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Proof that Root Two is Irrational: How to Prove that Root Two is Irrational Using Proof By Contradiction

Updated on July 18, 2012

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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    • Spirit Whisperer profile image

      Xavier Nathan 6 years ago from Isle of Man

      Hi topquark I wrote one called "Proof by Contradiction" about 4 months ago. Let me know what you think :)

    • profile image

      calculus-geometry 6 years ago

      One of my favorite proofs!

    • topquark profile image

      topquark 6 years ago from UK

      Did you do one on this topic? I haven't been around for a while. *goes to look*

    • Spirit Whisperer profile image

      Xavier Nathan 6 years ago from Isle of Man

      Copy cat! :)


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