The MM Species - Aesthetics and the Immovable Object - Burn Math Class: And Reinvent Mathematics for Yourself (2016)

Burn Math Class: And Reinvent Mathematics for Yourself (2016)

Act II

5. Aesthetics and the Immovable Object

5.5. The MM Species

We’ve got one more species to explore before the end of the chapter: MM. The MM species was defined to be the set of all machines that behave like this

f(xy) = f(x)f(y)

for all x and y. Let’s see if we can figure out what they look like. Since we don’t know what to do, it might help to do what worked before for the AM species: differentiating the definition of these beasts with respect to one of the variables, say y. This gives

xf′(xy) = f(x)f′(y)

where the prime stands for differentiation, thinking of y as the variable. This equation has to be true for all x and y, but that’s more information than we want, so it might help to plug in specific values for x or y to see if we can reduce it to a more transparently meaningful condition. If we set x = 1, then we get

which I suppose tells us that f(1) = 1. Okay, what if we go back and set y = 1 instead of x = 1? This gives

xf′(x) = f(x)f′(1)

Since f′(1) is just some number we don’t know, we can rewrite this as

This is one of those places in mathematics where it’s not at all clear what to do, and in the process of playing around with this problem, anyone would probably get stuck here for awhile. Now, no one in their right mind would likely think to do this right away, but if (for whatever silly reason) we differentiate the “natural logarithm” of f(x), then it becomes a bit easier for our human minds to see what these equations are saying. Recall that we discovered above that

where q(x) is what textbooks call the “natural logarithm,” or “log base e,” and write as ln(x). Because we can abbreviate things however we want, the above equation says the same thing as any of the following:

And it is the last of these that turns out to help us here. Now, if we differentiate q(f(x)) with respect to x using the hammer for reabbreviation, we get

And if we use the fact we discovered in equation 5.44, we can turn the f′(x) on the far right of the above equation into cf(x)/x, which gives us

Notice that the on the right side of the above equation can be thought of as the derivative of the natural logarithm q(x). As such, we can use this fact and the property of logarithms c · q(x) = q(xc) to write

We’ve arrived at a statement of the form “the derivative of (one thing) equals the derivative of (another thing).” But if that’s true, then it has to be the case that “(one thing) = (another thing) + (some number).” Why? Well, for the slopes of two machines to be exactly the same everywhere, the two machines have to be the same everywhere, except for a vertical shift of their graphs up or down. These general considerations tell us that we can use equation 5.45 to conclude

q(f(x)) = q(xc) + A

where A is some number we don’t know. But not knowing A is the same as not knowing its logarithm, so we can express our ignorance of it equally well by writing q(B) instead of A, where B is some other number we don’t know. This trick allows us to write

q(f(x)) = q(xc) + q(B) = q(Bxc)

And finally, feeding both sides of this equation to the opposite machine of q (namely, ex), we obtain

f(x) = Bxc

Plugging in x = 1 gives f(x) = B, but we discovered earlier1 that f(1) = 1 for all members of the MM species, so it must be the case that B = 1. Putting it all together, we have discovered that all members of the MM species have the form

In the line immediately after equation 5.43.

f(x) = xc

where different choices of c give us different members of the MM species. Since we’re already familiar with these machines, we don’t need to spend any more time discussing them.