Addition

Historically, the number system expanded as it became apparent that certain problems of interest had no solution in the then-current system. Fractions were included to deal with the problem of dividing a whole thing into a number of parts. Common fractions are numbers expressed as a ratio, such as 2/3, 7/9, and 3/2. When both parts of the fraction are integers, the result is a rational number. Each rational number may be thought of as representing a number of pieces; the numerator (top number) tells how many pieces the fraction represents; the denominator (bottom number) tells us how many pieces the whole was divided into. Suppose a cake is divided into two pieces, after which one half is further divided into six pieces and the other half into three pieces, making a total of nine pieces. If you take one piece from each half, what part of the whole cake do you get? This amounts to a simple counting problem if both halves are cut into the same number of pieces, because then there are a total of six or 12 equal pieces, of which you take two. You get either 2/6 or 2/12 of the cake. The essence of adding rational numbers, then, is to turn the problem into one of counting equal size pieces. This is done by rewriting one or both of the fractions to be added so that each has the same denominator (called a common denominator). In this way, each fraction represents a number of equal size pieces. A general formula for the sum of two fractions is a/b + c/d = (ad + bc)/bd.