Question

explain steps of proving irrationality​

Answer 1

Answer:

The steps to prove the irrationality of ANY number are essentially the same. You start by assuming your number can be written as the irreducible fraction of integers a/b where b =/= 0, then manipulate to show it CAN be reduced, thereby creating a contradiction.

Step-by-step explanation:

: Proving irrationality

Let’s look at Euclid’s proof that the square root of two is irrational.

Let us assume that the root of two is rational and represent it as such:

$p$ and qq are assumed to be co-prime integers, i.e., the fraction is fully reduce

p2=2q2p2=2q2

We see that pp must be even, since its square is even, making qq odd since they are co-prime. We can then let p=2mp=2m and q=2n+1q=2n+1 to represent an even and odd number respectively.

(2m)2=2(2n+1)2(2m)2=2(2n+1)2

Expansion of the squares yields:

4m2=2(4n2+4n+1)4m2=2(4n2+4n+1)

Divide through by 2 and partially factor the right side:

2(m2)=2(2(n2+n))+12(m2)=2(2(n2+n))+1

Let u=m2u=m2 and v=2(n2+n)v=2(n2+n) and substitute:

2u=2v+12u=2v+1

Here we absurdly have an even natural number being equal to an odd one. This impossibility means without a doubt that the assumption we made regarding the rationality of the square root of two is false. We can also look at it this way by dividing through by 2:

u=v+12u=v+12

This of course is also impossible for natural numbers, to be one half unit apart. This is what's known as proof by reductio ad absurdium. You begin with an assumption and then reduce the statement to an absurdity, thereby proving the assumption to be false.

This is essentially the proof given by Euclid in his The Elements.