I'm going to start a maths degree at Oxford next week and I thought a good project for the year would be to write about an interesting bit of maths that I had found that week.
How to go about answering such a question?
Before we talk about the line and the square, we'll consider a different question. Are there more numbers between 0 and 1 or between 0 and 100? At first it seems obvious, and there is a seemingly fool proof argument: Suppose there are 
numbers between 0 and 1, between 0 and 100 there must be 
numbers. Hence there are more numbers between 0 and 100, since 100 > 1. This is a flawed argument, and to explain why we'll need to introduce the idea of sets.
A bit of set theory
A set is just a collection of things - usually numbers. For example the set 
could be assigned like so: 
- just a set of the first 5 integers. This is an example of a finite set. Sets can be finite or infinite, with an example of an infinite set being the natural numbers: 
. The fancy N is the standard notation for the set of natural numbers.
All sets have other sets associated with them called subsets. A subset is a set that contains some (or all) of the members of another set. A proper subset is a subset containing strictly fewer members of another set. Eg: 
with the 
symbol meaning 'is a subset of.' Here all the elements of X are natural numbers, and it has fewer elements so it is a proper subset of the natural numbers.
A key property of infinite sets (which finite sets don't possess) is that there exists a one to one mapping from at least one proper subset to the set its self. This is almost the formalisation of the idea that infinity + 1 = infinity. For example, consider the set 
and the set 
. Since the set Y is a mapping from X which is a proper subset of the natural numbers, and Y is the natural numbers, this shows that there are infinitely many natural numbers.
Going back to the discussion of the number of numbers, the sets we are dealing with are infinite so there is no finite value of that we can assign, so it isn't clear that > . Also, with our new found knowledge of sets we could show that there exists a mapping from 
to 
namely 
and hence there are the same number of numbers between 0 and 1 as there are between 0 and 100. If you claim otherwise, then consider the fact that it is impossible to pick a number between 0 and 100 that doesn't have an associated number between 0 and 1. For all the numbers you pick, you will never run out of numbers between 0 and 1. (Here the fancy R is the set of all real numbers, so think fractions and irrational numbers. A rigours definition of the real numbers is perhaps a topic for another post.) An exercise for you to try would be to find a mapping from the interval 0 to 1 to the entire real numbers - to show that the total number of numbers is the same as the number of numbers between 0 and 1.
Back to the line and the Square
I'll use sets to define exactly what we mean. The line could be the set of numbers and the square could be the set of points 
. And the question is, does there exist a mapping from X to Y? Let's first have a look at possible mappings. One is
which actually ends up being part of the unit circle. The problem with this is that there is a cyclicity to this set - after 
the trigonometric functions just output the same numbers. The next logical thing to try would be 
but this still has the same problem and will do for all rational coefficients. This is because for 
where 
, it can be shown that there is a solution to this pair of simultaneous equations: 
. However, if one chooses irrational coefficients to define the points like this: 
(multiplying the coefficients or applying some function to t can get around the fact that t is now an unbounded real number) then you can prove that there exists no solution to the previous simultaneous equation.
The proof
Suppose that for there exists a solution to these simultaneous equations: 
where k and j are integers. Then, dividing the first equation by the second equation we get 
which is false since the left hand side is irrational where as the right hand side is rational. This means that there is no t such that at the same time such that 
and 
are both multiples of 
and hence the trigonometric functions in the point are always out of phase. Therefore the set 
is a mapping from [0, 1] to the square of side length 2 centred at the origin. So there are the same number of points on a line as there are in a square - the set is of the same size.
Conclusion
I think this went off the rails a bit in terms of mathematical complexity - it was difficult to write the end bit in a beginner friendly tone - feel free to ask any questions. But hopefully the idea of one to one mappings representing the number of numbers is interesting and beneficial to you.
