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    Revisiting Irrational Numbers

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6th class > > Revisiting Irrational Numbers

Revisiting Irrational Numbers

Recall, a number ‘s’ is called irrational if it cannot be written in the form ,pq where p and q are integers and q ≠ 0. Can we prove that some numbers are irrational? How can we prove that some numbers cannot be written in the form of pq.

To prove that a number like 2 is irrational we need some help. We first need to prove the following:

Let p be a prime number. If p divides a2, then p divides a, where a is a positive integer.

We know from before that every number can be represented by its prime factors.

Let a=p1p2p3...pn, where p1,p2 etc are primes.

Then we have a2=(p1p2p3....pn)(p1p2p3....pn)=p12p22p32*....pn2.

We are given the fact that p divides a2. That means from Fundamental Theorem of Arithmetic, p is a prime factor of a2 and p is one of p1,p2,p3,....pn.

But we assumed a=p1p2p3....pn. And we just proved that p is one of p1,p2,p3,....pn.

From this we can clearly see p divides a.

Let us now try to prove 2 is irrational.

Let us say 2 is rational.

That means we have a p and q such that 2=pq.

Suppose p and q have a common factor other than 1. Then, we divide by the common factor to get ,2=ab where a and b are coprime.

Basically we try to remove the common factors between p and q so that there are no common factors excpet 1 which makes a and b as coprimes.

Squaring on both sides of the equation we get 2=a2b2

That is 2b2=a2. Therefore 2 divides a2 which means 2 divides a(which we proved in the above theorem).

If 2 divides a then we can write a=2c for some integer c.

Substituting for a we get 2b2=2·c2

That is 2b2=4c2

This leads to b2=2c2

That means b2 is divided by 2 which means b is divided by 2.

So a is divided by 2 and b is divided by 2. But this is not possible as a and b are coprimes. We have reached an impossible state in our world. We reached this state because we made a wrong assumption. The assumption that 2 is rational. Since this assumption is wrong we are left with only one alternative.

And that is, 2 is irrational.

This approach of proving things is known as proof by contradiction.