Editing Multiplication table (section)

===Modern times===
In his 1820 book ''The Philosophy of Arithmetic'',<ref>{{cite book |last=Leslie |first=John |year=1820 |title=The Philosophy of Arithmetic; Exhibiting a Progressive View of the Theory and Practice of Calculation, with Tables for the Multiplication of Numbers as Far as One Thousand |publisher=Abernethy & Walker |location=Edinburgh}}</ref> mathematician [[John Leslie (physicist)|John Leslie]] published a table of "quarter-squares" which could be used, with some additional steps, for multiplication up to 1000 × 1000. Leslie also recommended that young pupils memorize the multiplication table up to 50 × 50. 

In 1897, [[August Leopold Crelle]] published ''Calculating tables giving the products of every two numbers from one to one thousand''<ref>{{cite web | url=https://archive.org/details/cu31924032190476/ | title=Calculating tables giving the products of every two numbers from one to one thousand and their application to the multiplication and division of all numbers above one thousand | date=1897 }}</ref> which is a simple multiplication table for products up to 1000 × 10000.

The illustration below shows a table up to 12 × 12, which is a size commonly used nowadays in English-world schools.

<div style="margin-left:4em">
{|class="wikitable" style="text-align: right;"
!style="width:7.14%"|×
!style="text-align: right; width:7.14%"|1
!style="text-align: right; width:7.14%"|2
!style="text-align: right; width:7.14%"|3
!style="text-align: right; width:7.14%"|4
!style="text-align: right; width:7.14%"|5
!style="text-align: right; width:7.14%"|6
!style="text-align: right; width:7.14%"|7
!style="text-align: right; width:7.14%"|8
!style="text-align: right; width:7.14%"|9
!style="text-align: right; width:7.14%"|10
!style="text-align: right; width:7.14%"|11
!style="text-align: right; width:7.14%"|12
|-
! style="text-align: right;" |1
| 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12
|-
! style="text-align: right;" |2
| 2 || 4 || 6 || 8 || 10 || 12 || 14 || 16 || 18 || 20 || 22 || 24
|-
! style="text-align: right;" |3
| 3 || 6 || 9 || 12 || 15 || 18 || 21|| 24 || 27 || 30 || 33 || 36
|-
! style="text-align: right;" |4
| 4 || 8 || 12 || 16 || 20 || 24 || 28 || 32 || 36 || 40 || 44 || 48
|-
! style="text-align: right;" |5
| 5 || 10 || 15 || 20 || 25 || 30 || 35 || 40 || 45 || 50 || 55 || 60
|-
! style="text-align: right;" |6
| 6 || 12 || 18 || 24 || 30 || 36 || 42 || 48 || 54 || 60 || 66 || 72
|-
! style="text-align: right;" |7
| 7 || 14 || 21 || 28 || 35 || 42 || 49 || 56 || 63 || 70 || 77 || 84
|-
! style="text-align: right;" |8
| 8 || 16 || 24 || 32 || 40 || 48 || 56 || 64 || 72 || 80 || 88 || 96
|-
! style="text-align: right;" |9
| 9 || 18 || 27 || 36 || 45 || 54 || 63 || 72 || 81 || 90 || 99 || 108
|-
! style="text-align: right;" |10
| 10 || 20 || 30 || 40 || 50 || 60 || 70 || 80 || 90 || 100 || 110 || 120
|-
! style="text-align: right;" |11
| 11 || 22 || 33 || 44 || 55 || 66 || 77 || 88 || 99 || 110 || 121 || 132
