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Degree: BSc Mathematics with Finance (Bachelors) - Exeter University
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Attempting to find the average of a/b and c/d, we have:
(a/b+c/d)/2 = [(ad+bc)/bd]/2 = (ad+bc)/2bd
As a,b,c,d are integers, we know that ad+bc and 2bd are also integers, as the addition and multiplication of integers will always lead to another integer.
Therefore, we can write x=ad+bc and y=2bd, where x and y are integers.
It follows that the average between a/b and c/d is now equal to x/y
As x and y are integers, x/y will be a rational number situated between a/b and c/d.
This proof applies to any integers a,b,c,d where b and d are non-zero integerssee more