# Multiplication table

For the Matthew Shipp album, see The Multiplication Table.
"Times table" redirects here. For a table of departure and arrival times, see Timetable (disambiguation).

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.

Visual representation of the different multiplication tables from 2 to 50

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.[citation needed]

## History

The Warring States decimal multiplication table of 305 B.C

The oldest known multiplication tables were used by the Babylonians about 4000 years ago. They used base 60.[1] The oldest known tables using base 10 are the decimal multiplication table on bamboo strips dating to about 305 BC, found in China.[1]

"Table of Pythagoras" on Napier's bones [2]

The table is sometimes attributed to Pythagoras. It is also called the Table of Pythagoras in many languages (for example French, Italian and apparently Russian long ago), sometimes in English.[3]

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)

In his 1820 book The Philosophy of Arithmetic,[4] 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.

01234567891011121314151617181920
11234567891011121314151617181920
2246810121416182022242628303234363840
33691215182124273033363942454851545760
448121620242832364044485256606468727680
55101520253035404550556065707580859095100
66121824303642485460667278849096102108114120
7714212835424956637077849198105112119126133140
881624324048566472808896104112120128136144152160
9918273645546372819099108117126135144153162171180
10102030405060708090100110120130140150160170180190200
11112233445566778899110121132143154165176187198209220
121224364860728496108120132144156168180192204216228240
1313263952657891104117130143156169182195208221234247260
1414284256708498112126140154168182196210224238252266280
15153045607590105120135150165180195210225240255270285300
16163248648096112128144160176192208224240256272288304320
171734516885102119136153170187204221238255272289306323340
181836547290108126144162180198216234252270288306324342360
191938577695114133152171190209228247266285304323342361380
2020406080100120140160180200220240260280300320340360380400

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 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.

## In abstract algebra

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

+01234
001234
112340
223401
334012
440123
×01234
000000
101234
202413
303142
404321

For other examples, see group, and octonion.

## Chinese multiplication table

The Chinese multiplication table consists of eighty-one sentences with five Chinese characters per sentence, making it is 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.

## Warring States Decimal multiplication bamboo slips

A bundle of 21 bamboo slips dated 305 B.C in the Warring States period in the Tsinghua Bamboo Slips(清华简）collection turns out to be the worlds earliest artifacts of decimal multiplication table.[5]

A diagram of Warring States decimal multiplication table to calculate 22 x 35

## 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.