Multiplication table

For the Matthew Shipp album, seeThe Multiplication Table.

"Times table" redirects here. For a table of departure and arrival times, seeTimetable (disambiguation).

Inmathematics,amultiplication table(sometimes, less formally, atimes table) is amathematical tableused to define amultiplicationoperationfor an algebraic system.

Multiplication table from 1 to 10 drawn to scale with the upper-right half labeled with prime factorisations

Thedecimalmultiplication 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

Pre-modern times

TheTsinghua Bamboo Slips,ChineseWarring Statesera decimal multiplication table of 305 BC

The oldest known multiplication tables were used by theBabyloniansabout 4000 years ago.However, they used a base of 60.The oldest known tables using a base of 10 are theChinesedecimal multiplication table on bamboo stripsdating to about 305 BC, during China'sWarring Statesperiod.

"Table of Pythagoras" onNapier's bones

The multiplication table is sometimes attributed to the ancient Greek mathematicianPythagoras(570–495 BC). It is also called the Table of Pythagoras in many languages (for example French, Italian and Russian), sometimes in English.TheGreco-RomanmathematicianNichomachus(60–120 AD), a follower ofNeopythagoreanism,included a multiplication table in hisIntroduction to Arithmetic,whereas the oldest survivingGreekmultiplication table is on a wax tablet dated to the 1st century AD and currently housed in theBritish Museum.

In 493 AD,Victorius of Aquitainewrote a 98-column multiplication table which gave (inRoman 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."

Modern times

In his 1820 bookThe Philosophy of Arithmetic,mathematicianJohn Lesliepublished a multiplication table up to 1000 × 1000, which allows numbers to be multiplied in triplets 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 12 × 12, which is a size commonly used nowadays in English-world schools.

×	1	2	3	4	5	6	7	8	9	10	11	12
1	1	2	3	4	5	6	7	8	9	10	11	12
2	2	4	6	8	10	12	14	16	18	20	22	24
3	3	6	9	12	15	18	21	24	27	30	33	36
4	4	8	12	16	20	24	28	32	36	40	44	48
5	5	10	15	20	25	30	35	40	45	50	55	60
6	6	12	18	24	30	36	42	48	54	60	66	72
7	7	14	21	28	35	42	49	56	63	70	77	84
8	8	16	24	32	40	48	56	64	72	80	88	96
9	9	18	27	36	45	54	63	72	81	90	99	108
10	10	20	30	40	50	60	70	80	90	100	110	120
11	11	22	33	44	55	66	77	88	99	110	121	132
12	12	24	36	48	60	72	84	96	108	120	132	144

Because multiplication of integers iscommutative,many schools use a smaller table as below. Some schools even remove the first column since 1 is themultiplicative 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 traditionalrote learningof multiplication was based on memorization of columns in the table, arranged as follows.

