Reductio: Argument to Absurdity is Mathematical Judo

 
Reductio is a form of argument process that was used by Aristotle and still remains very powerful way of proving or disproving a theory.

You start with a theory H that you claim is true

You then make a deduction that if H is true then F has to be false or True 

You then prove the  F to be True or Falseand hence you have proven that H is false

Lets do some Mathematical Judo
Theory:  The square root of 2 is a rational number
i.e √2 is a fraction m/n where m and n are whole numbers
let m/n be in its lowest form, that is to say that the numerator and denominator does not have any common factors it it did you could reduce the fraction to its lowest form by dividing by the factor   e.g if m/n was 9/15  then there is a common factor of 3 and hence in its lowest form you would get 3/5

Theory:  √2 is a fraction m/n where m and n are whole numbers with no common factors
Now if it is lowest form, then both m and n can not even numbers, if they were then you would be able to divide both by 2
so the statement:  both m and n are evenhas to be false
so as √2 =m/nthen 2 = m2/n2then 2n2=m2

regardless of if n2 is an odd or even 2 x n2  will be eventhis means that m2 = even  if m2  is even then m itself has to be EVENanother way of saying m is even is to say that m = 2k where k is any real number (odd or even)


so m2 = (2k)2
taking  2n2=m2 2n2=4k2 n2=2k2

as logic dictates 2 x any number will be EVEN so  n2=2k2  means that  n2  EVEN and hence n is EVEN


Now if we say H is true, then we have arrived at a falsehood or even a statement F being both true and false at the same time, an absudity. 
This means that H cannot be true, we proved the H to be false by assuming it was true.  Aristotle !!!


Ref:  How not to be wrong.  The power of Mathematical Thinking by Jordan Ellenberg