In mathematics, we normally describe a line as, for example, the set of points $\{(x,2x+1):x\in\mathbb R\}$:
Then the function $\phi:x\mapsto(x,2x+1)$ is an easy bijection between the line and $\mathbb R$.
But what if we defined it as the set of points $\{(x,2x+1):\mathbb Q\}$ instead? Then it would be in bijection with $\mathbb Q$ instead. So what would it look like then?
Oh. It's the same. The rational numbers form a dense set (for any rational numbers $x,y$ there is a rational number between them (e.g., $\frac{x+y}2$), so they do 'fill in' space in some way. But we would then be introducing more problems, such as:
The function $x^2-2$ would have no zeroes. That certainly doesn't coincide with our intuition from looking at the graph:
In addition, the function defined by taking
$$
f(x)=\begin{cases}0&x^2<2\\1&x^2>2\end{cases}
$$
would be continuous! In order to get round these, we introduce the real numbers, which are complete in the sense that any Cauchy sequence - i.e., any sequence that you would intuitively expect to converge to a value - does in fact converge.
Note that we could have used the algebraic numbers or some other countable set instead of the rational numbers, but would have run into similar problems. We could even use the definable numbers, which are a countable set: in that case, we wouldn't be able to construct these sorts of counterexamples to our intuition, as we could turn them into definitions of undefinable numbers; however, the definition of the real numbers isn't that hard, and it's what mathematicians are used to, and it allows us to prove lots of beautiful results, so that's what we use.
For a discussion on the use of real numbers in the form of a dialogue, see here. I'll just give the closing line.
Mathematician: So, finally, we arrive at the following justification for real numbers. 1. We must go further than just the rationals. 2. When we do so we introduce certain procedures that give us new numbers. 3. Formalizing these, we end up with the monotone-sequences axiom, or something equivalent to it. 4. This axiom is not as precise as it seems, since the notion of an arbitrary monotone sequence, even of rationals, is not precise. 5. There is no need to make it precise, because we know how to reason in terms of arbitrary sequences. 6. That allows us to define the real numbers we have a use for, even if it gives us a lot of junk as well. 7. In fact, we don't really know what junk it does give us, and it's not even clear that it makes sense to ask.
Best Answer
Who says the lines have to be staight or even represent a function in the form y = f(x) ?
Suppose that a line is instead defined parametrically where x = x(t) and y = y(t) and we require that x(t) and y(t) are continuous functions of the real variable t.
Let X be the set of all such functions {x(t)} and similarly Y = {y(t)}. If L is the set of possible lines then the cardinality |L| = |X x Y|
But X = Y = C the set of continuous functions of a real variable, and |L| = |C x C|
As a general result, for any set S we have |S x S| = S, so |L| = |C| and the only question is then what is |C|.
A particularly neat result is that a continuous function is compeltely specified by its values at rational points, so that it is equivalent to the set of continuous functions on Q -> Q.
The continuous functions on Q are a subset of all functions Q -> Q, i.e. each continuous function is a subset of Q x Q, so that C is a subset of P(Q x Q) and consequently |C| <= |P(Q x Q)| = |P(Q)| = |R|. I.e |L| <= |R|
It's also easy to see that we can inject R into L by a function f(x) = (x, 1) so that |R| <= |L|
Putting that together (Cantor Bernstein theorem) then |L| = |R| and there is a bijection between them.