Prop. 1. If a fraction be reduced to its lowest terms, so called,[61] that is, if neither, numerator nor denominator be divisible by any integer greater than unity, then no fraction of a smaller numerator and denominator can have the same value.
Let a/b be a fraction in which a and b have no common measure greater than unity: and, if possible, let c/d be a fraction of the same value, c being less than a, and d less than b. Now, since
| a | = | c | we have | a | = | b | ; |
| b | d | c | d |
let m be the integer quotient of these last fractions (which must exist, since a > c, b > d), and let e and f be the remainders. Then
| a | or | mc + e | = | c | = | mc |
| b | md + f | d | md |
Hence,
| e | and | mc | must be equal, for if not, |
| f | md |
| mc + e | would lie between | mc | and | e | , |
| md + f | md | f |
instead of being equal to the former. Hence,
| a | = | e | ; |
| b | f |