√2 Is Not Rational!

In this paper, we will prove that the square root of 2 is not a rational number. And we will use the proof that Euclid used about 2300 years ago and is considered one of the top 10 proofs in the history of mathematics.

Before the proof, we first need to know what we are dealing with. So, what makes a number a rational number?
By definition, rational numbers are numbers that can be written as the fraction of two integers, a and b (b ≠ 0). The set of rational numbers is indicated as Q (Quoziente).

Now, let’s proceed with the proof.

We will prove that √2 is not a rational number by assuming that it is a rational number and finding a contradiction. Say that √2 was indeed rational. Then we can express √2 in the form of a/b where a and b are relatively prime with each other. When we take the square root of both sides we see the following.

Now, on the last equation, we can see that the left-hand side contains an odd number of 2’s in it. We know this because square numbers are made up of prime numbers with even exponents. That is what makes them square numbers. Therefore b² contains an even number of 2’s in itself. And we have one more 2 (multiplied with b²) on the left-hand side which makes an odd number of divisor 2 on that side. However, by similar logic, the right-hand side only contains an even number of divisor 2, since the right-hand side only has a square number. (Remember again that all square numbers contain prime factors with even exponents!) Since if two expressions are equal, the number of divisor 2’s in these expressions must be the same but in this case, they are not! This is such a huge contradiction! Therefore, √2 is not a rational number!..

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