Burn Math Class: And Reinvent Mathematics for Yourself (2016)
Act II
5. Aesthetics and the Immovable Object
5.6. Reunion
Let’s summarize what we’ve done in this chapter.
1.Given the importance of addition and multiplication on our journey so far, we defined four species of machines based on how they interact with these operations. Using the abbreviations A and M for addition and multiplication, respectively, the four species were defined to be any machines that (1) turn A into A, (2) turn A into M, (3) turn M into A, (4) turn M into M.
2.We played around with the AA species and found that all its members looked like f(x) ≡ cx, where c is some fixed number.
3.We played around with the AM species and found that all its members looked like f(x) ≡ cx, where c is some number. Even better, we found that one of these machines had the surprising property of being its own derivative. Combining these two facts tells us that there must be some extremely special number e for which ex is its own derivative. All constant multiples of this machine are their own derivatives as well, but we found that ex is the only machine that is both its own derivative and a member of the AM species.
4.We played around with the MA species, and found that its members each had a partner in the AM species. That is, for each AM machine fc(x) ≡ cx, there had to be some MA machine gc(x) that did the opposite for all x, i.e.,
However, we had no idea what the members of the MA species “looked like” (i.e., we couldn’t describe them in terms of anything else we knew).
5.We found two different expressions for the extremely special number e.
These two expressions both agreed that the number e was approximately e = 2.71828182 . . .
6.We then returned to playing with the MA species (“logarithms,” in textbook jargon). We chose to write these machines as qc(x), because they act like the “power machines” pc(x) ≡ cx in reverse, and a reversed p looks kind of like a q.
7.We used what we had discovered about the MA species to derive the various “logarithm properties” that we hear about in mathematics courses. Along the way, we found that all of the MA machines were just constant multiples of each other, so we wouldn’t lose anything by simply picking our favorite and ignoring the rest. We chose qe(x), the opposite of our immovable object ex. Dropping the now-unneeded subscript, we called this machine q(x). This is the machine known as ln(x) in textbooks.
8.We figured out the derivative of q in two different ways. In both cases we found that:
As with the machines V and H from Chapter 4, we were able to determine the derivative of q before we learned how to write a description of it as a plus-times machine.
9.We attempted to use the Nostalgia Device on q, but we noticed that it wasn’t well-behaved at zero, so we shifted it over one unit, and examined the machine q(x + 1). In doing so, we found
This gave us a way of thinking about the “natural logarithm” q as an infinite plus-times machine. However, we’re not sure whether the expression will work for all x.
10.We then found that the members of the MM species all have the form f(x) = xc, where c is some fixed number.