|-
! style="text-align: right;" |12
| 12 || 24 || 36 || 48 || 60 || 72 || 84 || 96 || 108 || 120 || 132 || 144
|}
</div>

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]].{{cn|date=June 2024}}

<div style="margin-left:4em">
{|class="wikitable" style="text-align: right;"
|-
!style="text-align: right;"|1
| 1 || style="border-top: solid white 1px; border-right: solid white 1px; background: white" | || colspan=7, style="border-top: solid white 1px; border-right: solid white 1px; border-bottom: solid white 1px; background: white" |
|-
!style="text-align: right;"|2
| 2 || 4 || style="border-top: solid white 1px; border-right: solid white 1px; background: white" | || colspan=6, style="border-top: solid white 1px; border-right: solid white 1px; border-bottom: solid white 1px; background: white" |
|-
!style="text-align: right;|3
| 3 || 6 || 9 || style="border-top: solid white 1px; border-right: solid white 1px; background: white" | || colspan=5, style="border-top: solid white 1px; border-right: solid white 1px; border-bottom: solid white 1px; background: white" |
|-
!style="text-align: right;"|4
| 4 || 8 || 12 || 16 || style="border-top: solid white 1px; border-right: solid white 1px; background: white" | || colspan=4, style="border-top: solid white 1px; border-right: solid white 1px; border-bottom: solid white 1px; background: white" |
|-
!style="text-align: right;"|5
| 5 || 10 || 15 || 20 || 25 || style="border-top: solid white 1px; border-right: solid white 1px; background: white" | || colspan=3, style="border-top: solid white 1px; border-right: solid white 1px; border-bottom: solid white 1px; background: white" |
|-
!style="text-align: right;"|6
| 6 || 12 || 18 || 24 || 30 || 36 || style="border-top: solid white 1px; border-right: solid white 1px; background: white" | || colspan=2, style="border-top: solid white 1px; border-right: solid white 1px; border-bottom: solid white 1px; background: white" |
|-
!style="text-align: right;"|7
| 7 || 14 || 21 || 28 || 35 || 42 || 49 || style="border-top: solid white 1px; border-right: solid white 1px; background: white" | || style="border-top: solid white 1px; border-right: solid white 1px; border-bottom: solid white 1px; background: white" |
|-
!style="text-align: right;"|8
| 8 || 16 || 24 || 32 || 40 || 48 || 56 || 64 || style="border-top: solid white 1px; border-right: solid white 1px; background: white" |
|-
!style="text-align: right;"|9
| 9 || 18 || 27 || 36 || 45 || 54 || 63 || 72 || 81
|-
!style="width:7.14%"|×
!style="text-align: right; width:7.14%"|1
!style="text-align: right; width:7.14%"|2
!style="text-align: right; width:7.14%"|3
!style="text-align: right; width:7.14%"|4
!style="text-align: right; width:7.14%"|5
!style="text-align: right; width:7.14%"|6
!style="text-align: right; width:7.14%"|7
!style="text-align: right; width:7.14%"|8
!style="text-align: right; width:7.14%"|9
|}
</div>

The traditional [[rote learning]] of multiplication was based on memorization of columns in the table, arranged as follows.

