Inventing Calculus - The Infinite Power of the Infinite Magnifying Glass - Burn Math Class: And Reinvent Mathematics for Yourself (2016)

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

Act I

2. The Infinite Power of the Infinite Magnifying Glass

2.2. Inventing Calculus

2.2.1The Problem: Curvy Stuff Is Baffling

In Chapter 1, we exercised our inventing muscles by inventing the concepts of area (for rectangles) and steepness (for straight lines). Obviously, straight lines are straight, and rectangles are built from straight lines. Neither of these things is really “curvy.” However, as long as there was no curviness involved, we still found that we could talk about some pretty deep and interesting things. For example, when we invented the concept of area, we saw that it’s not hard to confidently talk about n-dimensional objects, or to convince ourselves that it makes sense to define the “n-volume” of an “n-dimensional cube” to be n, where is the length of one side (even though we can’t visualize anything in more than three dimensions).

What about curvy stuff, like circles? A circle is a lot easier to picture than an n-dimensional cube! And yet who among us can look at a circle and just intuitively “see” what a reasonable formula for its area is? None of us. You’ve probably been told what the area of a circle is. At some point in your life, someone told you that the area of a circle with radius r is πr2, where π is some bizarre number that’s slightly bigger than 3. But forget that. We haven’t invented that fact, and it’s not intuitively obvious to anyone. In fact, a fairly popular book called the Bible says that π is equal to 3, so it would appear that even deities have problems with curvy things.1 That’s why we’re approaching the subject in this backward way, inventing calculus before its prerequisites: because most of those prerequisites somehow involve curvy stuff, and curvy stuff is nearly impossible to deal with before we’ve invented calculus (and especially before we’ve learned how to invent things). So despite all we’ve done, we still haven’t got the faintest clue how to deal with curvy stuff on its own terms. Let’s put this to rest once and for all.

The line is 1 Kings, chapter 7, verse 23: “And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about.”

2.2.2The Embarrassing Truth

Here’s the central insight behind all of calculus. It’s kind of embarrassing how simple it is.

All of Calculus

If we zoom in on curvy stuff, it starts to look more and more straight.

What’s more, if we were to zoom in “infinitely far” (whatever that means), then any curvy thing would look exactly straight. But we know how to deal with straight stuff! At least a little bit. If we can learn to make sense of this idea of zooming in infinitely far — if we can invent an “infinite magnifying glass” — then we’ll be able to turn any problem involving curvy stuff (a hard problem) into a problem involving straight stuff (an easy problem). If we could do that, then maybe we’d be able to go back and reinvent for ourselves all of those unexplained facts they taught us in high school. Then we could forget them forever, and just reinvent them if we ever needed to.

Dear Reader,

Stop and take a breath.

This is where math starts to get interesting.

2.2.3The Infinite Magnifying Glass

The essence of mathematics is not to make simple things complicated, but to make complicated things simple.

—Stan Gudder

When we invented the concept of steepness for lines, we needed to pick two points, so that we could compare their horizontal and vertical positions. It didn’t matter which two points we picked, but we had to pick two. But for curvy things, picking two random points doesn’t seem to make sense, because if the steepness is constantly changing (as is typical for curvy things), then it seems like we’d get different answers depending on which two points we picked. That would make for a really ugly definition. What’s more, our brains seem to somehow know what steepness means at a single point. If we forget about mathematics and just stare at a curvy thing (say, this squiggle ), it’s very clear that some places are more steep than others, even though we don’t know how to use numbers to say how steep the different spots are. Is there any way to make sense of the idea of steepness at a single point of some general curvy thing? Well, if we had an infinite magnifying glass at our disposal, we could reduce this hard problem to an easy problem by zooming in on curvy things until they look straight.

Our Problem: If we have a curvy thing (for example, the graph of some machine M that isn’t just a line), is there any way of saying what we mean by the “steepness” of this curvy thing at a single point x?

So someone hands us a machine M and a number x, and we need to make sense of the concept of “steepness” at the point they handed us. Well, here’s one idea. Let’s look at the graph of M near x. That is, if we visualize x as some number on the horizontal axis, and if we visualize M(x) as some number on the vertical axis, then the point with horizontal coordinate x and vertical coordinate M(x) will live on the graph of the machine M. We can write this point as something like (x, M(x)), or however else we want. Now let’s look really hard at that point. If we had an infinite magnifying glass, then we could center it over this part of M’s graph and zoom in infinitely far. Then we’d see a straight line. Since we already invented the concept of steepness for lines, we could just apply that old concept to two points that are infinitely close to each other. What does it mean for two points to be infinitely close to each other? I don’t know! Let’s decide.

