Some years ago, a Certain Toy Corporation got into quite a bit of trouble for marketing a (girl) doll that spoke phrases. In particular, one phrase: “Math is hard.”
I’ve always argued that that phrase isn’t objectionable. Math is hard. So is throwing a three-point shot in basketball, doing a triple gainer, bowling a perfect game, and changing out a car engine. One of my favorite episodes of The Bernie Mac Show was when Bernie Mac made this very point: yes, math is hard…but you do hard things all the time, so why not do math? (Season 1, Episode 16, Mac 101)
So how do you get better at basketball? You practice, practice, practice. The same is true in math, and teachers often tell students this. And while that’s all true, and certainly good advice, it occurs to me there are two components to being “good at math.” The first is being good at doing math. Maybe you’ve just learned how to solve a quadratic equation, and so solving takes a little effort. But after you’ve solved a few hundred quadratic equations, it becomes second nature, and you can throw down the solution () without hesitation.
But the other component of being “good at math,” and ultimately what it means to be a mathematician, is being good at creating math. This is far more difficult. It’s the difference between doing 30 push-ups a day, and inventing a new calisthenic.
So how do you do that? Let’s consider two of the greatest mathematicians ever: Gauss and Euler. They actually talked about what it took to be a great mathematician, and the short form is this: Never solve a problem one time.
For example, in 1736, Euler proved a result of Fermat, namely that if is prime and , then is divisible by . Euler proved this, using an induction argument so obscure that it keeps being rediscovered by mathematicians, both great ones (Laplace and Cauchy) and obscure ones (me, actually…this may be the only time I’ve sat at the same table as Euler, Laplace, and Cauchy).
But Euler didn’t stop with one proof. About every ten years, he came up with a new way to prove the theorem. His re-examination of the problem led him to discover the -function (where is the number of numbers less than which are relatively prime to ) and generalized it to what is now called the Euler-Fermat Theorem: For any number and any number relatively prime to , the least value for which is divisible by is a divisor of .
Incidentally, this result is the basis of modern computer security (the RSA algorithm).
What about Gauss? In 1799, Gauss proved the Fundamental Theorem of Algebra,namely that a th degree polynomial with real coefficients has real and/or complex roots.
And then, over the course of his career, Gauss proved the Fundamental Theorem three more times, each time extending the result and developing new mathematics.
What’s the practical application? Let’s consider something really basic: multiplication of two numbers, say . We all know how to do this: we were taught how to do this computation in school. We can practice the multiplication algorithm by trying different products: today, tomorrow, the day after, and so on. If you do this, you will develop your skills at applying the standard algorithm.
On the other hand, suppose that instead of doing new products the way you were taught, what if you tried to find the same product using a completely different method? You know what the answer is supposed to be, so you’ll have a good way to check if your method works.
How might that work? In this case, is the sum of forty-seven s. So you could add . That’s one method of multiplying; one nice feature of it is that it’s something a first grader can do. (Granted, you’d probably have them do something easier, like , but the important thing is that they don’t have to know multiplication to be able to solve the problem “Find “)
Obviously, you don’t want to spend the next half hour adding forty-seven s together…but progress comes when someone asks “Can we find a better approach?” So you start thinking about how to improve the efficiency of your sums. Maybe tomorrow, you realize , so adding together forty-seven s is the same as adding together forty-seven s, forty-seven s, and forty-seven s.
And even that gets a little tricky, so the day after, you come up with a new insight that allows you to make the addition even more efficient.