Hi Steemicians. This is Math Owl. Today I want to show a simple proof for the existence of an irrational number. More specifically, we will show that 
Irrational and rational numbers
Irrational numbers are numbers which are not rational numbers. Now you might be wondering what a rational number is. A rational number is a number which can be expressed as the fraction of two whole numbers. Some examples of rational numbers are 1/3, 4/5, 2, 6/7. So what does it mean if a number is irrational? Well it simply means that it cannot be expressed as the fraction of two whole numbers.
Existence of irrational numbers
So do irrational numbers exist? You might have heard that numbers like
is an irrational number
We will use a handy proof method called proof by contradiction. Proof by contradiction works by assuming that the opposite is true and then showing that such an assumption leads to a contradiction. For example if we want to apply proof by contradiction to the statement: sharks are not owls. Then I would procceed by assuming the contrary: sharks are owls (1). Since owls can fly it follows from (1) that sharks can fly. But this is in contradiction with the fact that sharks are animals which cannot cannot fly. Then we arrive at the conclusion that sharks are not owls. Summarizing: If a shark is an owl (1) then sharks should be able to fly but sharks cannot fly; therefore, sharks are not owls.
We return to proving that
and such that there is no number which divides both p,q other than one. The latter means for example that we write 14/10 as 7/5.
Squarying both sides gives
Then rearranging terms yields
We now observe that p must be an even number! So there must exist an r such that 
But now we observe that q must also be an even number. Recall that there is no number which divides both p,q other than one. This gives us the desired contradiction which proves that
Proofs for other irrational numbers
It can also be shown that 
There are also numbers which are suspected to be irrational but for which we still do not have a prove. A famous example is the Euler-Maschernoni constant. Source
Further reading and source
This post is loosely based on the first chapter of Understanding Analysis by Stephen Abbott (Springer, 2002). This chapter also contains many other fun and fundamental mathematical concepts. It is a good place to start learning more about mathematics :)
Thank you!
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