Let’s write tiny to stand for a number that’s infinitely small. It’s not zero, but it’s also smaller than any positive number. If you’re worried about that idea, let’s talk about it in this footnote.2 Okay, time for some abbreviations. Let’s say the point where we zoomed in has horizontal coordinate xand vertical coordinate M(x), while the point infinitely close to it has horizontal coordinate x + tiny and vertical coordinate M(x + tiny). To say it another way:

You’re right to be worried! It’s not clear that this idea of infinitely small numbers makes any sense, but if we’re worried about it, we can just imagine that tiny is 0.00(etc)001, where there are maybe 100 or 1000 or 10, 000 zeros between the decimal point and the 1. Then instead of using an infinite magnifying glass, we’d just be using an extremely powerful magnifying glass. After zooming in like this, curvy things won’t be exactly straight, but they’ll be so close to straight that we could act as if they were straight and our answers would all be so accurate that we’d never notice the difference. Indeed, all of calculus could be done this way, so — resting assured that we always have this safer but less elegant method to fall back on if our “infinite magnifying glass” approach runs into problems — we can simply forge ahead with our more risky way of thinking, always knowing that we have a safety net.

All of these are just different abbreviations for the same idea, but the far right is the most important. Notice that the bottom right side of this long string of equations is (x + tiny) − x. The two x’s cancel each other, so we can rewrite this as:

There’s a picture of this idea in Figure 2.1.

Figure 2.1: Pick any point on a curvy thing and zoom in infinitely far. Once we’re zoomed in, we can just treat it like a straight line. For example, we can define the steepness (at the point we zoomed in) just by looking at the “rise over run” of two points that are infinitely close to each other.

2.2.4Does Our Idea Make Sense? Testing It on Some Simple Examples

All of this is getting a bit abstract, and we’ve just been making this stuff up as we go, so let’s stop for a reality check to make sure we haven’t gone off the rails. Whenever we invent a new concept, it’s always a good idea to test it on some simple example where we know what we expect.

The idea of infinitely small numbers lets us talk about the steepness of curvy things, but we’re not really sure it makes sense. However, it had better reproduce what we already know about straight things. If it doesn’t, then either our new concept is broken or else we didn’t invent what we meant to. Let’s see if it gives us what we expect.

Trying It Out on the Simplest Machines

First let’s test the idea on a really simple machine: M(x) ≡ 7. This is a machine that spits out 7 no matter what we feed it. If we “graph” this machine, it’s just a horizontal line, so its steepness is zero. Since we know what to expect, let’s compute its steepness using infinitely small numbers and see if we get zero. As before, we’ll use the abbreviation tiny to stand for a really tiny number, either infinitely small or just “as small as we want it to be,” depending on our philosophical preferences. Since M always spits out 7, we have:

Notice that we didn’t use any special facts about the number 7 in the above argument, so the same argument should work for any machine that always spits out the same number no matter what we feed it. Alright, so for all the machines that look like M(x) ≡ #, the idea of infinitely small numbers gave us what we expected. Onward!

Trying It Out on Lines

Let’s test our idea on another simple kind of machine: straight lines. In Chapter 1, we found that straight lines can be described by machines that look like M(x) ≡ ax+b. Let’s see if computing their steepness using infinitely small numbers gives us the answer we expect (namely, a).

Perfect! Our strange idea hasn’t let us down yet. Let’s see how it works in a less familiar situation.

Trying It Out on a Genuinely Curvy Thing

Okay, now let’s try this zooming in idea on something where it actually matters whether we zoom in or not: the Times Self Machine. This was the machine M(x) = x2 that we talked about in Chapter 1. Whatever number we feed it, it multiplies that number by itself and hands back the result. Let’s see what these infinitely small numbers give us when we try to compute the steepness at some point x, whose particular value we’ll choose to remain agnostic about.

We can use the obvious law from Chapter 1 to unwrap (x + tiny)2 into the form x2 + 2x(tiny) + (tiny)2. Then the above sentence becomes

Since tiny stands for a number that’s infinitely small, we’ll never be able to tell the difference between the answer we got, which is 2x + tiny, and an answer that’s infinitely close to it, namely, 2x. So if we accept this odd way of reasoning, we can write:

The Result of Zooming In Infinitely Far:

If M is the Times Self Machine, M(x) = x2,

then the steepness of M at x is 2x.

