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Lexical Leavings

by Tom Gally



Mark Spahn writes:

On page 33 of the (British) textbook Elementary Number Theory by Gareth A. Jones and J. Mary Jones, after an explanation of various tricks for determining quickly whether a number n is divisible by 2, 3, 5, and 11, is the sentence "For primes p ≠ 2, 3, 5 and 11, one simply has to divide p into n and see whether or not the remainder is 0." It is clear from the context that "divide p into n" means "divide n by p".

Personally, when referring to division I always say "divide n by p", never "divide p into n".

In a division expression like n/p = q, n is called the "dividend" (the number to be divided), p is called the "divisor", and the result q is called the "quotient".

Looking up what "into" means in a division context, I find in Webster's New World Dictionary, Third College Edition, that "into" is "used to indicate division".  [In the fourth edition, this is rephrased as "considered as a divisor of." --TG] This is useless as a definition without the accompanying example "3 into 21 is 7", which indicates (assuming that usage examples are true statements) that "into" is used in the form "divisor into dividend is quotient" and means "dividend divided by divisor is quotient".

Another point to note is that "into", like "plus", is what in math is called a "dyadic infix operator"; that is, it occurs between two numbers and operates on them to produce another number. Compare:
3 plus 21 is 24    3 + 21 = 24
3 minus 21 is -18    3 - 21 = -18
3 from 21 is 18    3 ? 21 = 18
3 times 21 is 63    3 x 21 = 63
3 divided by 21 is 1/7    3/21 = 1/7
3 into 21 is 7    3 ? 21 = 7
There is an operator symbol (found on all calculators) for each of the four operators "plus", "minus", "times", and "divided by", but there are no signs to denote the into or from operations. Among the four operations addition, subtraction, multiplication, and division, with subtraction and division it makes a difference in which order the arguments occur (3 + 4 = 4 + 3, but 3 - 4 ≠ 4 - 3). The inverse of the division operation is the into operation, in the sense that "A into B" is the same as "B divided by A". Similarly, the inverse of subtraction is from. But there are no mathematical symbols for the operators into or from.

The corresponding Japanese expressions are 加減乗除 = addition, subtraction, multiplication, division (加法, 減法, 乗法, and 除法, or 足し算, 引き算, 掛け算, and 割り算).

The results of these operations are called the sum (和), difference (差), product (積), quotient (商).
3 plus 4 is 7         3 + 4 = 7         3 足す 4 は 7
8 minus 6 is 2        8 - 6 = 2         8 引く 6 は 2
6 from 8 is 2          6 ? 8 = 2         6 ? 8 は 2
3 times 4 is 12       3 x 4 = 12         3 掛ける 4 は 12
24 divided by 6 is 4         24/6 = 4         24 割る 6 は 4
6 into 24 is 4        6 ? 24 = 4         6 ? 24 は 4
Let's check these readings with native speakers. In particular, how are schoolkids taught to read "3 + 4 = 7"? And I hope someone will sit down one day and write a short article about the wording that goes through the mind of a Japanese while doing arithmetic.  For example, the operation 3 x 3 = 9 seems to be memorized with the wording "サザン[三三]が九" (三省堂国語辞典, 第5版, entry "さ【三】").

I wonder whether short mental wordings like this make it easier to do arithmetic fast than wordings like "3 times 3 is nine". (Is it even possible to do arithmetic without imagining a pronunciation? Do abacists think of words as they manipulate their beads, or would that just slow them down?)
(September 1, 2003)