# Why is the set of rational numbers dense, and set of integers numbers not?

#### jenniferruurs

If we have two sets:
1. Set one is the set of rational numbers with the usual less-than ordering
2. Set two is the set of integers numbers with the usual less-than ordering

Why is the set of rational numbers dense, and set of integers numbers not?

Density is that for all choices of x and y with x < y there is a âˆˆ A with x < a < y.

Can somebody show me this via an example?

#### idontknow

Because two rational numbers are more close to each other than two integer numbers.
Simply there are more values of $$\displaystyle a$$ that can be found in $$\displaystyle (x,y)$$.

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#### jenniferruurs

Because two rational numbers are more close to each other than two integer numbers.
Simply there are more values of $$\displaystyle a$$ that can be found in $$\displaystyle (x,y)$$.
What do you mean with this, can you give an example maybe?

#### topsquark

Math Team
What do you mean with this, can you give an example maybe?
You can always find a (countably) infinite number of rational numbers between any two rational numbers. You can't do that with the integers.

-Dan

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#### jenniferruurs

You can always find a (countably) infinite number of rational numbers between any two rational numbers. You can't do that with the integers.

-Dan
So for the integers an example would be that there is no number between 2 and 3. Is this correct?

#### Benit13

Math Team
So for the integers an example would be that there is no number between 2 and 3. Is this correct?
Well... yeah!

#### JeffM1

So for the integers an example would be that there is no number between 2 and 3. Is this correct?
Yes, but it might be more illuminating to say that there are three integers between 3 and 7, but you cannot say that there is a finite number of rational numbers between 1/7 and 1/3.

#### Maschke

You can always find a rational between any two rationals.

You can't always find an integer between any two. For example there's no third integer between 2 and 3.

2 people