# How to prove √X is irrational number

1. Mar 5, 2010

### SOHAWONG

when X is even number,it's easy to prove
but how about the condition which X is odd number?
I have no idea of this

2. Mar 5, 2010

### Hurkyl

Staff Emeritus
$\sqrt{4}$ is irrational?

3. Mar 5, 2010

4. Mar 5, 2010

### Char. Limit

So in other words...

$$\sqrt{x}$$ is irrational iff x=/=n^2 for n belonging to the integer set.

5. Mar 5, 2010

### SOHAWONG

yes, but how to prove?

6. Mar 6, 2010

### Tinyboss

Fundamental theorem of arithmetic. Assume p^2/q^2=x with gcd(p,q)=1, and see what has to divide what.

7. Mar 6, 2010

### SOHAWONG

what does gcd mean?

8. Mar 6, 2010

### Tinyboss

Greatest common divisor. If gcd(p,q)=1, it means the fraction p/q is in lowest terms.

Look at the proof for sqrt(2), and adapt it. Remember that "even" just means "is divisible by 2", so that if you're checking a number other than 2, you won't be thinking about "even" anymore.

9. Mar 6, 2010