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May 2013

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A proof that the square root of 2 is irrational

How do we know that square root of 2 is an irrational number? In other words, how do we know that √2 wouldn't have a pattern in the decimal sequence? Maybe the pattern is very well hidden and is really long, billions of digits? Even if you check it till million first digits, maybe the pattern is just longer than you were able to print the digits out with your computer?

Here is where mathematical proof comes in. The proof that √2 is indeed irrational is usually found in college level texts, but it isn't that difficult to follow. It does not rely on computers at all, but instead it is a "proof by contradiction"—if √2 WERE a rational number, then we'd get a contradiction. I encourage you to let your high school students study this proof since it is very illustrative of a typical proof in mathematics and is not very hard to follow.

The proof that square root of 2 is irrational:

Let's suppose √2 were 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 the lowest terms, since that can obviously be done with any fraction. Notice that in order for a/b to be in its simplest terms, both a andb must be not be even. One or both must be odd. Otherwise, you could simplify.

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 can 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 if you don't believe that!

Okay, if a itself is an even number, then a is 2 times some other whole number, or a = 2k where k is this other number. We don't need to know exactly what k is; it won't matter. Soon is coming 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 b² is even, from which follows again that b itself is an even number!!!

WHY is that a contradiction? Because we started the whole process saying that a/b is simplified to the lowest terms, and now it turns out that a and b would both be even. So √2 cannot be rational.

More information:

Calculators and Irrational Numbers
When I square the square root of 11 on any calculator, I get the answer 11 (exactly). That seems to indicate that the square root of 11 is a rational number, but it's not. Can you explain this?

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?

Adding Rational and Irrational Numbers
How can you prove that a rational number added to an irrational number results in an irrational number?

First 100 Irrational Numbers
Could you please tell me the first 100 irrational numbers?

More proofs that square root of 2 is irrational

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New! Times Tales is now on DVD!

The fast, FUN, and easy way to learn multiplication. Learn the upper times tales in two sittings using mnemonic stories.