<div style="margin-left:4em">
{|
!style="text-align: right; width:5%"|
!style="text-align: right; width:5%"|
!style="text-align: right; width:5%"|
!style="text-align: right; width:5%"|
!style="text-align: right; width:5%"|
|-
|
{{figure space}}0&nbsp;×&nbsp;0&nbsp;=&nbsp;0<br>
{{figure space}}1&nbsp;×&nbsp;0&nbsp;=&nbsp;0<br>
{{figure space}}2&nbsp;×&nbsp;0&nbsp;=&nbsp;0<br>
{{figure space}}3&nbsp;×&nbsp;0&nbsp;=&nbsp;0<br>
{{figure space}}4&nbsp;×&nbsp;0&nbsp;=&nbsp;0<br>
{{figure space}}5&nbsp;×&nbsp;0&nbsp;=&nbsp;0<br>
{{figure space}}6&nbsp;×&nbsp;0&nbsp;=&nbsp;0<br>
{{figure space}}7&nbsp;×&nbsp;0&nbsp;=&nbsp;0<br>
{{figure space}}8&nbsp;×&nbsp;0&nbsp;=&nbsp;0<br>
{{figure space}}9&nbsp;×&nbsp;0&nbsp;=&nbsp;0<br>
10&nbsp;×&nbsp;0&nbsp;=&nbsp;0<br>
11&nbsp;×&nbsp;0&nbsp;=&nbsp;0<br>
12&nbsp;×&nbsp;0&nbsp;=&nbsp;0<br>
|
{{figure space}}0&nbsp;×&nbsp;1&nbsp;=&nbsp;0<br>
{{figure space}}1&nbsp;×&nbsp;1&nbsp;=&nbsp;1<br>
{{figure space}}2&nbsp;×&nbsp;1&nbsp;=&nbsp;2<br>
{{figure space}}3&nbsp;×&nbsp;1&nbsp;=&nbsp;3<br>
{{figure space}}4&nbsp;×&nbsp;1&nbsp;=&nbsp;4<br>
{{figure space}}5&nbsp;×&nbsp;1&nbsp;=&nbsp;5<br>
{{figure space}}6&nbsp;×&nbsp;1&nbsp;=&nbsp;6<br>
{{figure space}}7&nbsp;×&nbsp;1&nbsp;=&nbsp;7<br>
{{figure space}}8&nbsp;×&nbsp;1&nbsp;=&nbsp;8<br>
{{figure space}}9&nbsp;×&nbsp;1&nbsp;=&nbsp;9<br>
10&nbsp;×&nbsp;1&nbsp;=&nbsp;10<br>
11&nbsp;×&nbsp;1&nbsp;=&nbsp;11<br>
12&nbsp;×&nbsp;1&nbsp;=&nbsp;12<br>
|
{{figure space}}0&nbsp;×&nbsp;2&nbsp;=&nbsp;0<br>
{{figure space}}1&nbsp;×&nbsp;2&nbsp;=&nbsp;2<br>
{{figure space}}2&nbsp;×&nbsp;2&nbsp;=&nbsp;4<br>
{{figure space}}3&nbsp;×&nbsp;2&nbsp;=&nbsp;6<br>
{{figure space}}4&nbsp;×&nbsp;2&nbsp;=&nbsp;8<br>
{{figure space}}5&nbsp;×&nbsp;2&nbsp;=&nbsp;10<br>
{{figure space}}6&nbsp;×&nbsp;2&nbsp;=&nbsp;12<br>
{{figure space}}7&nbsp;×&nbsp;2&nbsp;=&nbsp;14<br>
{{figure space}}8&nbsp;×&nbsp;2&nbsp;=&nbsp;16<br>
{{figure space}}9&nbsp;×&nbsp;2&nbsp;=&nbsp;18<br>
10&nbsp;×&nbsp;2&nbsp;=&nbsp;20<br>
11&nbsp;×&nbsp;2&nbsp;=&nbsp;22<br>
12&nbsp;×&nbsp;2&nbsp;=&nbsp;24<br>
|
{{figure space}}0&nbsp;×&nbsp;3&nbsp;=&nbsp;0<br>
{{figure space}}1&nbsp;×&nbsp;3&nbsp;=&nbsp;3<br>
