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 is an irrational number. This proof is a good example of a proof which relies more on logic than computation. But before I can start I need to explain what irrational numbers are.

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
are irrational but this requires a proof. I will show that is an irrational number.

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 is an irrational number. As announced we use proof by contradiction. So suppose that is not an irrational number. Hence, we suppose that is a rational number. Then there must exist whole numbers p,q such 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 . Inserting this in the previous equation we get

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 is an irrational number.

Proofs for other irrational numbers

It can also be shown that are irrational numbers but these proofs are a bit harder. Proofs can be found here Proof pi and Proof e

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

Thank you!

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