Here I collect a few proofs that I find particularly simple and beautiful, yet point to deep mathematical results.
Infinite sums
Take 1, add a half, then again half of that (which means a quarter), and so on. Keep adding half of what you added before, and do this an infinite number of times. What is the result?
Geometrically, the answer becomes clear: take a rectangle of sides 1 and 2, like this:
If you split it in two along the middle, then the area of the (red) left side is 1. Then you split the remaining part on the right in the middle. Now you have added 1/2 (the orange bit). Again split what is left in half, now you have added 1/4 (dark yellow). And once more, then you have added 1/8 (bright yellow). If you keep doing this, you are in fact performing the sum we had above. And it becomes clear that the result will be 2 (the area of the entire rectangle), since you keep adding more and more of this rectangle, without ever being able to exceed the rectangle. So now you have performed an infinite sum.
Scaling symmetry, the logarithmic spiral and the golden ratio
Take a rectangle which is such that it can be divided into a square and a smaller rectangle that has the same proportions as the large rectangle:
More precisely, if the large side has size g and the small side has size 1, then the ratio of their sides is g/1=g. That means that the small side of the small rectangle must have size 1/g, so that once again the ratio of large to small side is g. From the figure one can easily find an equation for g, namely g-1=1/g. This equation only has one positive solution, and it is the golden ratio
So now we have a large rectangle containing a smaller rectangle of the same proportion. We can now split up the small rectangle in the same way again, dividing it into a square and an even smaller rectangle again of the same proportions:
That the smallest rectangle has the same proportions is automatic, since everything is simply scaled down by a factor of g. But now we can keep doing this, always splitting any small rectangle into a square and an even smaller rectangle of the same proportions. In this way we get a figure that is self-similar (we can imagine also doing the reverse, and adding progressively larger squares to the big rectangle we started with). In other words, this figure remains identical if we zoom in or zoom out – it has a scaling symmetry.
One can also inscribe a spiral onto the figure by linking opposite corners of the squares, as done above. This is a logarithmic spiral, which is also self-similar. It is a shape that is common in nature, because it grows in proportion to what is already there. It governs the shape of various shells (like the nautilus), snail’s houses, sunflowers and Romanesco broccoli.
How many fractions are there?
If we look at the integers: 0, 1, 2, 3, 4, and so one, it is clear that there are an infinite number of them. The rational numbers are those that can be written as fractions, for example 6/11, 5/9, 147/12, and so on. To many people, it seems natural to guess that there are many more fractions than there are natural numbers. However, there is an argument, due to Georg Cantor, that shows that there are just as many fractions as natural numbers. It goes as follows: one can list all possible fractions by writing them in a grid, such that the numerator increases by one as you go to the right, and the denominator increases by one as you go down, see this figure:
Now one can find a path, as shown by the arrows, that puts these fractions into correspondence with the series of natural numbers:
The point is that one can list all fractions in exactly the same way as one can list all natural numbers. In other words, there are infinitely many fractions, but it is the same infinity as that of the natural numbers.
If we say “the same infinity”, this is because other infinities also exist. For example, if we look at all real numbers (say, all points on a line), then there are in fact more of them than natural numbers. It is a higher infinity. How can one show that? Again, Cantor had a clever argument. Suppose there exists a list of all real numbers (here we restrict to all numbers between 0 and 1). It would presumably look a bit like the numbers in black here:
Now form a new number, which you get by taking the first digit from the first number, the second digit from the second number, and so on (in green). This new number can be transformed, for example by adding 1 to each digit (if one has a 9, one transforms it into a 0). This new, transformed number is special, because it cannot be equal to one of the numbers we already had, since it has at least one digit that is different from every other number on the list. But the list was already supposed to contain all numbers between 0 and 1. Hence we have a contradiction, and we conclude that no such list of all real numbers between 0 and 1 exists! Such an infinite list, which one could have put in one-to-one correspondence with the integers (or the fractions) does not exist, because there are infinitely more real numbers. There are hierarchies of infinities!