**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:

(source)

Irrational numbers can be represented as – , 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:

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:

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:

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

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 **b **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.

Hands down my favorite exercise, and the story that this truth haunted a man to his grave is just too good to pass up!

LikeLiked by 1 person