142. If the fractions to be added or subtracted do not have a common denominator, they should be reduced to equivalent fractions having a least common denominator; then proceed according to the rule in 2141.

A similar proof holds for the difference of the two fractions. Compare 139 and 140. Thus,

In case the denominators b and d have common factors, for example,

Since the denominators do not have common factors use the

NOTE. In case the denominators are binomials, it will always simplify the reduction to arrange the fractions so that all the denominators are arranged in the order of the descending or the ascending powers of some letter.

59.

+

60.

x2−(b+c) x + be

(x + y) (x + z) + (y + z) (y + x) + (z+x) (z + y)'

x + b
x2-(a+c)x+ac

+

+

x + c
x2 - (a+b)x+ai
z2-xy

MULTIPLICATION OF FRACTIONS

143. The rule for the multiplication of two fractions is: multiply the numerators together for a new numerator and the denominators for a new denominator.

a

b

It has already been proved in 263, equation 1, that if and are two quotients, which are integers, then

This formula is also satisfactory when one of the quotients is an integer (138, 1, 2).

d

As

In case, however, both and ene are fractions (see 126), we can not speak of multiplying them together without defining what we mean by the term multiplication, for, according to the usual meaning of this term, the multiplier must be a whole number. already explained in 72, the so-called rule of multiplication of fractions is really a definition of what we find it convenient to understand by the multiplication of fractions. And this definition is so fashioned that in case one of the fractions which we may desire to multiply together is an integer in a fractional form, or when both are integers, the result of the definition coincides with the consequences deduced from the common use of the word multiplication.

a

The symbolic definition of the quotient a by b, e. g.,, and its operation as described in 172, formula, justify the rule for the product of two fractions and as expressed by the equation

NOTE. The following verbal definition also will show clearly the connection between the meaning of the word multiplication when applied to integers, and when applied to fractions. When the product of an integer a by b is formed, the operation may be described as follows: What is done with unity to form b, one must do with a to find b To obtain b from unity the unit is repeated b times; hence to find b times a the number a is repeated b times. Therefore, if one desires to find the product of C by by adopting the same definition as above, it follows that what is done with unity to produce one must do with to find

times a.

d'

a

times

a b

с

с

d

a

a

с

by the

To obtain from unity one divides the unit into d equal parts, and takes c of them; hence, to find the product of by fraction is divided into d equal parts, and c of such parts are

b

d