Irrationality of √2

Brief explanation: This is a quick follow-up on our last exercise. If you haven’t seen it, check it here!

— — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —

Question: Prove that the square root of two is an irrational number.

— — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —

Comprehending: This is a ‘classical’ proof, given in most calculus books. I decided to put it in here because of one thing in particular: sometimes, to prove something, you assume the contrary is true until you reach an impossible situation. This is called reductio ad absurdum and was described by G.H. Hardy as one of a mathematician’s finest weapons[1]This is one of the most simple proofs that use this kind of argument.

Remember what we studied in the last exercise: an even number is divisible by two. This will be a key fact in our proof.

By the way, let’s quickly review what an irrational number is. An irrational number is a number that cannot be expressed has a division of integers, meaning it can only be represented by infinite decimals. The following graph may help you:

Number-systems.svg.png

(source)

Irrational numbers can be represented as \mathbb{R} – \mathbb{Q}, the space between these two groups.

Now, let’s get to the proof.

— — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — — —

Solving: First, assume √2 is a rational number. This is the most important step towards the proof.

If √2 is rational, then:

1.png

This is just the definition of being a rational number, being the quotient of two integers, a and b, with no common divisor. If we continue this line of thought:

2.png

We just took the square of both sides and arrived at a curious result. This means a^2 is an even number. Because of the result of our last problem, a is also a square number! This means that:

3.png

Because a is even. We can use these two equations two obtain another curious result:

4.png

Again, what does this means? It means b^2 is also even! By our last exercise, b is also even! But wait…

If a and are both even, how is it possible that they have no common divisor? You could divide them by 2, right? Hence, we arrived at a contradiction. This means √2 cannot be a rational number! If it’s not a rational number, it can only be irrational. Our proof is now complete.

Advertisements

One thought on “Irrationality of √2

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s