MATHS - 3.REDUCTIO-AD-ABSURDUM

TEACHING REDUCTIO-AD-ABSURDUM.

Maths is a science .It is universally true always. That is 2 +1=3 … every where …every time. But teaching is an art. It is individual. That is one student may read well if given encouragement, but another may not behave the same , he may need intensive coaching ..yet another may need even occasional stick!! Behaviour may change for the same person with time and place.

Let us see with an example how teaching can effect the understanding of a big sounding method in Maths.

REDUCTIO-AD-ABSURDUM : Name looks scary and in-fact many students to day do not know it by this name though they were taught and using it from 8/9^th.class.This is how our Teacher explained….

Your Mother told “Don’t jump from a wall ! If you jump from a wall, you will break your bones”. This is a theorem to be proved. So let us prove it by assuming that the conclusion (to be proved ) is wrong. So jump from the wall.. you will break a bone…So our assumption is wrong….Hence the theorem is proved…

This is a method of proving a theorem by contradiction. We start by assuming that the conclusion (to be proved) is wrong. Then we end up with a contradiction to a known truth or the hypothesis (given data). Hence we conclude that our assumption is wrong….that is the theorem is correct.

This is a very powerful method with a wide variety of applications. You are aware of this method in solving theorems in geometry and in proving irrational numbers etc. Now let us see its application to a seemingly tough and not usually heard of problem but well within our scope.

You know a rational number as that which can be expressed as p / q where p and q are integers. You also know that square root of 2 is about 1.414….etc and it is an irrational number because it can not be expressed as p / q where p and q are integers. Now the question is can an irrational number raised to an irrational number power be a rational number ? In-fact the the theorem to be proved is that an irrational number raised to an irrational number power can be a rational number .

At the out set it looks baffling as we do not even know how to raise an irrational number to an irrational number without using logarithms or a calculator.

But let us remember that we have a powerful tool in REDCUTIO - AD – ABSURDUM . Let us put it to use .

Let us assume that the theorem is wrong .That is an irrational number raised to an irrational number power can not be a rational number .

We know square root of 2 is an irrational number. Let us raise it to again square root of 2 power. As per our assumption it is an irrational number. Now let us raise this irrational number to square root of 2 power which should once again be an irrational number as per our assumption.

Look now what we get …

[{(square root of 2)^sq.rt(2)}^sq.rt(2)] = [{sq.rt(2)} ^{sqrt(2)*sqrt(2)}] = [sqrt(2)]^2=2

We get 2 as the answer .

It is a perfectly rational number….even an integer….so our assumption is wrong . So the theorem that an irrational number raised to an irrational number power can be a rational number is correct. Got It ? How does it look ?

Now , one question about your understanding of this proof .What if that our first step , that is , square root of 2 raised to square root of 2 power is an irrational number is not correct ?

Well my Dear Watson as Sherlock Homes used to say , then there is no problem …the theorem is already proved at the first stage i