Mathematical table
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193908113
2008-02-25T11:58:00Z
Dingar
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Before [[calculator]]s 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 table]]s, which most people know from their early [[mathematics]] classes:
<center>
{| class="wikitable" style="text-align:center;"
! ×
! 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
|}
</center>
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==
[[Image:Abramowitz&Stegun.page97.agr.jpg|thumb|Part of a table of [[common logarithm]]s from [[Abramowitz and Stegun]].]]
Tables of [[trigonometric function]]s were first known to be made by [[Hipparchus]]. Tables of [[common logarithm]]s and [[antilogarithm]]s were used to do rapid multiplications, divisions, and exponentiations, including the extraction of ''n''th roots. Tables of [[special function]]s 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 engine]]s 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 [[HP-35|scientific calculator]]s, most mathematical tables went out of use.
Creating tables is a common code [[optimization (computer science)|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 [[Lookup table|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.
==See also==
* [[Table (information)]]
* [[Abramowitz and Stegun]] ''Handbook of Mathematical Functions''
* [[Jurij Vega]]
* [[matrix (mathematics)|Matrix]]
* [[Random number table]]
* [[Truth table]]
== External links ==
* [http://www.farfarfar.com/math/reference/generator/ Mathematical Table Generator] -- generates addition, subtraction, multiplication, division, exponentiation, and trigonometric tables using JavaScript.
[[Category:Mathematical tools]]
[[Category:Mathematical tables|*]]
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