I’ve talked a fair amount in some of the earlier posts about the idea of a proof. Now that we’ve developed a conceptual underpinning of what that means, we can see one in action. I hope my readers enjoy this as much as I do, as what we will discuss here is among my favorites.
In case you’re worried, I’m not going to start talking about the proofs you may have seen from high school geometry. While these technically count as proofs, they are a far cry from what most proofs today look like. Every proof will justify what it says, but almost never is this done in the painful and boring ‘two column’ setup that I have heard many talk about, or in some kind of highly structured list. Proofs are usually presented in more of a narrative form, actually.
Just because a proof is highly logical does not mean it is not beautiful. Just as in every other form of art, precision does not stand at odds with beauty. Most mathematics today is done with sentences and paragraphs explaining and guiding the reader through the ideas. When this is done well, the person reading not only can understand why each step in the progress is correct, but is able to perceive the overall direction of the proof. As a graduate students, one of the ways I know I understand a proof is if I can give it a “plot summary,” capturing the progression of ideas with minimal reference to details. As you read through and study a proof, things should begin to make more sense as you go, or at least they should come together at the end. The best proofs will have a sort of climax that represent a very powerful or surprising flash of insight, and the experienced reader will immediately notice one of these when it occurs, just as you can tell when a movie is reaching its climax.
Just as someone who reads a lot of literature can perceive different sorts of depth and beauty in writing, so an experienced reader of mathematics sees different kinds of beauty in a proof – the most common words of this sort that I hear are beautiful, elegant, and slick. I haven’t every actually asked other mathematicians about whether they consider these aesthetic terms synonyms or not (I’d love to hear some comments from others on that!) but I personally think these are all subtly different. Maybe one day I’ll be able to put into words what I mean by each of those. But regardless of whether these words actually have different aesthetic meanings, my point is that math is not merely black-and-white. There is rich color to be found, if you know where to look and keep an open mind. There’s a lot of beautiful math out there. One of my primary career aspirations is to make this beauty accessible to everyone I come across.
So, to begin on that mission, I will now try my hand at a presentation of one of the most famous equations of all – the Pythagorean Theorem.
This is a very ancient piece of mathematics, that was proved by a civilization that developed mathematical proof in a way that had lasting impact, ancient Greece. The theorem is named after Pythagoras, who either proved this himself or whose school of pupils brought forth a proof. Today, this is how we normally phrase the theorem:
Pythagorean Theorem: Suppose that the sides of a right-angled triangle have lengths a, b, and c, with c the length of the longest side. Then the following equation is always true:
Since the Greeks phrased all of their mathematical work directly in terms of geometry, they would have used the areas of actual squares as the parts of this equation. I don’t want to try to actually use their phrasing, because that might lead to confusion, but the following pictures should give a good illustration.
This is the framework within which a Greek mathematician would have thought about the Pythagorean theorem. It is worth saying that it is far from obvious that this should be true, and I think that ought to make it interesting to us. Why should there be any equation relating these three sides, and why this one in particular? In all areas of life, we get bored by something that is too obvious, but less obvious things catch our attention. It seems to me that non-obviousness is at least part of why we like comedy, magic, and sports – the punchlines we don’t see coming make us laugh the most, we love magic because it amazes and surprises us, and we love the upset victory, the Cinderella story, and the impossible goal in sports in large part because they surprise us.
To me, something similar should happen with great mathematical ideas. We don’t talk about the fact that 1+1=2 very often – it is just as true as the Pythagorean theorem, but it is too obvious to be of much interest. Another reason this is more surprising than 1+1=2 is because it is a broader statement. This is not a mere calculation. Your calculator can’t do this. This is not a calculation about a particular shape, this is a statement about the way that all right triangles are. So we learn far more from the Pythagorean theorem than we ever could from building lots of triangles and measuring their sides, because there would always be more triangles we hadn’t measured yet.
But how do you know this is true? Well, this is where proofs come in. There are literally entire books published that contain nothing but hundreds of different and creative proofs of this statement, including a proof by Albert Einstein and one by American President James Garfield, plus the original proof discovered and written down by the Greeks themselves. The proof that I have chosen to present is the one that I find the simplest and most enjoyable.
Proof of the Pythagorean Theorem:
This proof centers around two key ideas and a little cleverness. The two main ideas we will need are (1) Two shapes that have the same dimensions have the same area, and (2) if two shapes have the same area, and if we remove from both shapes identical pieces, then the new shapes still have the same area. These two statements are just part of what area is, so this is a good starting point. Now, we need to use these ideas in a clever way. We will do so with two conveniently-drawn squares. Consider first “Square 1”:
The green triangles in this image are all identical to each other – in fact, these are right triangles with side lengths a, b, and c. There are also two smaller squares, with side lengths a and b respectively. Square 1 itself has side length a+b. We will use this information to work out some areas later.
Now, lets look at a different square, which we will call “Square 2″:
Square 2 also has four green triangles, which are identical to the green triangles in Square 1. There is also a square of side length c in the inside of Square 2. Finally, notice that Square 2 also has side length a+b.
Now, we can compare these two squares. Square 1 and Square 2 are both squares with side length a+b, and so are identical shapes. So, using idea (1) from the beginning, we know that the area of Square 1 is the same as the area of Square 2. We can also look at the pieces of these two squares. Both squares have four identical green triangles, and these have the same area. Since Square 1 and Square 2 have the same area, and idea (2) tells us that we can remove the four green triangles from each and the resulting shapes will still have the same area as each other. But what happens when we remove the green triangles from Squares 1 and 2? The remaining portions of Square 1 are the two small squares with side lengths a and b, which has total area a2+b2 and the remaining portion of Square 2 is the square with side length c, which has area c2. Since these total areas must be the same, we now know that a2 + b2 = c2.
There it is! I encourage you to reread the proof a few times if you don’t quite feel like you understand. It often takes multiple readings to get a handle on what is happening. Take a look at the logic I used – are there gaps you can fill in? (For example, I opted to not explain why Square 1 and Square 2 actually are squares – can you explain this on your own?) When all is considered, this argument truly is airtight. I hope you will appreciate the beauty of the reasoning, it’s simplicity and cleverness. If not, then perhaps a different idea from math will be more appealing to you.
I hope people walk away with a greater understanding of what math is about, and perhaps some joy too. And feel free to write me about anything from your math class you’re curious about and would like to see proved!