EUCLID’S PROOF THAT √2 IS IRRATIONAL NUMBER

 

Readers who have read the earlier article “ Irrational Action” will know about the birth of the irrational number (and the consequent death of the discoverer). Readers who have not read that article can go to that article by clicking on the above link; otherwise, you have not missed much and you need not worry.You can go there and read later.

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An irrational number is one that cannot be expressed as ‘p/q’ where both ‘p’ and ‘q’ are integers. How can you conclusively prove that a specific number is definitely an irrational number as one can always argue that there may be a very large numerator and denominator leading to that value?

 

{In Mathematics, as in the court of law, the burden of proof, beyond any shadow of doubt, lies with the man making any claim. Hence if I say √2 is irrational, I need to jolly well prove it conclusively.}

 

It was Euclid, [yeah, the same chap who troubled us in Geometry] who gave a wonderful proof that √2 is irrational.

 

To achieve that, he employed a novel method developed by early Greeks called “Reductio Ad Absurdum”. {We all feel naively that scientists, and especially Mathematicians, are straight laced. Along with Euclid’s proof, I will simultaneously prove nothing is farther from the truth.}

 

Shred of the entire hullabaloo that surrounds it, " Reductio Ad Absurdum " goes like this. First Euclid will state just the opposite of what he wants to prove. {Exactly like our current politicians.} The unwary student or the reader in all innocence will go along with him. After that he will take you step by step along with him and show that it is foolish to have agreed with him in the beginning and nothing can be more absurd than our assumption. But you are not told when he will employ this trick. Next time, if you try to disagree with him, he may end up proving that it is absurd to have disagreed with him by proving it conclusively.

 

Hence next time, you see any mathematical proof starting with the words “ Let us assume”, be on guard.

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THE PROOF:

 

We need to be aware of the following facts before we go to the proof:

 

1)      If we take any number, say-p, and multiply that number by 2, the resultant number is an even number. -Well, this happens to be the definition of an even number.

 

2)      Supposing we have an even number, say- p, then the square of that number is also even. (We are multiplying two even numbers, you see!) The converse is also true. If we come across any even square number, we know that the square root of that number needs to be even as well.

(The same property exists for odd numbers as well; but for our purpose here, that is irrelevant)

 

3)      Supposing we have any number expressed in the form of  ‘p/q’, then we can reduce the values of ‘p’ and ‘q’ in such a way that the fraction is expressed in the lowest form by removing any common divisors from both the numerator and denominator. Every fraction ‘p/q’ with p and q as whole numbers will have this property. (For example, 8/12 can be reduced to 4/6 and then to 2/3; we can go no further.)

 

Armed with this knowledge and knowing the ‘ absurdum trick’ I have warned you about, we can now go through the Euclid’s proof.

 

Let us assume that√2 is a rational number; i.e., it can be expressed as ‘p/q’ and let us carry on our arguments and see what happens.

 

√2 = p/q   ……………………..(I)  

 

     2 = p²/ q²

 

 Hence, 2 q² = p² ………………..(II)

 

Now we can see that p² as well as p need to be even numbers. (Refer point 2)

 

Since p is an even number, we can write it as equal to 2m where ‘m’ is a whole number.

 

i.e., p = 2m; hence p² = 4 m²  …………………………(III)

 

Hence, 2q² = 4m²; q² =2 m²……………………………………...(IV)

 

From the equation (IV), we now know that q is also even. Now ‘q’ can be expressed as ‘2n’ where ‘n’ is some other whole number. i.e., q = 2n

 

Hence, q² = 4 n² ………………………………………..(V)

 

Substituting in (I), we get √2 = p/q = 2m/2n = m/n. Thus we find that we have reduced p/q to m/n. But then, we can go on doing this again and again and reduce further and further with no end in sight. However we know that as per our point (3), it cannot be so. Any rational number, after being divided by all the common divisors of the numerator and denominator will get reduced to its lowest form, whereas √2 expressed as p/q refuses to do so. Hence our assumption that √2 is rational at the beginning of this argument must have been a wrong one.

 

It means that√ 2 must be an irrational number.