0 × 0 = 0  1 × 0 = 0  2 × 0 = 0  3 × 0 = 0  4 × 0 = 0  5 × 0 = 0  6 × 0 = 0  7 × 0 = 0  8 × 0 = 0  9 × 0 = 0 10 × 0 = 0 11 × 0 = 0 12 × 0 = 0	0 × 1 = 0  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	0 × 2 = 0  1 × 2 = 2  2 × 2 = 4  3 × 2 = 6  4 × 2 = 8  5 × 2 = 10  6 × 2 = 12  7 × 2 = 14  8 × 2 = 16  9 × 2 = 18 10 × 2 = 20 11 × 2 = 22 12 × 2 = 24	0 × 3 = 0  1 × 3 = 3  2 × 3 = 6  3 × 3 = 9  4 × 3 = 12  5 × 3 = 15  6 × 3 = 18  7 × 3 = 21  8 × 3 = 24  9 × 3 = 27 10 × 3 = 30 11 × 3 = 33 12 × 3 = 36	0 × 4 = 0  1 × 4 = 4  2 × 4 = 8  3 × 4 = 12  4 × 4 = 16  5 × 4 = 20  6 × 4 = 24  7 × 4 = 28  8 × 4 = 32  9 × 4 = 36 10 × 4 = 40 11 × 4 = 44 12 × 4 = 48
0 × 5 = 0  1 × 5 = 5  2 × 5 = 10  3 × 5 = 15  4 × 5 = 20  5 × 5 = 25  6 × 5 = 30  7 × 5 = 35  8 × 5 = 40  9 × 5 = 45 10 × 5 = 50 11 × 5 = 55 12 × 5 = 60	0 × 6 = 0  1 × 6 = 6  2 × 6 = 12  3 × 6 = 18  4 × 6 = 24  5 × 6 = 30  6 × 6 = 36  7 × 6 = 42  8 × 6 = 48  9 × 6 = 54 10 × 6 = 60 11 × 6 = 66 12 × 6 = 72	0 × 7 = 0  1 × 7 = 7  2 × 7 = 14  3 × 7 = 21  4 × 7 = 28  5 × 7 = 35  6 × 7 = 42  7 × 7 = 49  8 × 7 = 56  9 × 7 = 63 10 × 7 = 70 11 × 7 = 77 12 × 7 = 84	0 × 8 = 0  1 × 8 = 8  2 × 8 = 16  3 × 8 = 24  4 × 8 = 32  5 × 8 = 40  6 × 8 = 48  7 × 8 = 56  8 × 8 = 64  9 × 8 = 72 10 × 8 = 80 11 × 8 = 88 12 × 8 = 96	0 × 9 = 0  1 × 9 = 9  2 × 9 = 18  3 × 9 = 27  4 × 9 = 36  5 × 9 = 45  6 × 9 = 54  7 × 9 = 63  8 × 9 = 72  9 × 9 = 81 10 × 9 = 90 11 × 9 = 99 12 × 9 = 108
0 × 10 = 0  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 10 × 10 = 100 11 × 10 = 110 12 × 10 = 120	0 × 11 = 0  1 × 11 = 11  2 × 11 = 22  3 × 11 = 33  4 × 11 = 44  5 × 11 = 55  6 × 11 = 66  7 × 11 = 77  8 × 11 = 88  9 × 11 = 99 10 × 11 = 110 11 × 11 = 121 12 × 11 = 132	0 × 12 = 0  1 × 12 = 12  2 × 12 = 24  3 × 12 = 36  4 × 12 = 48  5 × 12 = 60  6 × 12 = 72  7 × 12 = 84  8 × 12 = 96  9 × 12 = 108 10 × 12 = 120 11 × 12 = 132 12 × 12 = 144

This form of writing the multiplication table in columns with complete number sentences is still used in some countries, such as Bosnia and Herzegovina,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	↓
	4	5	6
	7	8	9				6		8
←						←
	0		5					0
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 atelephone 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.

In abstract algebra

Tables can also define binary operations ongroups,fields,rings,and otheralgebraic systems.In such contexts they are calledCayley tables.

For every natural numbern,addition and multiplication inZ_n,the ring of integers modulon,is described by annbyntable. (SeeModular arithmetic.) For example, the tables forZ₅are:

+	0	1	2	3	4
0	0	1	2	3	4
1	1	2	3	4	0
2	2	3	4	0	1
3	3	4	0	1	2
4	4	0	1	2	3

×	0	1	2	3	4
0	0	0	0	0	0
1	0	1	2	3	4
2	0	2	4	1	3
3	0	3	1	4	2
4	0	4	3	2	1

For other examples, seegroup.

Hypercomplex numbers

Hypercomplex numbermultiplication tables show the non-commutativeresults of multiplying two hypercomplex imaginary units. The simplest example is that of thequaternionmultiplication table.

