Problem: Show that is an irrational number (can’t be expressed as a fraction of integers).
Solution: Suppose to the contrary that
for integers
, and that this representation is fully reduced, so that
. Consider the isosceles right triangle with side length
and hypotenuse length
, as in the picture on the left. Indeed, by the Pythagorean theorem, the length of the hypotenuse is
, since
.
Swinging a
-leg to the hypotenuse, as shown, we see that the hypotenuse can be split into parts
, and hence
is an integer. Call the point where the
and
parts meet
. If we extend a perpendicular line from
to the other leg, as shown, we get a second, smaller isosceles right triangle. Since the segments
and
are symmetrically aligned (they are tangents to the same circle from the same point), they too have length equal to
. Finally, we may write the hypotenuse of the smaller triangle as
, which is also an integer.
So the lengths of the sides of the smaller triangle are integers, but by triangle similarity, the hypotenuse to side-length ratios are equal: , and obviously from the picture the latter numerator and denominator are smaller numbers. Hence,
was not in lowest terms, a contradiction. This implies that
cannot be rational.
This proof is a prime example of the cooperation of two different fields of mathematics. We just translated a purely number-theoretical problem into a problem about triangle similarity, and used our result there to solve our original problem. This technique is widely used all over higher-level mathematics, even between things as seemingly unrelated as topological curves and groups. Finally, we leave it as an exercise to the reader to extend this proof to a proof that whenever is not a perfect square, then
is irrational. The proof is quite similar, but strays from nice isosceles right triangles
Great proof. Any idea who proved it? Must be some Greek mathematician….
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Hippamus probably
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Awesome proof. This is a great blog.
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