A Little Algebra

Here's a problem that's nearly impossible to solve until you use a (very) little algebra, and then you can generate an infinite number of solutions:

Can you think of two numbers that give you the same result when you add them as when you multiply them? For example, 2 + 2 = 2 x 2, and there's another trivial solution, but are there any solutions to this when the numbers aren't the same? What if you include fractions or decimals?

If you try to guess and check your way to a solution, it seems impossible. But believe it or not, there are an infinite number of pairs that work. All you have to do is use a little algebra.

Translate this sentence into symbols: "Multiplying two numbers, x and y, is the same as adding x and y."

xy = x + y

I agree, this doesn't automatically lead to a list of solutions for me, either, but plug in a number for y and solve for x:

3x = 3 + x

2x = 3

x = 3/2

Sure enough, 3 (3/2) = 3 + 3/2 = 4.5

Plug in 4:

4x = 4 + x

3x = 4

x = 4/3

4 (4/3) = 4 + 4/3 = 5 1/3

See a pattern yet? It's pretty easy to generate tons of pairs of numbers that work now, having taken the most elementary step into algebra land. And that's the only reason to go there, in my opinion: to solve a real problem.

Series Join The Series

It was always a mystery to me why in Calculus we stopped learning the fun geometrical stuff like curves, areas and volumes and all of a sudden started learning about series. The professor never mentioned why this topic was necessary. So much for context. He also couldn't teach series very well, and I had to take the class over during the summer to graduate.

In the spirit of teaching an all-encompassing, connected Calculus course, I decided to add another lesson to the Fun Calculus Program on series. Actually, series were already introduced in "The Hunt for e" when Newton extended Pascal's Triangle (in the opposite direction!) to find the area under the hyperbola y = 1/x.

I dare you to ask your math teacher what series have to do with Calculus. "Well, they're on the AP exam," is the response you're likely to get, and that's true, too. But the real reason is that polynomials are child's play to differentiate and integrate, and many other functions are not. In Visual Complex Analysis, Professor Tristan Needham of USF writes that

Newton's 1665 version of the calculus was different from the one we learn today: its essence was the manipulation of decimal expansions in arithmetic. The symbolic calculus...was also perfectly familiar to Newton, but apparently it was of only incidental interest to him. After all, armed with his power series, Newton could evaluate an integral like ∫ e^-x2dx just as easily as ∫ sin x dx. Let Leibniz try that!

That very integral (∫ e^-x2dx) stumps most calculators, and I assume most students as well. But in the Series module participants in The Fun Calculus Program learn how to derive the series for e^x, and simply substitute -x² for x.

Personally, I love how Calculus tools evolved to make easy work of previously unsolvable problems, and series fit right into that toolbox.