Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Proving there is no smallest positive number

  1. Oct 1, 2011 #1
    1. The problem statement, all variables and given/known data
    "True or false: there is a smallest positive number. Explain."

    2. Relevant equations
    N/A, but for practice I'll try my hand at phrasing it mathematically:

    3. The attempt at a solution
    My issue with the question is mathematically proving it - I'm a bit paranoid because I've been losing a lot of marks on communication and I don't think it'll be enough for me in this particular class to simply say that the statement is false because there is an infinite amount of numbers between 0 and 1. So, I was thinking it could be proven in a way similar to how we prove there is no largest real number...
    Let z be the smallest positive real number such that 0<z<x where x[itex]\in[/itex](0,∞):
    let x=z-1
    0<-1 which is not true. Therefore, the statement is false and there is no smallest positive number.
    Is this a logical argument? This is my first course in proofs, and I'm a freshman, so I don't feel very confident in constructing my arguments. Mainly I would just like some feedback, and if I'm doing something wrong, could someone hint towards the correct argument...? Any response is much appreciated : )
  2. jcsd
  3. Oct 1, 2011 #2
    If z<x, why does x=z-1 ? I would try a contradiction. Let x= the smallest positive number. Then there is no number z such that x>z>0. Let z=x/2.... its a little course in the phrasing but you see what I'm trying to do?
  4. Oct 1, 2011 #3
    Right, that is a much better argument...I suppose I just misunderstood the proof that there is no largest real number which I came across in my calculus text. : \
  5. Oct 1, 2011 #4
    I do that all the time. Flip a sign here, switch all for exists there, and before you know it, you're proving the wrong thing. It got me once on a test >_<
  6. Oct 1, 2011 #5


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    I see that you apparently understand the problem now.

    One way to approach it would be to ask yourself, if given a positive number, x, what number is between x and zero?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook