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In [[mathematics]], a '''multiplication table''' (sometimes, less formally, a '''times table''') is a [[mathematical table]] used to define a multiplication [[binary operation|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 countries using the [[Imperial system]] of measurement, such as the United States, it is often considered helpful to memorize the table up to 12 × 12.

{| class="wikitable" style="text-align:right;"
! ×
! 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
|}

==Traditional use==
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
<div style="-moz-column-count:2;">
   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 x 10 = 100
  11 x 10 = 110
  12 x 10 = 120
  13 x 10 = 130
  14 x 10 = 140
  15 x 10 = 150
  16 x 10 = 160
  17 x 10 = 170
  18 x 10 = 180
  19 x 10 = 190
  100 x 10 = 1000
</div>

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:

# Look at the 7 in the first picture and follow the arrow.
# 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.
# 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.
# 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.
# Proceed in the same way until the last number, 3, which corresponds to 63.
# Next, use the 0 at the bottom. It corresponds to 70.
# Then, start again with the 7.  This time it will correspond to 77.
# 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 [[group (mathematics)|group]]s, [[field (mathematics)|field]]s, [[ring (mathematics)|ring]]s, and other [[Abstract algebra|algebraic systems]]. In such contexts they can be called [[Cayley table]]s. 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. It is thought by many{{Who|date=November 2009}} that electronic calculators have made it unnecessary or counter-productive to invest time in memorizing the multiplication table. NCTM made it clear in their 2006 [[Principles and Standards for School Mathematics#Curriculum Focal Points|Focal Points]] that basic mathematics facts must be learned, though there is no consensus on whether rote memorization is the best method.

==See also==
* [[Vedic square]]

[[Category:Multiplication]]
[[Category:Mathematics education]]

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