Smallest positive irrational number

[kolley]

Homework Statement

Prove that there is no smallest positive irrational number

Homework Equations

The Attempt at a Solution

I have no idea how to do this, please help walk me through it.

 

[Mark44]

Homework Statement

Prove that there is no smallest positive irrational number

Homework Equations

The Attempt at a Solution

I have no idea how to do this, please help walk me through it.

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.

 

[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.

 

[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.

 

[nsnayak]

You could try using the density of rationals in R

 

[HallsofIvy]

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