Quaternion multiplication table

↓ × →	1	i	j	k
1	1	i	j	k
i	i	−1	k	−j
j	j	−k	−1	i
k	k	j	−i	−1

For further examples, seeOctonion § Multiplication,Sedenion § Multiplication,andTrigintaduonion § Multiplication.

Chinese and Japanese multiplication tables

Main article:Chinese multiplication table

Mokkandiscovered atHeijō Palacesuggest that the multiplication table may have been introduced to Japan through Chinese mathematical treatises such as theSunzi Suanjing,because their expression of the multiplication table share the characterNhưin products less than ten.Chinese and Japanese share a similar system of eighty-one short, easily memorable sentences taught to students to help them learn the multiplication table up to 9 × 9. In current usage, the sentences that express products less than ten include an additional particle in both languages. In the case of modern Chinese, this isĐến(dé); and in Japanese, this isが(ga). This is useful for those who practice calculation with asuanpanor asoroban,because the sentences remind them to shift one column to the right when inputting a product that does not begin with atens digit.In particular, the Japanese multiplication table uses non-standard pronunciations for numbers in some specific instances (such as the replacement ofsan rokuwithsaburoku).

The Japanese multiplication table

×	1ichi	2ni	3san	4shi	5go	6roku	7shichi	8ha	9ku
1in	in'ichi ga ichi	inni ga ni	insan ga san	inshi ga shi	ingo ga go	inroku ga roku	inshichi ga shichi	inhachi ga hachi	inku ga ku
2ni	ni ichi ga ni	ni nin ga shi	ni san ga roku	ni shi ga hachi	ni go jū	ni roku jūni	ni shichi jūshi	ni hachi jūroku	ni ku jūhachi
3san	san ichi ga san	san ni ga roku	sazan ga ku	san shi jūni	san go jūgo	saburoku jūhachi	san shichi nijūichi	sanpa nijūshi	san ku nijūshichi
4shi	shi ichi ga shi	shi ni ga hachi	shi san jūni	shi shi jūroku	shi go nijū	shi roku nijūshi	shi shichi nijūhachi	shi ha sanjūni	shi ku sanjūroku
5go	go ichi ga go	go ni jū	go san jūgo	go shi nijū	go go nijūgo	go roku sanjū	go shichi sanjūgo	go ha shijū	gokku shijūgo
6roku	roku ichi ga roku	roku ni jūni	roku san jūhachi	roku shi nijūshi	roku go sanjū	roku roku sanjūroku	roku shichi shijūni	roku ha shijūhachi	rokku gojūshi
7shichi	shichi ichi ga shichi	shichi ni jūshi	shichi san nijūichi	shichi shi nijūhachi	shichi go sanjūgo	shichi roku shijūni	shichi shichi shijūku	shichi ha gojūroku	shichi ku rokujūsan
8hachi	hachi ichi ga hachi	hachi ni jūroku	hachi san nijūshi	hachi shi sanjūni	hachi go shijū	hachi roku shijūhachi	hachi shichi gojūroku	happa rokujūshi	hakku shichijūni
9ku	ku ichi ga ku	ku ni jūhachi	ku san nijūshichi	ku shi sanjūroku	ku go shijūgo	ku roku gojūshi	ku shichi rokujūsan	ku ha shichijūni	ku ku hachijūichi

Warring States decimal multiplication bamboo slips

A bundle of 21 bamboo slips dated 305 BC in theWarring Statesperiod in theTsinghua Bamboo Slips( Thanh Hoa giản ) collection is the world's earliest known example of a decimal multiplication table.

A modern representation of the Warring States decimal multiplication table used to calculate 12 × 34.5

Standards-based mathematics reform in the US

In 1989, theNational 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 asInvestigations in Numbers, Data, and Space(widely known asTERCafter its producer, Technical Education Research Centers) omitted aids such as multiplication tables in early editions. NCTM made it clear in their 2006Focal Pointsthat 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.

See also

• Vedic square
• IBM 1620,an early computer that used tables stored in memory to perform addition and multiplication