# Irrationality of Difference of two numbers

Proof of Irrationality

How can I prove that the square root of 8 minus the square root of 3 is an irrational number using the fact that the square root of 3 is an irrational number? I know I need to use a proof by contradiction, but I am stuck after that.

## Homework Statement

$$\sqrt{8}$$-$$\sqrt{3}$$ is an irrational number.
Use the fact that $$\sqrt{3}$$ is an irrational number to prove the following theorem.

## Homework Equations

A rational number can be written in the form $$\frac{p}{q}$$ where p and q are integers in lowest terms.

## The Attempt at a Solution

I know that I need to use a proof by contradiction to solve this problem. Therefore, we would assume that $$\sqrt{8}$$-$$\sqrt{3}$$ is an rational number and that $$\sqrt{3}$$ is an irrational number and try and reason to a contradiction. I am stuck and don't know how to get to the contradiction.

MathematicalPhysicist
Gold Member
well it's actually asking to prove that
(sqrt(2)+sqrt(3))+sqrt(2)
is irrational.
assume it equals p/q where (p,q)=1
then multiply by sqrt(8)-sqrt(3)
youll get that sqrt(8)-sqrt(3)=5q/p
and you also have sqrt(8)+sqrt(3)=p/q
so you get that 4sqrt(2)=5q/p+p/q
which yields: sqrt(2)=(5q/p+p/q)/4 which is a contradiction.
you could have easily have done it with sqrt(3) instead but it doesnt matter.

Last edited:
MathematicalPhysicist
Gold Member
well, i had mistaken sqrt(8)-sqrt(3) with sqrt(8)+sqrt(3) but it doenst matter the same idea will work as well as in this case.

matt grime
Homework Helper
Suppose that it is, then this implies that the square root of 8 is what?

WWGD
Gold Member
Just to add something to review my number theory.Another idea:

I think a nice general result is that for x a pos. integer, sqr(x) is rational iff x is a perfect square (an integer, of course). Think x=a^2/b^2 , so a^2x=b^2 . Then think of what the factorization of x needs to satisfy in order for a^2x to be a perfect square.

x=p_1^e_1...p_ne^n .

Then , re your problem, think of what happens when you square your expression.

HallsofIvy