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So the full problem is

Prove that between every two rational numbers $a/b$ and $c/d$ that:

  1. There is a rational number
  2. There are an infinite number of rational numbers

I am having some trouble on which approach I should use, but I tried to solve part 1.

Part 1) So I believe I proved that between every two rational numbers $a/b$ and $c/d$ that there is a rational number

I have my base case as $a,b,c,d$ all being integers.

My solution:

For some number $x$, $$ x = \frac{\dfrac a b + \dfrac c d}{2} = \frac{ \dfrac{ad}{bd} + \dfrac{cb}{bd}} 2 = \frac{ \dfrac{ad + cb}{bd} } 2 = \frac{ad + cb}{2bd} $$

I concluded that ad + cb must be an integers because addition/subtraction of integers leads to an integer. Same case for $2db$.

Not sure if this is correct though

Part 2) Prove that there are an infinite number of rational numbers

I am confused onto how to solve this part, where to start, and how to use the answer from part 1 into this problem.

I've been toiling over this, so your input is greatly appreciated. :)

Community
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Rohit Tigga
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  • Your first part looks good. For the second part, let your function for the midpoint of two rational numbers be $f(a,b)$. Then: you know 0 and 1 are rational, and $f(0,1)$ is rational, and $f(0,f(0,1))$ is rational, and so on... ad infinitum. – Jeff Snider Jan 29 '14 at 20:53
  • Part 2 : If there were finitely many rational numbers, you could sort them to their value. Then, no rational number would be between two consecutive numbers in this list, contradicting the statement in part 1. – Peter Jan 29 '14 at 20:54
  • Well if you can prove that there are at least two rational numbers $x$ and $y$, you know that you can find a rational at their midpoint call it $z_1$. Then consider $x$ and $z_1$. You know that these are rational and so you can find a rational between them. You can continue this ad infinitum. – Cameron Williams Jan 29 '14 at 21:02

1 Answers1

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Hint, between one of the original rationals and the one you just found you can find another rational number, and between one of the original rationals and the one you just found you can find another rational number...

Think induction.

kleineg
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