{{figure space}}2&nbsp;×&nbsp;3&nbsp;=&nbsp;6<br>
{{figure space}}3&nbsp;×&nbsp;3&nbsp;=&nbsp;9<br>
{{figure space}}4&nbsp;×&nbsp;3&nbsp;=&nbsp;12<br>
{{figure space}}5&nbsp;×&nbsp;3&nbsp;=&nbsp;15<br>
{{figure space}}6&nbsp;×&nbsp;3&nbsp;=&nbsp;18<br>
{{figure space}}7&nbsp;×&nbsp;3&nbsp;=&nbsp;21<br>
{{figure space}}8&nbsp;×&nbsp;3&nbsp;=&nbsp;24<br>
{{figure space}}9&nbsp;×&nbsp;3&nbsp;=&nbsp;27<br>
10&nbsp;×&nbsp;3&nbsp;=&nbsp;30<br>
11&nbsp;×&nbsp;3&nbsp;=&nbsp;33<br>
12&nbsp;×&nbsp;3&nbsp;=&nbsp;36<br>
|
{{figure space}}0&nbsp;×&nbsp;4&nbsp;=&nbsp;0<br>
{{figure space}}1&nbsp;×&nbsp;4&nbsp;=&nbsp;4<br>
{{figure space}}2&nbsp;×&nbsp;4&nbsp;=&nbsp;8<br>
{{figure space}}3&nbsp;×&nbsp;4&nbsp;=&nbsp;12<br>
{{figure space}}4&nbsp;×&nbsp;4&nbsp;=&nbsp;16<br>
{{figure space}}5&nbsp;×&nbsp;4&nbsp;=&nbsp;20<br>
{{figure space}}6&nbsp;×&nbsp;4&nbsp;=&nbsp;24<br>
{{figure space}}7&nbsp;×&nbsp;4&nbsp;=&nbsp;28<br>
{{figure space}}8&nbsp;×&nbsp;4&nbsp;=&nbsp;32<br>
{{figure space}}9&nbsp;×&nbsp;4&nbsp;=&nbsp;36<br>
10&nbsp;×&nbsp;4&nbsp;=&nbsp;40<br>
11&nbsp;×&nbsp;4&nbsp;=&nbsp;44<br>
12&nbsp;×&nbsp;4&nbsp;=&nbsp;48<br>
|-
|
{{figure space}}0&nbsp;×&nbsp;5&nbsp;=&nbsp;0<br>
{{figure space}}1&nbsp;×&nbsp;5&nbsp;=&nbsp;5<br>
{{figure space}}2&nbsp;×&nbsp;5&nbsp;=&nbsp;10<br>
{{figure space}}3&nbsp;×&nbsp;5&nbsp;=&nbsp;15<br>
{{figure space}}4&nbsp;×&nbsp;5&nbsp;=&nbsp;20<br>
{{figure space}}5&nbsp;×&nbsp;5&nbsp;=&nbsp;25<br>
{{figure space}}6&nbsp;×&nbsp;5&nbsp;=&nbsp;30<br>
{{figure space}}7&nbsp;×&nbsp;5&nbsp;=&nbsp;35<br>
{{figure space}}8&nbsp;×&nbsp;5&nbsp;=&nbsp;40<br>
{{figure space}}9&nbsp;×&nbsp;5&nbsp;=&nbsp;45<br>
10&nbsp;×&nbsp;5&nbsp;=&nbsp;50<br>
11&nbsp;×&nbsp;5&nbsp;=&nbsp;55<br>
12&nbsp;×&nbsp;5&nbsp;=&nbsp;60<br>
|
{{figure space}}0&nbsp;×&nbsp;6&nbsp;=&nbsp;0<br>
{{figure space}}1&nbsp;×&nbsp;6&nbsp;=&nbsp;6<br>
{{figure space}}2&nbsp;×&nbsp;6&nbsp;=&nbsp;12<br>
{{figure space}}3&nbsp;×&nbsp;6&nbsp;=&nbsp;18<br>
{{figure space}}4&nbsp;×&nbsp;6&nbsp;=&nbsp;24<br>
{{figure space}}5&nbsp;×&nbsp;6&nbsp;=&nbsp;30<br>
{{figure space}}6&nbsp;×&nbsp;6&nbsp;=&nbsp;36<br>
