Thursday, July 02, 2015

Prime Factorization, Perfect Squares and Irrational Numbers

Moving on from yesterday's discussion about Rational Numbers, what about irrational numbers, numbers which cannot be written as a ratio of two integers?

Most whole numbers have square roots which are irrational numbers, but not everything with a radical is irrational. The square root of a perfect square is perfectly rational. So how can you tell if a number is a perfect square without a calculator?

One way is through prime factorization. (Remember those factor trees from a long time ago. C'mon, they were fun to do -- and you can do them again.... just not when you're typing in a blog. Then, they're kind of a pain, but I'll try.)

Take a number such as 60. It's prime factorization is 2 X 2 X 3 X 5, or 22 X 3 X 5.
If we were to square 60, we'd multiply 60 X 60, but we could also multiply 22 X 3 X 5 X 22 X 3 X 5.
That number (3600) would have a prime factorization of 24 X 32 X 52.

Notice what happened to the exponents. They've all doubled from 1 to 2 or 2 to 4. Every time you square a number, the exponents of its prime factors double. So if a number has been square, then all of the exponents of its prime factors will be even numbers because they are multiples of two.

Going back to our original number, is the square root of 60 a rational number?

It could only be a rational number if 60 were a perfect square, and it can only be a perfect square if all the exponents of its prime factors are even. However, the prime factorization is 22 X 3 X 5. Only one factor is even, so it is not a perfect square and the square root of 60 is irrational.

But wait! There's more!

As long as we've done the legwork, there is one more thing that we can do. Radicals that are irrational can be simplified. This is done by factoring out the largest perfect square. If we look back at the prime factorization, 22 X 3 X 5, we can see that there are two factors of 2.

So the square root of 60 is the same as (the square root of 22) X (the square root of 3 X 5). The square root of 22 is just 2.
That means that the square root of 60 is (2) times (the square root of 3 X 5), or 2(radical 15).

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