For an introduction to this series, see this article. Briefly, I show examples of beautiful math that use only elementary mathematics.
What are square roots?
A square root is a number that, squared (multiplied by itself) equals a given number. For example, the square root of 9 is 3, because 3*3 = 9.
What are irrational numbers?
An irrational number is not a number that is emotional! It’s a number that cannot be expressed as a fraction, or ratio. Most of the numbers we deal with every day are rational. Some are integers (whole numbers), others are fractions. The integers are discrete – there is no whole number between two consecutive whole numbers; after 2 comes 3, and nothing is in between. The fractions are not discrete. Between any two fractions I can always give another number. One way to do this is to take the average of the two numbers. For instance, one number that is in-between 1/2 and 3/4 is the average of those two numbers, which is 5/8. In addition, integers can be expressed as fractions. For example 2 = 6/3 or 2/1. This might give you the idea that the fractions cover all the numbers. The ancient Greeks thought so. But they were wrong.
Someone from the Pythagorean school tried to figure out how long the diagonal of a unit square was; that is, he looked at a square that was 1 unit (say 1 foot, although the Greeks used other measurements) on a side. He drew a line from one corner to the opposite corner. How long is that line. By the Pythagorean theorem he knew that it was the square root of 2. But no fraction seemed to fit. e.g, if you try 8/5, you get 8/5*8/5 = 64/25 which is more than 2. If you try 7/5 you get 7/5*7/5 = 49/25 which is less than 2. But just because no fraction seemed to fit didn’t mean there wasn’t one. They needed proof. And they came up with one.
Proof that the square root of 2 is irrational
This is a reductio ad absurdum proof. That is, we assume the opposite of what we wish to prove and then derive nonsense. So, assume that some fraction a/b is the square root of 2. Reduce this fraction to its lowest terms (make sure that no =number is a factor of both). If a/b is 2.5 then a/b*a/b = 2. a/b*a/b = a2/b2. So
- 2 = a2/b2 . Now multiply both numerator (top) and denominator (bottom) by b2 to get
- 2b2 = a2 Now divide both sides by 2 to get
- b2 = a2/2. So b2 is even.
Now comes the key insight: First, if b2 is even then b must be even. Why? Well any odd number can be represented as 2n + 1. Squaring that gives 4n2 + 2n + 1 (for more on this, see this article). Second, any even number squared is divisible by 4. Why? Well, any even number can be represented as 2n, and 2n*2n = 4n2 which must be divisible by 4. So, b2 is divisible by 4; and that means that q2 must be even (since anything that is half of something that is divisible by 2. But if a and b are both even, they have a common factor (namely 2). But we started by making sure they didn’t! Contradiction! So, the square root of 2 cannot be rational!
Play! What about cube roots? What about numbers other than 2? Are they rational or irrational? Can you prove it?