Results 1 to 2 of 2

Thread: Rational and irrational numbers

  1. #1
    Oct 2008

    Rational and irrational numbers

    True or false

    Between any two unequal irrational numbers there is a rational number.

    I'm leaning towards true, but I can't come up with a convincing argument.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Nov 2008

    What you think is right, what is said is that $\displaystyle \mathbb{Q}$ is dense in $\displaystyle \mathbb{R}.$

    Let $\displaystyle a,b,\ a<b$ be two (unequal) reals. Then $\displaystyle d=b-a>0.$ Since the sequence $\displaystyle \left(\frac{1}{k}\right)_{k\in\mathbb{N}^*}$ strictly decreases and has limit $\displaystyle 0,$ there is a $\displaystyle n\in\mathbb{N}$ such that $\displaystyle \frac{1}{n}<d.$

    Let $\displaystyle m\in\mathbb{Z}$ be the greatest integer such that $\displaystyle \frac{m}{n}\leq a.$ Then $\displaystyle a<\frac{m+1}{n}=\frac{m}{n}+\frac{1}{n}\leq a+\frac{1}{n}<a+d<b,$ therefore $\displaystyle a<\frac{m+1}{n}<b$

    Since $\displaystyle m,n,\ n \neq 0$ are integers, $\displaystyle \frac{m+1}{n}$ is a rationnal; we've proved that between two unequal reals lies a rational number. In particular, there's always a rational number between two unequal irrational numbers.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Irrational and Rational Numbers
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: Sep 25th 2011, 02:32 PM
  2. Rational and Irrational numbers
    Posted in the Algebra Forum
    Replies: 9
    Last Post: Apr 20th 2010, 03:00 PM
  3. Induction on irrational and rational numbers
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: Feb 2nd 2010, 03:34 AM
  4. Replies: 8
    Last Post: Sep 15th 2008, 04:33 PM
  5. Rational and Irrational numbers
    Posted in the Math Topics Forum
    Replies: 11
    Last Post: May 23rd 2007, 08:50 AM

Search Tags

/mathhelpforum @mathhelpforum