First, let's start with an assumption (actually two assumptions). The golden ratio is between 1 and 2. Neither of these assumptions is difficult to prove. Now let's assume there are positive integers x and y such that the golden ratio is equal to . Then 1 < < 2 so y < x < 2y which gives 0 < x-y < y. We may either assume at this point that is in reduced form, or we may skip that assumption (the finishing touches of the proof are different, but the middle section is identical in either case). If we do make the assumption that is reduced, it is worth noting that the reduced form of a rational number is the form with the smallest positive denominator.
The definition of the golden ratio implies that
Now a little algebra:
x2=xy + y2
x2-xy = y2
Thus if is equal to the golden ratio, then there is another representation of the same number that has a smaller positive denominator. If we assumed earlier that is reduced, then we have a contradiction. Otherwise, note that we could apply the same logic to to find a fraction also equal to the golden ratio with a still smaller positive denominator. Since there are only a finite number of positive integers less than y, this process cannot continue indefinitely which gives us a contradiction.
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