Tables for Multiplication

The combination "multiplication tables" is most likely to be associated with a fixture of elementary math classes, often called "times tables". However, at various times books of tables have been published that did not include the products of numbers, but were nonetheless composed to facilitate finding such products.

Squares

Tables of quarter squares have been published in 1690 by Johann Ludolff and in 1817 by Antoine Voisin. These could be used for number multiplication in at least two ways:

$\begin{align}\displaystyle ab &= \frac{(a+b)^2}{4} - \frac{(a-b)^2}{4}, \\ &= 2\bigg[\frac{a^2}{4}+\frac{b^2}{4}-\frac{(a-b)^2}{4}\bigg]. \end{align}$

The second formula is more cumbersome, but, for the same table of squares, admits factors twice as large compared to the first one. Both identities may successfully serve as tools of mental arithmetic, provided numbers involved are not too large and the squares of integers are either memorized or can be computed fast. In particular, integer squares can be computed successively with $(n+1)^{2}=n^{2}+(2n+1).$

Triangular numbers

Triangular numbers have the form $\displaystyle T_{n}=\frac{n(n+1)}{2}.$ There are several identities that combine triangular numbers into products (with implied constraints on $a$ and $b)$:

$\begin{align}\displaystyle ab &= T_{a-1}+T_{b}-T_{a-b-1},\\ &= T_{a}+T_{b-1}-T_{a-b},\\ &= T_{a-n}+T_{b+n-1}-T_{a-b-n}-T_{n-1},\,\mbox{for any}\,n\\ &= T_{(a-1)/2+b}-T_{(a-1)/2-b},\\ &= T_{a/2+b-1}-T_{a/2-b-1}+b,\\ &= T_{a+b}-T_{a}-T_{b}.\\ \end{align}$

A table of 20,000 triangular numbers has been published by L. De Joncourt in 1762; the table by A. Arnaudeau (with most of the above identities) published in 1896 contained 200,000 triangular numbers.

Logarithms

The fundamental property of logarithms, $\mbox{log}ab=\mbox{log}a+\mbox{log}b,$ also replaces multiplication with addition, and was actually devised by John Napier (1614) for exactly that purpose. Unlike the two methods above, the implied calculations are necessarily approximate and are best used for calculations with a fixed number of digits.

References

  1. Julian Havil, John Napier: Life, Logarithms, and Legacy Princeton University Press, 2014

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