This article is a semi-continuation of my previous post regarding the sordid state of Rutgers University’s ‘Liberal Arts Math’ curriculum. Since people seemed to like it, I thought I’d switch gears away from the Kafka-eqsue world of collegiate bureaucracies and instead look at more examples of ‘beautiful’ math. Keep in mind that, just like with any other art form, what’s omg-inducing to me may be meh-inducing for you—so if you don’t find one, or even both, of the following two examples pretty, don’t fret or think math isn’t for you, our tastes are probably just different.
An Infinitely Long List
To start off the adventure, I want to show you a stunning result from calculus but, unfortunately, I can’t prove it. It’s not that the proof is too mysterious or confusing or anything like that; it’s just that it involves a part of math called mathematical analysis—which is a bit beyond the scope of this article. Nevertheless, if you take me at my word and assume the proof to be solid, I think the result is amazing enough to stand on its own.
Notice the pattern? Here’s a hint: … If you didn’t get it, that’s okay—don’t worry about it. In English the pattern sounds like this: the bottom number for each fraction is the
square of a number greater than or equal to one, in sequence—which makes this an infinitely long list. Maybe if I write it like this it’ll be easier to see the pattern:
It never ends. Ever. It just goes on, and on, and on, and on… So my question is this: what happens when I add together all the numbers in this infinitely long list? Won’t I just get infinity, since there are infinitely many numbers? Well, let’s prod the list a little bit and see what happens. I’m going to start adding the numbers together, one by one. I’ve circled the answers in blue.
Notice anything? The answers circled in blue are indeed getting bigger, but the number by which they increase every time I add a term is getting smaller. So by the time we get to terms to the tune of 1/ten billion, the answer is only increasing by .00000000001. As the list goes on and on, the number which I add onto the total gets smaller and smaller. So when I add all the terms of the list together, I seem to be approaching some number which is somewhere between 1 and 2. Can you figure out what that number could be? I’ll let you think.
Did you get it? If you did, please waltz on over to closest university you can find and start working towards a PhD in Pure Mathematics, because you’re probably a genius. The problem we’re exploring is historically important enough to have its own name: The Basel Problem. It was posed by Pietro Mengoli in 1644 and had eluded so many smart people for so long that anyone who could find the answer would be rewarded with instant academic and public fame. Of course any nerdy shmuck could sit there and calculate the terms like we did, but mathematicians wanted what’s called a closed form solution; in others words, a nice looking answer, like 3/2 or something like that. No one could seem to figure it out—not the Bernoullis, not Leibniz, nobody! Expect Euler, of course. Leonard Euler, one of the greatest minds to ever grace the earth, found the solution when he was only 28 years old. The answer is this:
Soak that in for a second. You take this list, this infinitely long list, add up all the terms, and you get the ratio of the circumference of a circle to its diameter squared, divided by six. Plugging this into the calculator, we see the result is 1.644934… on and on and on. The first time I saw this result, I promptly began working towards a B.A in Mathematics, I kid you not. I’m sad that I can’t really go through the proof here, which is equally striking, but I just had to share this result.
Part of the joy in math is precision. Start with some definitions that you or your buddy has come up with; then try to show how the definitions are related, use these definitions to get results, etc, etc. So when I say that the sum of all the terms in an infinitely long list equals something, I’m using the definitions of ‘equality’ and ‘adding’ and so forth that mathematicians have made up (which is the only kind there is, anyway).
With that in mind, let me confess that I’ve cheated you a bit by omitting the mathematical idea of a ‘limit’. It would have been more exact for me to say something like ‘in the limit as n approaches infinity, the terms of one divided by n squared equals pi squared over six’. So really, the way I phrased it originally, when you add up all the terms in that list, you get closer and closer to pi squared over six; you never really reach it. Instead, you just get arbitrarily close—as close as you want. You are only allowed to say ‘equals’ when you are talking about limits. In a way though, the idea of getting closer and closer without ever really reaching is the definition of a limit. I hope this all made some sense.
The Irrationality of √2
Next, let’s mull over another famous math problem: how do we prove that a number is irrational? That is, how do we prove that a number can’t be expressed as a ‘simple’ fraction? Irrational numbers go on forever without any discernible pattern. Or in other words, they don’t repeat. We can’t possibly look at all the infinite decimal places in the number and say ‘see, it doesn’t repeat!’ So what’s left to do? This problem was discovered and solved by the ancient Greeks thousands of years ago. The solution is short, simple and sweet—in other words, elegant.
First, you have to notice that all rational numbers can be written as ‘simple’ fraction. For example:
- 1.5 = 3/2
- 4.5 = 9/2
- 3.33333333333… = 10/3
Pi is not rational because its decimal values never end, and never repeat. To prove that √2 is irrational, let’s first assume that it is rational, and then show that this assumption leads to a logical contradiction. This method of proof, aptly enough, is called proof by contradiction.
- Suppose √2 is rational. Then it can be written as the simple fraction , where a and b are integers. Stated mathematically, we are saying that √2 = a/b. Assume, also, that is the most reduced form of the fraction. In other words, a and b have no common factors. The fraction is written in its simplest form. Remember, we’re assuming that √2 is rational, not irrational.
- Note that since √ 2 = a/b , 2 = a²/b². All we just did was square both sides. Now we can rewrite this all as 2*b² = a². All I did there was bring b over to the 2 side of the equation.
- Note that a² is even, since the definition of an even number is that it is divisible by two.
- Also note that if the square of a number is even, the number itself must be even (if you don’t believe me, try it), and therefore a is even. So we can rewrite a as a = 2*k, where k is some random integer. By extension, we can also write 4*k² = a² , again just by squaring both sides. Plugging this into the formula 2*b² = a² , we get 4*k² = 2*b² , which simplifies to 2*k² = b² .
- So b² is even. Therefore b must be even, by the same logic we used for a. So we can rewrite b as b = 2*z, where z is another arbitrary integer.
- Finally, we take our two expressions for a and b and see that a/b = 2k/2z , which is a contradiction, because our initial assumption was that a/b was an irreducible fraction, i.e. no common factors, but we see here that a and b have a common factor of 2. Therefore √ 2 is irrational.
How cool is that? We just proved the irrationality of a number using nothing but algebra 101 and a few basic definitions. I hope this proof sent a tickle up your spine like it did for me the first time I saw it.
I think I’ll conclude on the exact same note I concluded my last article with. Tell me which is more interesting: the problems we just solved, or “comput[ing] the future value of assets and value of a deferred annuity”?
Leo Kozachkov (Staff Writer, Rutgers University) Leo Kozachkov is an undergraduate at Rutgers University, studying physics and mathematics. He is currently working as an Aresty Research Assistant under Professor Thomas V. Papathomas. He enjoys writing, drawing, creating/playing music, going on long walks with his beloved dog, and reading/hoarding books.His grandest hopes are to discover a new physical law, have a mathematical theorem feature his last name, and to write many books.