The (almost really) Complete Works of Lewis Carroll

Long Multiplication Worked with a Single Line of Figures

To the Editor of the Educational Times

Sir,—If the following brief method of working Long Multiplication should prove to be new, I hope you may think it worth publishing:—

Suppose we wish to multiply 56248 by 3726. We set the sum in the usual way, thus:—

We then write out the upper line, backwards, on the lower edge of a separate slip of paper, placing a mark over the unit-digit, as a guide to the eye: with this slip we cover the upper line of the given sum, bringing the marked digit over the unit of the lower line, thus:—

We then take the product of the digits which are in the same vertical line (viz., 8, 6); this gives us 48; we write the unit of this (viz., 8) vertically under the scored digit, and “carry” the 4, thus:—

We then shift the slip one place to the left, thus:—

We then add together the carried digit and the products of the digits which are in the same vertical lines, and write the result as before. The mental process being, “$4+24=28$, $+16=44$; set down 4 and carry 4.”

We then shift the slip again, and proceed as before; the mental process being, “$4+12=16$; $+8=24$; $+56=80$; set down 0 and carry 8.”

We then shift the slip again, and so on; the last step being reached when the sum stands thus, with 5 to carry:—

Hence the mental process of the last step is, “$5+15=20$; set it down.” We then remove the slip, and the result appears thus:—

A similar method will serve for multiplying decimals: all we have to remember is, to bring the marked digit of the slip vertically over whatever decimal place we wish to carry the working to. For example, if we wish to multiply together ⋅63624 and ⋅25873; and if, in order to have the answer correct to 3 places, we wish to carry the working to 4 places, we set the sum thus:—

We then write 426360 on a separate slip of paper, and place it so that its marked digit comes vertically over the 4th decimal place in the answer, thus:—

The mental process of the first step will be “$0+48=48$; $+15=63$; $+12=75$; set down 5 and carry 7.”

We then shift the slip to the left and proceed as before, the last step being reached when the sum stands thus, with 1 to carry:—

Hence the mental process of the last step is “$1+0=1$; set it down.” We then remove the slip, and the result appears thus:—

Hence the answer, correct to 3 places, will be ⋅164. This method seems to me not only to save space and time, but also to avoid the risk of mistakes involved in writing all the intermediate lines of figures required in the old method, as well as the constant risk of losing one’s place while carrying the eye obliquely from one figure to another figure several rows above it.

Your obedient servant,