Elements of arithmetic

by Augustus De Morgan

51. You may find, in this way, either 6 times 7, or 7 times 6, and for both you find 42. That is, six sevens is the same number as seven sixes. This may be shewn as follows: Place seven counters in a line, and repeat that line in all six times. The number of counters in the whole is 6 times 7, or six sevens, if I reckon the rows from the top to the bottom; but if I count the rows that stand side by side, I find seven of them, and six in each row, the whole number of which is 7 times 6, or seven sixes. And the whole number is 42, whichever way I count. The same method may be applied to any other two numbers. If the signs of (23) were used, it would be said that 7 × 6 = 6 × 7.

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52. To take any quantity a number of times, it will be enough to take every one of its parts the same number of times. Thus, a sack of corn will be increased fifty-fold, if each bushel which it contains be replaced by 50 bushels. A country will be doubled by doubling every acre of land, or every county, which it contains. Simple as this may appear, it is necessary to state it, because it is one of the principles on which the rule of multiplication depends.

53. In order to multiply by any number, you may multiply separately by any parts into which you choose to divide that number, and add the results. For example, 4 and 2 make 6. To multiply 7 by 6 first multiply 7 by 4, and then by 2, and add the products. This will give 42, which is the product of 7 and 6. Again, since 57 is made up of 32 and 25, 57 times 50 is made up of 32 times 50 and 25 times 50, and so on. If the signs were used, these would be written thus:

7 × 6 = 7 × 4 + 7 × 2.
50 × 57 = 50 × 32 + 50 × 25.

54. The principles in the last two articles may be expressed thus: If a be made up of the parts x, y, and x, ma is made up of mx, my, and mz; or,

ifa = x + y + z.

ma = mx + my + mz,

or,m(x + y + z) = mx + my + mz.

A similar result may be obtained if a, instead of being made up of x, y, and z, is made by combined additions and subtractions, such as x + y-z, x- y + z, x-y-z, &c. To take the first as an instance: