I. 15 + 38 means that 38 is to be added to 15, and is the same thing as 53. This is the sum of 15 and 38, and is read fifteen plus thirty-eight (plus is the Latin for more).
II. 64-12 means that 12 is to be taken away from 64, and is the same thing as 52. This is the difference of 64 and 12, and is read sixty-four minus twelve (minus is the Latin for less).
III. 9 × 8 means that 8 is to be taken 9 times, and is the same thing as 72. This is the product of 9 and 8, and is read nine into eight.
IV. 108/6 means that 108 is to be divided by 6, or that you must find out how many sixes there are in 108; and is the same thing as 18. This is the quotient of 108 and 6; and is read a hundred and eight by six.
V. When two numbers, or collections of numbers, with the foregoing signs, are the same, the sign = is put between them. Thus, that 7 and 5 make 12, is written in this way, 7 + 5 = 12. This is called an equation, and is read, seven plus five equals twelve. It is plain that we may construct as many equations as we please. Thus:
| 12 | ||
| 7 + 9 - 3 = 12 + 1; | — | - 1 + 3 × 2 = 11, |
| 2 |
and so on.
24. It often becomes necessary to speak of something which is true not of any one number only, but of all numbers. For example, take 10 and 7; their sum[4] is 17, their difference is 3. If this sum and difference be added together, we get 20, which is twice the greater of the two numbers first chosen. If from 17 we take 3, we get 14, which is twice the less of the two numbers. The same thing will be found to hold good of any two numbers, which gives this general proposition,—If the sum and difference of two numbers be added together, the result is twice the greater of the two; if the difference be taken from the sum, the result is twice the lesser of the two. If, then, we take any numbers, and call them the first number and the second number, and let the first number be the greater; we have
(1st No. + 2d No.) + (1st No. - 2d No.) = twice 1st No.
(1st No. + 2d No.) - (1st No. - 2d No.) = twice 2d No.
The brackets here enclose the things which must be first done, before the signs which join the brackets are made use of. Thus, 8-(2 + 1) × (1 + 1) signifies that 2 + 1 must be taken 1 + 1 times, and the product must be subtracted from 8. In the same manner, any result made from two or more numbers, which is true whatever numbers are taken, may be represented by using first No., second No., &c., to stand for them, and by the signs in (23). But this may be much shortened; for as first No., second No., &c., may mean any numbers, the letters a and b may be used instead of these words; and it must now be recollected that a and b stand for two numbers, provided only that a is greater than b. Let twice a be represented by 2a, and twice b by 2b. The equations then become