Prop. 1. If a fraction be reduced to
its lowest terms, so called,1
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
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
| 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,
so that if a fraction whose numerator and denominator have
no common measure greater than unity, be equal to a fraction of lower numerator
and denominator, it is equal to another in which the numerator and
denominator are still lower. If we proceed with
| a | = | e | in a similar manner, we find |
|---|
| b | f |
| a | = | g | where g < e , h < f , |
|---|
| b | h |
and so on. Now, if there be any process which perpetually
diminishes the terms of a fraction by one or more units at every step, it must
at last bring either the numerator or denominator, or both, to 0. Let