This section is from the book "The American House Carpenter", by R. G. Hatfield. Also available from Amazon: The American House Carpenter.

As an example in subtraction, let the quantities represented by + b - a - f+ c, be taken from the quantities represented by + a+b - c - f This may be written -

(+ a + b - c - f) - (+b - a - /+ c),

an expression showing that the quantities enclosed within the second pair of parentheses are to be subtracted from those included within the first pair. Let the quantities represented in the first pair of parentheses for convenience be represented by A, or, a + b - c - f = A. Now, by the terms of the problem, we are required to subtract from A the quantities enclosed within the second pair of parentheses. To do this take first the positive quantity, b, and subtract it or indicate the subtraction, thus -

A-b;

we will then subtract the positive quantity c, or indicate the subtraction, thus -

A - b - c.

We have yet to subtract - a and - f two negative quantities.

The method by which this can be accomplished may be discovered by considering the requirements of the problem. The plus quantities b and c, before being subtracted from A, were required to have the two negative quantities and f deducted from them. It is evident, therefore, that in subtracting b and c, before this deduction was made, too much has been taken from A, and that the excess taken is equal to the sum of a and f. To correct the error, therefore, it is necessary to add just the amount of the excess, or to add the sum of a and f, or annex them by the plus sign, thus -

A - b - c + a+f.

To test the correctness of the operation as here performed, let numerals be substituted for the symbols; let a = 2, b = 3, c= 1, f = 1/2; then the given quantities to be subtracted, -

(+b - a - f+c:), become -

(+3-2-1/2+1), which reduces to -

(4 - 2 1/2) = 1 1/2

Thus the quantity to be substracted equals 1 1/2. Applying the numerals to the above expression -

A - b+a+f - c becomes -

A - 3 + 2 + 1/2 - 1 = A - 4+2 1/2 == A - 1 1/2.

A correct result; it is the same as before. Restoring now the symbols represented by A, we have for the whole expression -

+ a+ b - c - f - b +a+f - c,

which, by cancelling (Art. 403) and by adding like symbols with like signs, reduces to -

2 a - 2 c.

To test this result, let the quantity which was represented by A have the proper numerals substituted, thus:

+ a + b - c - f,

+2+3 - 1 - 1/2 = 5 - 1 1/2 = 3 1/2.

The sum of the given quantity required to be subtracted was before found to amount to 1 1/2, therefore -

A - 1 1/2 becomes -

3 1/2 - 1 1/2 == 2.

And the result by the symbols as above was -

2 a - 2 c, which becomes -

2x2 - 2x1,

or -

4 - 2 = 2;

a result the same as before, proving the work correct. An examination of the signs in the above expression, which denotes the problem performed, will show that the sign of each symbol which was required to be subtracted has been changed in the operation of subtraction. Before subtracting they were -

(+b - a - f+c);

after subtraction they are -

( - b + a+ f - c).

By this result we learn, that to subtract a quantity we have but to change its sign and annex it to the quantity from which it was required to be subtracted.

Example: Subtracts - b from c+d. Answer, c + d - a + b.

If numerals be substituted, say a = 7, b = 4, c = 5, and d=g, then -

c+d becomes 5+9= 14, a - b " 7 - 4=: 3,

c + d - (a - b) - 14 - 3= 11.

So, also, -

c + d - a + b becomes -

5+9 - 7 + 4= 11.

405. - Algebraic Fractions: Added and Subtracted. - When algebraic fractions of like denominators are to be added or subtracted, the same rules (Arts. 385 and 386) are to be observed as in the addition or subtraction of numerical fractions - namely, add or subtract the numerators for a new numerator, and place beneath the sum or difference the common denominator.

For example, what is the sum of a/b,c/b,d/b?

For this we have - a + c + d /b

Subtract c/d from b/d, For this we have -

b-c/d.

What is the algebraical sum of -

b/d' c/d' - n/d' and - r/d'?

For these we have -

b+c - n - r/d.

To exemplify this, let b represent 9, c = 8, n = 2, r = 3, and d= 12.

Then, for the algebraic sum, we have -

9+8-2-3/12 = 12/12 = 1.

Now, taking the positive and negative fractions separately, we have -

9 + 8 = 17

12+12 12;

and -

-2 -2 = -5

12 12 12

Together -

17 -5 = 12 = 1,

12 12 = 12

as before.

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