2.2.5What Just Happened? Infinitesimals vs. Limits

To this day calculus all over the world is being taught as a study of limit processes instead of what it really is: infinitesimal analysis. As someone who has spent a good portion of his adult life teaching calculus courses for a living, I can tell you how weary one gets of trying to explain the complex and fiddling theory of limits.

—Rudy Rucker, Infinity and the Mind

Sometimes it is useful to know how large your zero is.

—Author unknown

If this idea of infinitely small numbers scares you a little, you’re not alone! After Isaac Newton invented calculus, people racked their brains for more than a century trying to figure out how arguments like this kept giving the right answers. After all, they seemed so obviously ridiculous. Either the number tiny is zero or it isn’t! How can we act like it’s not zero, and then a few lines later act like it is zero?

Over the years, people have invented all sorts of mathematical Rube Goldberg contraptions — ways of “formalizing” calculus — in order to help them make sense of exactly what is going on here. That’s good! It’s always helpful to make sure the crazy ideas we’re inventing make sense, and we should be happy that people have done it. But the idea of infinitely small numbers is so beautiful that it’s a shame to hide them under all sorts of contraptions, especially since they actually work! In fact, physicists tend to be less scared of using the concept of infinitely small numbers directly. They do their calculations just like we did, and they get the same answers that mathematicians do, but often with much less work.3 Now, some of these contraptions take the idea of infinitely small numbers seriously, while others try to avoid the idea altogether. The second type is much more common, so I’ll mention the first type first, just for the sake of heresy.

This difference gets even bigger as we move to more advanced contexts, as we’ll see throughout the book.

One of the contraptions that can be used to formalize the idea of infinitely small numbers is called an “ultrafilter.” Ultrafilters are pretty complicated, and never mentioned in the introductory textbooks. Even though we won’t talk about them after this, it’s good to know that they exist, because it means that there’s at least one precise way of making sense of calculus that takes the idea of infinitely small numbers seriously.

However, the contraption you’ll see in all the standard intro textbooks is called a “limit,” and it’s much simpler, but it serves the same purpose: it lets mathematicians get all the benefits of using infinitely small numbers, without giving the idea of infinitely small numbers any of the credit.

Here’s the basic idea of the “limit” contraption. Instead of thinking of the number tiny as a number that’s infinitely small, let’s just think of it as a number that is as small as we want it to be. That is, by not writing down a particular number like tiny = 0.00001, we can instead choose to remain agnostic about its value and just run through the same calculation. To put it another way, we can think of the number tiny as having a knob attached to it that we can turn at will, choosing to make it as small as we want, as long as it’s not exactly zero. But notice that this entire discussion assumed that the steepness of M at x could be figured out by pretending it was a perfectly straight line. Now, that assumption is going to be true only when we’ve turned the “tiny” knob all the way down to zero (i.e., once we’ve zoomed in infinitely far). So instead of what we wrote, you would see something like this in a standard textbook. Stare at all of these strange new symbols for a moment, and then I’ll try to explain how it’s essentially the same as what we did.

What’s going on here? Let’s translate it.

First, they’re using the letter h instead of the word tiny. I’m not sure why they use h, but I assume it stands for “horizontal,” since a tiny change in x will be a tiny change on the horizontal axis. Second, instead of writing the phrase “Steepness of M at x” like we did, the textbooks abbreviate this as M′(x). That’s fine, and it’s a lot quicker to write.4 Third, there’s the weird thing on the left of each piece that looks like this:

Basically, the apostrophe is just an abbreviation that means “Zoom in on M(x) and find the slope as if it were a line.” They pronounce M′(x) by saying “M prime of x.”

This is the contraption that allows us to avoid thinking about infinitely small numbers if we want to. The symbol above is pronounced “the limit as h goes to 0,” and it means something like this:

What the abbreviation limh→0[stuff] means:

Calculate everything inside me (the stuff) as if h were a regular everyday number, not an infinitely small number. Then, when we’ve gotten rid of all the h’s on the bottom of stuff (so that we don’t have to worry about what it means to divide by zero), we imagine turning the h knob down so that h gets smaller and smaller and smaller. For example, something like 3 + h will get closer and closer to 3 as you turn the h knob closer and closer to zero. A big complicated thing like 79x999 + 200x2h + h5 will get closer and closer to 79x999 as you turn the h knob down to zero.

