Again, if c be less than a and b,
(a - c) - (b - c) = a - b.
The brackets cannot be here removed as in (36). That is, p- (q-r) is not the same thing as p-q- r. For, in the first, the difference of q and r is subtracted from p; but in the second, first q and then r are subtracted from p, which is the same as subtracting as much as q and r together, or q + r. Therefore p-q-r is p-(q + r). In order to shew how to remove the brackets from p -(q-r) without altering the value of the result, let us take the simple instance 12-(8-5). If we subtract 8 from 12, or form 12-8, we subtract too much; because it is not 8 which is to be taken away, but as much of 8 as is left after diminishing it by 5. In forming 12-8 we have therefore subtracted 5 too much. This must be set right by adding 5 to the result, which gives 12-8 + 5 for the value of 12-(8-5). The same reasoning applies to every case, and we have therefore,
p - (q + r) = p - q - r.
p - (q - r) = p - q + r.
By the same kind of reasoning,
a - (b + c - d - e) = a - b - c + d + e.
2a + 3b - (a - 2b) = 2a + 3b - a + 2b = a + 5b.
4x + y - (17x - 9y) = 4x + y - 17x + 9y = 10y - 13x.
42. I want to find the difference of the numbers 57762 and 34631. Take these to pieces as in (29) and
57762 is 5 ten-th. 7 th. 7 hund. 6 tens and 2 units.
34631 is 3 ten-th. 4 th. 6 hund. 3 tens and 1 unit.
| Now 2 units exceed | 1 unit | by 1 unit. |
| 6 tens | 3 tens | 3 tens. |
| 7 hundreds | 6 hundreds | 1 hundred. |
| 7 thousands | 4 thousands | 3 thousands. |
| 5 ten-thousands | 3 ten-thous. | 2 ten-thous. |