Mathematical table

Mathematical table

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Mathematical table

Before calculators were cheap and plentiful, people would use mathematical tables —lists of numbers showing the results of calculation with varying arguments— to simplify and drastically speed up computation. Most common are multiplication tables, which most people know from their early mathematics classes:

×	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 the result of 7×8, one looks in the left column to seven, then across the "seven-line" to eight. The easily found answer is 56. To find 9×3, one would swap the factors and find the equal product 3×9 (27) by the same technique.

History and use

Part of a table of common logarithms from Abramowitz and Stegun.

Tables of trigonometric functions were first known to be made by Hipparchus. Tables of common logarithms and antilogarithms were used to do rapid multiplications, divisions, and exponentiations, including the extraction of nth roots. Tables of special functions are still used. For example, the use of tables of values of the cumulative distribution function of the normal distribution remains commonplace today, especially in schools.

Mechanical special-purpose computers known as difference engines were proposed in the 19th century to tabulate polynomial approximations of logarithmic functions – i.e. to compute large logarithmic tables. This was motivated mainly by errors in the logarithmic tables made by the human 'computers' of the time. Early digital computers were developed during World War II in part to produce specialized mathematical tables for aiming artillery. From 1972 onwards, with the launch and growing use of scientific calculators, most mathematical tables went out of use.

Creating tables is a common code optimization technique, and works as well for computers as humans. In computers, use of such tables is done in order to speed up calculations in those cases where a table lookup is faster than the corresponding calculations (particularly if the computer in question doesn't have a hardware implementation of the calculations). In essence, one trades computing speed for the computer memory space required to store the tables.

External links

Mathematical Table Generator -- generates addition, subtraction, multiplication, division, exponentiation, and trigonometric tables using JavaScript.

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Mathematical table

History and use

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