{{figure space}}7&nbsp;×&nbsp;6&nbsp;=&nbsp;42<br>
{{figure space}}8&nbsp;×&nbsp;6&nbsp;=&nbsp;48<br>
{{figure space}}9&nbsp;×&nbsp;6&nbsp;=&nbsp;54<br>
10&nbsp;×&nbsp;6&nbsp;=&nbsp;60<br>
11&nbsp;×&nbsp;6&nbsp;=&nbsp;66<br>
12&nbsp;×&nbsp;6&nbsp;=&nbsp;72<br>
|
{{figure space}}0&nbsp;×&nbsp;7&nbsp;=&nbsp;0<br>
{{figure space}}1&nbsp;×&nbsp;7&nbsp;=&nbsp;7<br>
{{figure space}}2&nbsp;×&nbsp;7&nbsp;=&nbsp;14<br>
{{figure space}}3&nbsp;×&nbsp;7&nbsp;=&nbsp;21<br>
{{figure space}}4&nbsp;×&nbsp;7&nbsp;=&nbsp;28<br>
{{figure space}}5&nbsp;×&nbsp;7&nbsp;=&nbsp;35<br>
{{figure space}}6&nbsp;×&nbsp;7&nbsp;=&nbsp;42<br>
{{figure space}}7&nbsp;×&nbsp;7&nbsp;=&nbsp;49<br>
{{figure space}}8&nbsp;×&nbsp;7&nbsp;=&nbsp;56<br>
{{figure space}}9&nbsp;×&nbsp;7&nbsp;=&nbsp;63<br>
10&nbsp;×&nbsp;7&nbsp;=&nbsp;70<br>
11&nbsp;×&nbsp;7&nbsp;=&nbsp;77<br>
12&nbsp;×&nbsp;7&nbsp;=&nbsp;84<br>
|
{{figure space}}0&nbsp;×&nbsp;8&nbsp;=&nbsp;0<br>
{{figure space}}1&nbsp;×&nbsp;8&nbsp;=&nbsp;8<br>
{{figure space}}2&nbsp;×&nbsp;8&nbsp;=&nbsp;16<br>
{{figure space}}3&nbsp;×&nbsp;8&nbsp;=&nbsp;24<br>
{{figure space}}4&nbsp;×&nbsp;8&nbsp;=&nbsp;32<br>
{{figure space}}5&nbsp;×&nbsp;8&nbsp;=&nbsp;40<br>
{{figure space}}6&nbsp;×&nbsp;8&nbsp;=&nbsp;48<br>
{{figure space}}7&nbsp;×&nbsp;8&nbsp;=&nbsp;56<br>
{{figure space}}8&nbsp;×&nbsp;8&nbsp;=&nbsp;64<br>
{{figure space}}9&nbsp;×&nbsp;8&nbsp;=&nbsp;72<br>
10&nbsp;×&nbsp;8&nbsp;=&nbsp;80<br>
11&nbsp;×&nbsp;8&nbsp;=&nbsp;88<br>
12&nbsp;×&nbsp;8&nbsp;=&nbsp;96<br>
|
{{figure space}}0&nbsp;×&nbsp;9&nbsp;=&nbsp;0<br>
{{figure space}}1&nbsp;×&nbsp;9&nbsp;=&nbsp;9<br>
{{figure space}}2&nbsp;×&nbsp;9&nbsp;=&nbsp;18<br>
{{figure space}}3&nbsp;×&nbsp;9&nbsp;=&nbsp;27<br>
{{figure space}}4&nbsp;×&nbsp;9&nbsp;=&nbsp;36<br>
{{figure space}}5&nbsp;×&nbsp;9&nbsp;=&nbsp;45<br>
{{figure space}}6&nbsp;×&nbsp;9&nbsp;=&nbsp;54<br>
{{figure space}}7&nbsp;×&nbsp;9&nbsp;=&nbsp;63<br>
{{figure space}}8&nbsp;×&nbsp;9&nbsp;=&nbsp;72<br>
{{figure space}}9&nbsp;×&nbsp;9&nbsp;=&nbsp;81<br>
10&nbsp;×&nbsp;9&nbsp;=&nbsp;90<br>
11&nbsp;×&nbsp;9&nbsp;=&nbsp;99<br>
12&nbsp;×&nbsp;9&nbsp;=&nbsp;108<br>
|-
|
