Give an example of a dense linear ordering

Question

Can someone give an example of a dense linear ordering. I know what it needs to satisfy, but an example would be great for understanding.

Answer 1

I'll give you an example and a counter example.
Example : take $\Bbb{R}$ with the usual order. It is a linear order, and if you take 2 elements in $\Bbb{R}$, say $a$ and $b$, with $a<b$ then $\frac{a+b}{2}$ is between $a$ and $b$ in term of order, and it is also a real number so the density is fulfiled.
Counterexample : look at $\Bbb{Z}$ with the usual order. It is a linear order, but it is not dense : if you look at 1 and 2, there is no element between them.

Answer 2

Take the set of rational numbers $\mathbb{Q}$ with the usual order relation $<$. It is:

1. a strict order (i.e. a transitive and irreflexive binary relation),
2. linear (i.e. for any $p, q \in \mathbb{Q}$, either $p = q$ or $p < q$ or $q< p$,
3. dense (i.e. for any $p, q \in \mathbb{Q}$, if $p < q$ then $p < \frac{p+q}{2} < q$, where $\frac{p+q}{2} \in \mathbb{Q}$).

Note that the set of natural numbers $\mathbb{N}$ with the usual order relation $<$ is a linear ordering that is not dense, because between $0$ and $1$ there is no natural number.