Cartesian Plane and Number Lines

Question

good day. May I ask how many number lines are there in a cartesian plane? I'm clueless. Thank you so much

Answer 1

There is an unique line through any two points of the Euclidean plane. By assigning $0$ to one, and $1$ to the other point, you uniquely specify an uniform numbering of the line, which in my interpretation of the word makes it a number line.

Therefore there are as many number lines as there are pairs $(p,q)$ of points with $p\ne q$. Now from set theory, we know that there are exactly as many such pairs as there are real numbers.

Now you might use another definition. Noting that you speak of the "cartesian plane", I assume you've already been given a coordinate system. So maybe you only consider a line to be a number line if it goes through the origin and has its $0$ there. In that case, the number line is specified by one single point other than the origin. But again, there are as many such points as there are real numbers, so we again get the same number of number lines.

You might also add the additional requirement that the distance between $0$ and $1$ on the number line is $1$. In that case, the number line is uniquely determined by the angle to the $x$ axis. But again, the angle is a real number from the interval $[0,2\pi)$ (other intervals of the same length can be chosen by convention), and there are as many real numbers in an interval as there are in total, so again we get as many number lines as there are real numbers.

Now you might also restrict it further to define a number line to just be one of the axes of the Cartesian coordinate system. In that case, there would of course be only two number lines. But if that had been the definition you had in mind, you probably wouldn't have asked to begin with.

So to summarize:

For any likely definition of "number line", there are as many number lines as there are real numbers.