Here you can read a step-by-step proof with simple explanations for the fact that the square root of 2 is an irrational number. It is the most common proof for this fact and is by contradiction.
How do we know that square root of 2 is an irrational number? In other words, how do we know that √2 doesn't have a pattern in its decimal sequence? Maybe the pattern is very well hidden and is really long, billions of digits?
Here is where mathematical proof comes in. The proof that √2
is indeed irrational is usually found in college level math texts, but it isn't that difficult to follow. It does not rely on computers at all, but instead is a "proof by contradiction": if √2
WERE a rational number, we'd get a contradiction. I encourage all high school students to study this proof since it illustrates so well a typical proof in mathematics and is not hard to follow.
A proof that the square root of 2 is irrational
Let's suppose √2 is a rational number. Then we can write it
√2 = a/b where a, b are whole numbers, b not zero.
We additionally assume that this a/b is simplified to lowest terms, since that can obviously be done with any fraction. Notice that in order for a/b to be in simplest terms, both of a and b cannot be even. One or both must be odd. Otherwise, we could simplify a/b further.
From the equality √2
= a/b it follows that 2 = a2/b2, or a2 = 2 · b2. So the square of a is an even number since it is two times something.
From this we know that a itself is also an even number. Why? Because it can't be odd; if a itself was odd, then a · a would be odd too. Odd number times odd number is always odd. Check it if you don't believe me!
Okay, if a itself is an even number, then a is 2 times some other whole number. In symbols, a = 2k where k is this other number. We don't need to know what k is; it won't matter. Soon comes the contradiction.
If we substitute a = 2k into the original equation 2 = a2/b2, this is what we get:
|2|| = ||4k2/b2|
|2*b2|| = ||4k2|
b2|| = ||2k2
This means that b2 is even, from which follows again that b itself is even. And that is a contradiction!!!
WHY is that a contradiction? Because we started the whole process assuming that a/b was simplified to lowest terms, and now it turns out that a and b both would be even. We ended at a contradiction; thus our original assumption (that √2 is rational) is not correct. Therefore √2 cannot be rational.