Why is the square root of 2 irrational?

Answer 1

By definition an irrational number is one that cannot be expressed in a fraction x/y where x and y are integers (whole numbers). If the screen on your calculator went on forever and you typed in square root of 2, the result would run forever (just like pi). Because the square root of two does not end (the value after the decimal continues) it cannot be expressed in the form of a fraction x/y therefore it is an irrational number.

Answer 2

The proof is by the method of reductio ad absurdum.

We start by assuming that sqrt(2) is rational.

That means that it can be expressed in the form p/q where p and q are co-prime integers.

Thus sqrt(2) = p/q.

This can be simplified to 2*q^2 = p^2

Now 2 divides the left hand side (LHS) so it must divide the right hand side (RHS).

That is, 2 must divide p^2 and since 2 is a prime, 2 must divide p.

That is p = 2*r for some integer r.

Then substituting for p gives, 2*q^2 = (2*r)^2 = 4*r^2

Dividing both sides by 2 gives q^2 = 2*r^2.

But now 2 divides the RHS so it must divide the LHS.

That is, 2 must divide q^2 and since 2 is a prime, 2 must divide q.

But then we have 2 dividing p as well as q which contradicts the requirement that p and q are co-prime.

The contradiction implies that sqrt(2) cannot be rational.

Answer 3

Yes because it can't be expressed as a fraction