April 12, 2012
You might remember this definition of rational numbers from school.
A rational number can be expressed as a fraction p/q where p and q are integers, q is not zero, and p and q have no factors in common.
In other words, p/q is a fraction that can’t be reduced.
We like rational numbers. Rational numbers, as we also learned in school, can be expressed as decimals that terminate or repeat.
What if we want to live in a world of rational numbers?
Add two rational numbers together and we get a rational number. The same is true of subtractions, multiplication, and division.
The Greeks noticed that if you have a square that has length one on each side, the diagonal was not rational.
The Pythagorean theorem tells us that the length of the diagonal squared is two. So the length of the diagonal is the square root of two.
Another way to write the square root of two is two raised to the power of (1/2). It turns out that the square root of two is an irrational number.
So this would mean that a rational number raised to a rational power is irrational.
Tomorrow we’ll look at one of my favorite proofs – the proof that you can raise an irrational number to an irrational power and get a rational number – let’s finish today with a quick look at the classic proof of how we know the square root of two is irrational.
If the square root of two is rational then it can be written as p/q where p and q are integers with no factors in common.
This means that 2 = p^2/q^2 or, multiplying both sides by q^2, we see that 2 q ^2 = p^2.
Now, this means that p^2 is an even number. The only way for p^2 to be even is if p is even. So p is an even number and therefore it can be written as two times another number. In other words there is an integer r so that p = 2 r so p^2 = (2 r)^2 = 4 r^2.
2 q^2 = 4 r^2. Dividing both sides by 2 we see that q^2 = 2 r^2. This tells us that q^2 is an even number which tells us that q is an even number.
So this tells us that p/q wasn’t in lowest terms since both p and q were divisible by 2. Therefore the square root of two is not a rational number.
You have to love a proof like that.