# Homework Help: Smallest positive irrational number

1. Apr 30, 2010

### kolley

1. The problem statement, all variables and given/known data

Prove that there is no smallest positive irrational number

2. Relevant equations

3. The attempt at a solution

2. Apr 30, 2010

### Staff: Mentor

Do you know how to do a proof by contradiction? If so, assume that there is a smallest positive irrational number, and then produce another one that's even smaller.

3. Apr 30, 2010

### kolley

I thought the way to do it was by contradiction. But I'm confused as to how to produce a generalized irrational number, and then like you say, get one smaller than that.

4. May 1, 2010

### Tedjn

Label your arbitrary irrational number a. Starting with a, can you think of a way to get a number smaller than a? There may be many ways to do this. Once you have a candidate idea in mind, try to prove that the number you get is always irrational.

5. May 1, 2010

### nsnayak

You could try using the density of rationals in R

6. May 1, 2010

### HallsofIvy

If r is a positive irrational number, then r/2 is a smaller positive irrational number.