

### Multiplication by 9

Nine is unique in that it is the only digit for which the multiples change by one for both the ten’s and the one's place.  This enabled the development of lots of tricks/methods to calculate the 9s.  This is the simplest. It uses only ONE number. ONE number stays in the brain's 'register' for two of the easiest operations in math.  What's more, it is based on Make10.  Make10 is a FOUNDATIONAL math skill that we WANT students to practice (not using their hands).  There are dozens of Make-10 exercises.  This one is easier than most (since there is no remainder) AND it results in students learning the 9s.  The calculation is so fast that memorization is no longer necessary.  Takes less than a minute to teach this to a student that knows Make10.  Just add a simple count back.

Both steps are on the Multiplier:

1. Subtract 1 for the 1st digit. For 9 x 4, 4 - 1 = 3.
2. Make10 with the Multiplier for the 2nd digit = 6.

Note for those unfamiliar with Make10. In this eg, we added 6 to 4 to make10. After K and 1st graders learn Make10, they do not do it by going through an equation. They fill the 'space' between the starting number and 10. The mind snaps forward to 10. That's why this method is so fast. You can now calculate faster than you can say or write - so no need to take 1 or 2 hours to memorize the 9-multiples.

Hands Method for 9-Muliples
1. Number the fingers and thumbs from 1 to 10 from left to right.
2. Bend the finger or thumb corresponding to the number.
3. The number of fingers or thumb to the left of the bend gives the tens digit (if none, the digit is zero).
4. The number of fingers or thumb to the right of the bend gives the units digit (if none, the digit is zero).

## In abstract algebra

Tables can also define binary operations on groups, fields, rings, and other algebraic systems. In such contexts they are called Cayley tables. Here are the addition and multiplication tables for the finite field Z5:

• for every natural number n, there are also addition and multiplication tables for the ring Zn.

For other examples, see group, and octonion.

## Chinese multiplication table

The Chinese multiplication table consists of eighty-one sentences with four or five Chinese characters per sentence, making it easy for children to learn by heart. A shorter version of the table consists of only forty-five sentences, as terms such as "nine eights beget seventy-two" are identical to "eight nines beget seventy-two" so there is no need to learn them twice. A minimum version by removing all "one" sentences, consists of only thirty-six sentences, which is most commonly used in schools in China. It is often in this order: 2x2=4, 2x3=6, ..., 2x8=16, 2x9=18, 3x3, 3x4, ..., 3x9, 4x4, ..., 4x9, 5x5,...,9x9

## Warring States decimal multiplication bamboo slips

A bundle of 21 bamboo slips dated 305 BC in the Warring States period in the Tsinghua Bamboo Slips (清華簡) collection is the world's earliest known example of a decimal multiplication table.

## Standards-based mathematics reform in the US

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, 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. In recent years, a number of nontraditional methods have been devised to help children learn multiplication facts, including video-game style apps and books that aim to teach times tables through character-based stories.