This is a perfectly fine way to do the same thing we were doing when we imagined zooming in infinitely far, but it can be pretty confusing if they don’t explain why they’re doing it. In one sense it’s making things simpler because we don’t have to worry about the meaning of an infinitely small number. But in another sense it makes things harder because it’s not always obvious to students why they have to learn all about these odd things called “limits,” especially since we’re usually taught about limits before we hear about ideas like reducing curvy problems to straight problems, and defining the slope of curvy things by zooming in. That is, they teach us the behavior of these limit things before we have any reason to care, and before they tell us the reason that limits were invented in the first place. So it’s not surprising that people think calculus is confusing.

It’s less confusing if we realize that limits are just one of several (optional!) contraptions that let us avoid worrying about the meaning of infinitely small numbers if we want to. Throughout the book, we’ll occasionally use limits, and we’ll occasionally use infinitely small numbers, just so you get used to both. Fortunately, we’ll always get the same answers using either method, so you’re free to pick whichever one you prefer.

2.2.6A Laundry List of Abbreviations

In the previous section, we used the phrase “Steepness of M at x” as an abbreviation for the process of zooming in on a curvy thing and computing its steepness as if it were a line. That’s a lot of words. Let’s look at some common ways of abbreviating the idea. All of these mean the same thing:

Figure 2.2: The steepness

Figure 2.2: The steepness (or “derivative”) of a machine M is sometimes abbreviated by writing . This is why.

1.The Steepness of M at x

2.The Derivative of M at x

This is definitely the most common name for the concept. It’s a noun, and the corresponding verb is “differentiate,” which means “figure out the derivative.”

3.M′(x)

This abbreviation emphasizes the fact that we can think of the steepness as a machine in its own right. M′(x) stands for a machine that works like this: when we hand it some number x, the machine M′(x) spits out the slope of the original machine M at the same point x.

4.

As much as I complain about the abbreviations that textbooks usually use, this one is pretty good once you get used to another abbreviation that’s not as good. Just for a moment, let’s rename our machine V to stand for “vertical,” and let’s use H instead of x to talk about the stuff it eats. So we’ll write V (H) instead of M(x), but only for the next few paragraphs. We’re doing this because we’re thinking of drawing the graph of the machine, so that its output is drawn in the vertical direction and its input is drawn in the horizontal direction. In the first chapter, we wrote h and v to stand for the differences in horizontal and vertical location between two points. Our h and v would normally be called something like ΔH and ΔV in the standard textbooks, where Δstuff stands for “a difference in stuff between one place and another.” This is why (when the machine Vhappens to be a line) you’ll sometimes see its slope written like this:

This is just “rise over run,” or what we called in the first chapter. Now, the symbol Δ is the Greek letter d (it’s really more like D, but stay with me), so it sort of makes sense to use Δ as an abbreviation for a “difference” between two things, or a “distance” between two points, which is what the textbooks do: ΔV stands for the vertical distance between two places, and ΔH stands for the horizontal distance between them. So all of the following are different ways of abbreviating the steepness of the machine V, as long as it’s just a straight line:

Now, while Δstuff stands for a change in stuff between two points, it basically always stands for a regular change involving regular numbers, not an infinitely small change involving infinitely small numbers. But now that we’ve started to invent calculus, we suddenly find that we’d like to distinguish between normal changes (when we’re not zoomed in) and infinitely small changes (when we are zoomed in). Here’s an occasion where the standard notation does something really nice: to change an expression involving regular numbers to an expression involving infinitely small ones, just change the Greek alphabet to the Latin alphabet (i.e., change Δ to d). So if we use the abbreviation d(stuff) to stand for an infinitely small change in stuff between two infinitely close locations, then we can write a string of equations similar to the ones above, but this time they’ll also apply to curvy things like V (H) = H2, instead of only applying to straight lines:

This is why you’ll often see the derivative of a machine M written as . See Figure 2.2 for a picture of this idea. Similarly, we’ll see later that when the textbooks pass from the letter Σ (the Greek S, which stands for the word “sum”) to its corresponding Latin letter S (they actually write , which kind of looks like an S), they’re doing a similar trick. In both cases, the passage from Greek letters to their Latin equivalents signifies the passage from a mundane expression involving regular numbers, to an expression involving numbers that are infinitely small. To be clear, Greek letters in mathematics certainly don’t always have this nice interpretation. They’re used for all sorts of different things. But at least in the above two cases (unlike others we’ll see later) the standard notation was designed extremely well.

In summary, all of the above abbreviations are referring to exactly the same idea, so it might seem like we have more than we need. But we’ll soon see how switching back and forth between different abbreviations has a surprising ability to make complicated things seem simple, and vice versa.