Proof that Root Two is Irrational: How to Prove that Root Two is Irrational Using Proof By Contradiction
Proving that the square root of two is irrational is an example of a proof by contradiction.
The basic method is to suppose that square root two is rational and then show that this leads to a contradiction (an equation that is obviously not true).
Prove that root two is irrational
A rational number is one that can be written as a ratio of two integers (a fraction):
Rational number = p / q , where p and q are integers (whole numbers).
An irrational number can't be written in this way.
1. Suppose that square root two is rational
√2 = p / q
where p and q are integers. Assume that this fraction is expressed in its simplest form: i.e.,p and q have no common factors (this will be important later).
2. Show that this leads to a contradiction
First, square both sides:
2 = p2 / q2
q2 = 2 p2
Now, we can draw some conclusions about p and q.
q2 must be even, because it is equal to 2 times an integer.
Therefore, q must also be even. (Because: even x even = even; odd x odd = odd)
Because q is even, we can write q it as 2 times an integer. We call this integer k.
q = 2 k
q2 = (2 k)2 = 4 k2
We put this back into the equation q2 = 2 p2 to get:
4 k2 = 2 p2
Rearrange to get an equation for p2
p2 = 2 k2
This means that p2 is even (because it is equal to two times an integer). Therefore, p is also even.
This is where the contradiction lies. If p is even, and q is even, then they both have q as a factor. Our original assumption stated that p and q had no common factors.
Our assumption that the square root of two is rational has led to a contradiction. Therefore, it must be untrue.
We have proved that the square root of two is irrational.
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