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.
- 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. |
Therefore, by (40, Principle II.) all the first column together exceeds all the second column by all the third column, that is, by
2 ten-th. 3 th. 1 hund. 3 tens and 1 unit,
which is 23131. Therefore the difference of 57762 and 34631 is 23131, or 57762-34631 = 23131.
- Suppose I want to find the difference between 61274 and 39628. Write them at length, and
61274 is 6 ten-th. 1 th. 2 hund. 7 tens and 4 units.
39628 is 3 ten-th. 9 th. 6 hund. 2 tens and 8 units.
If we attempt to do the same as in the last article, there is a difficulty immediately, since 8, being greater than 4, cannot be taken from it. But from (40) it appears that we shall not alter the difference of two numbers if we add the same number to both of them. Add ten to the first number, that is, let there be 14 units instead of four, and add ten also to the second number, but instead of adding ten to the number of units, add one to the number of tens, which is the same thing. The numbers will then stand thus,
6 ten-thous. 1 thous. 2 hund. 7 tens and 14 units.2 3 ten-thous. 9 thous. 6 hund. 3 tens and 8 units.