## THE GOLDEN RATIO IS IRRATIONAL

### An Algebraic Proof

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:

x^{2}=xy + y^{2}

x^{2}-xy = y^{2}

x(x-y)=y^{2}

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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Date last modified: 7/1/04

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