{{figure space}}0&nbsp;×&nbsp;10&nbsp;=&nbsp;0<br>
{{figure space}}1&nbsp;×&nbsp;10&nbsp;=&nbsp;10<br>
{{figure space}}2&nbsp;×&nbsp;10&nbsp;=&nbsp;20<br>
{{figure space}}3&nbsp;×&nbsp;10&nbsp;=&nbsp;30<br>
{{figure space}}4&nbsp;×&nbsp;10&nbsp;=&nbsp;40<br>
{{figure space}}5&nbsp;×&nbsp;10&nbsp;=&nbsp;50<br>
{{figure space}}6&nbsp;×&nbsp;10&nbsp;=&nbsp;60<br>
{{figure space}}7&nbsp;×&nbsp;10&nbsp;=&nbsp;70<br>
{{figure space}}8&nbsp;×&nbsp;10&nbsp;=&nbsp;80<br>
{{figure space}}9&nbsp;×&nbsp;10&nbsp;=&nbsp;90<br>
10&nbsp;×&nbsp;10&nbsp;=&nbsp;100<br>
11&nbsp;×&nbsp;10&nbsp;=&nbsp;110<br>
12&nbsp;×&nbsp;10&nbsp;=&nbsp;120<br>
|
{{figure space}}0&nbsp;×&nbsp;11&nbsp;=&nbsp;0<br>
{{figure space}}1&nbsp;×&nbsp;11&nbsp;=&nbsp;11<br>
{{figure space}}2&nbsp;×&nbsp;11&nbsp;=&nbsp;22<br>
{{figure space}}3&nbsp;×&nbsp;11&nbsp;=&nbsp;33<br>
{{figure space}}4&nbsp;×&nbsp;11&nbsp;=&nbsp;44<br>
{{figure space}}5&nbsp;×&nbsp;11&nbsp;=&nbsp;55<br>
{{figure space}}6&nbsp;×&nbsp;11&nbsp;=&nbsp;66<br>
{{figure space}}7&nbsp;×&nbsp;11&nbsp;=&nbsp;77<br>
{{figure space}}8&nbsp;×&nbsp;11&nbsp;=&nbsp;88<br>
{{figure space}}9&nbsp;×&nbsp;11&nbsp;=&nbsp;99<br>
10&nbsp;×&nbsp;11&nbsp;=&nbsp;110<br>
11&nbsp;×&nbsp;11&nbsp;=&nbsp;121<br>
12&nbsp;×&nbsp;11&nbsp;=&nbsp;132<br>
|
{{figure space}}0&nbsp;×&nbsp;12&nbsp;=&nbsp;0<br>
{{figure space}}1&nbsp;×&nbsp;12&nbsp;=&nbsp;12<br>
{{figure space}}2&nbsp;×&nbsp;12&nbsp;=&nbsp;24<br>
{{figure space}}3&nbsp;×&nbsp;12&nbsp;=&nbsp;36<br>
{{figure space}}4&nbsp;×&nbsp;12&nbsp;=&nbsp;48<br>
{{figure space}}5&nbsp;×&nbsp;12&nbsp;=&nbsp;60<br>
{{figure space}}6&nbsp;×&nbsp;12&nbsp;=&nbsp;72<br>
{{figure space}}7&nbsp;×&nbsp;12&nbsp;=&nbsp;84<br>
{{figure space}}8&nbsp;×&nbsp;12&nbsp;=&nbsp;96<br>
{{figure space}}9&nbsp;×&nbsp;12&nbsp;=&nbsp;108<br>
10&nbsp;×&nbsp;12&nbsp;=&nbsp;120<br>
11&nbsp;×&nbsp;12&nbsp;=&nbsp;132<br>
12&nbsp;×&nbsp;12&nbsp;=&nbsp;144<br>
|
|
|}
</div>

This form of writing the multiplication table in columns with complete number sentences is still used in some countries, such as Colombia, Bosnia and Herzegovina,{{citation needed|date=December 2016}} instead of the modern grids above.