Smallest positive irrational number

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Homework Help Overview

The discussion revolves around the proof that there is no smallest positive irrational number, a concept in real analysis and number theory.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore proof by contradiction, questioning how to generate a smaller irrational number from an assumed smallest one. There is discussion on the generalization of irrational numbers and methods to demonstrate their properties.

Discussion Status

Participants are actively engaging with the problem, offering various approaches and considerations. Some have suggested specific methods, such as using the density of rationals or manipulating a given irrational number to find a smaller one.

Contextual Notes

There is an emphasis on understanding the properties of irrational numbers and the implications of their density within the real numbers. Participants are navigating the challenge of constructing a proof without definitive guidance on the steps involved.

kolley
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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.
 
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kolley said:

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.
 
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.
 
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.
 
You could try using the density of rationals in R
 
If r is a positive irrational number, then r/2 is a smaller positive irrational number.
 

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