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A proof that the square root of 2 is irrational

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 = (2k)2/b2
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.

See also

What is a mathematical proof?

Irrational Numbers; Rational Square Roots
How can you tell whether root 10 is a terminating or repeating decimal, or an irrational number? Are some square roots rational?

More proofs that square root of 2 is irrational
Provided by Cut-the-Knot.org.

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