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# Mathematical table

Before calculators were cheap and plentiful, people would use mathematical tables to drastically speed up computation. These are lists of numbers showing the results of calculation with varying variables. The most common is the table of multiplication, which most people know and love from their early math classes.

 x×y 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

To find 7×8, you'd look left to seven, then down to eight. The answer is 56.

Tables of trigonometric functions were first known to be made by Hipparchus, and were used up until the 1980s when calculators included this functionality. Tables of common logarithms and antilogarithms were used to perform tasks such as exponentiation, and sometimes as a short-cut for multiplication. Other tables are still commonly used, particularily in statistics, such as for the normal distribution.

Mechanical special-purpose computers known as difference engines were constructed in the 19th century to tabulate polynomial approximations of logarithmic functions – i.e. to compute large logarithmic tables. This was motivated mainly by the abundancy of errors in the logarithmic tables made by the human 'computers' of the time.

Creating tables is a common code optimization technique, and works as well for computers as humans.

Also see truth table.