# Every root of 2 is irrational.

1. Jan 15, 2010

### Char. Limit

I was thinking on the square root of 2 being irrational proof... and I got the idea that you could use the same idea for every root higher than two. The cube root, the quartic root, the quintic root, etcetera. (Obviously assuming the roots are natural numbers.)

As a reassurance I'm not crazy, is this correct?

2. Jan 15, 2010

### Hurkyl

Staff Emeritus
You are not crazy. At least for this particular reason.

3. Jan 15, 2010

### Char. Limit

Good to know.

Is there a way to prove that other roots that aren't natural numbers are irrational?

4. Jan 15, 2010

### Hurkyl

Staff Emeritus
Prime factorization.

5. Jan 15, 2010

### Hurkyl

Staff Emeritus
Also, rational root* theorem.

*: as in, root of a polynomial

6. Jan 15, 2010

### Char. Limit

So... are all roots of two that aren't 1 irrational?

7. Jan 15, 2010

### HallsofIvy

Staff Emeritus
Almost. Of course the "1" root of 2, 21/1= 2 is rational! If n is a positive integer, greater than 1, then 21/n is irrational.