numbers that cannot be expressed as a ratio 'p over q' of two integers
p and q. How do we know that such numbers even exist? In fact, this
isn't totally obvious: in ancient Greece, the cult led by Pythagoras
preached that all numbers, and all of reality in general, could be
expressed as ratios of integers. Around the year 500 B.C.,
Pythagoras' disciple Hippasus of Metapontum conclusively proved that
such numbers do exist, and that the square root of two is irrational.
Rather than incorporating this new knowledge into their beliefs, the
Pythagoreans declared Hippasus a heretic. According to some legends,
he made his discovery of irrational numbers while travelling at sea
with a group of fellow cult members, and was thrown overboard before
they reached land. As a result, Hippasus never published his proof,
or at least never published it in any work that survived to the
present day, though the knowledge passed secondhand to various
So, how do we know that the square root of 2 is irrational? The
most common proof taught these days starts with the assumption that
you have expressed root-2 as a fraction 'p over q', with this fraction
in lowest terms, and then shows that both p and q must be even. Thus
the original fraction was not in lowest terms, and we have a
contradiction. To see this, look at the equation 'p/q = root-2'.
Square both sides, to get p-squared/q-squared = 2. Then p-squared =
2q-squared. This shows p-squared is even, and for that to be true, p
must be even, since the square of an odd number is always odd. But if
p is even, then p-squared is not only even, but also divisible by 4--
so we can divide both sides of 'p-squared = 2q-squared' by 2, and see
that q must be even by the same line of reasoning.
Browsing this topic on the net, I was surprised to learn that this
concise and convincing algebraic proof was almost certainly not the
one created by Hippasus. I probably shouldn't have been surprised--
the ancient Greeks were very devoted to geometric constructions, and
didn't have modern algebra at their disposal. The irrationality of
root-2 can also be proven by an elegant geometric construction, which
is probably much closer to what Hippasus did. The idea is that we
again assume we have expressed root-2 as p-over-q as a minimum value
in lowest terms, and draw a right triangle with legs of size q and
hypotenuse of size p. Then we show that we can construct a triangle
with a hypotenuse of size 2q-p and legs of size p-q. Since their
ratio expresses the same fraction we started with in lower terms, we
again have a contradiction, and again have shown that our initial
assumption that the ratio existed must have been mistaken.
As for the details of this geometric construction: I think if I
haven't lost you already, I definitely will in trying to verbally
explain the details of a geometric construction in the podcast
format. Here it is in one sentence: mark off a point at distance p
along the hypotenuse from one of the acute angles, and a point at
distance p-q from the right angle along one leg, and you should see
our magic triangle. But if you're lazy, you can just go to the
wikipedia page listed in the show notes and look at the picture.
So, in short, we have seen both an algebraic and a geometric proof
that the square root of 2 is irrational. I always find it fun when I
see something that I already know proven by a totally different
method. Especially when I'm not thrown off a boat for it.
And this has been your math mutation for today.