Burn Math Class: And Reinvent Mathematics for Yourself (2016)
Act III
6. Two in One
Interlude 6: Slaying Sharp
Best Served Cold
(Author is frantically working on something, as Reader enters the interlude. All manner of paraphernalia is strewn about the room. Author is pacing back and forth next to a table, frowning at some papers.)
Reader: What’s all this? What’s happening?
(Author is absorbed in whatever he is doing.)
Reader: Hellooooo???
(Author continues working frantically.)
Reader: Author!!!
Author: (Startled) Oh, hi. Good to see you.
Reader: So, what are we doing today?
Author: Settling the score.
Reader: What score? What are you talking about?
Author: Revenge!
Reader: Revenge? Against who? Everyone we’ve met has been really nice.
Author: I know!
Reader: Are you doing okay? You seem different recently. . .
(Mathematics enters the room.)
Author: Finally! You’re here!
Mathematics: WHAT’S GOING ON? THIS HAD BETTER BE IMPORTANT.
Author: Oh, it is!
Mathematics & Reader: (Simultaneously) WhAt ArE WE dOiNg?
(Author smiles.)
Author: We’re slaying 
.
Reader: Finally!
Mathematics: I THOUGHT WE WOULD NEVER GET AROUND TO THIS!
Author: Come on now, team. Watches synchronized?
Reader: Check.
Mathematics: I DON’T HAVE A WATCH. THE VOID IS TIMEL—
Author: Helmets securely fastened?
Reader: Check.
Mathematics: I JUST GOT HERE ABOUT HALF A PAG—
Author: We may have to use the Nostalgia Device, so it’ll need to be close at hand the entire time. First things first. We’ve got to form a plan of attack or else this thing is going to defeat us.
Mathematics: ALLOW ME!
A Plan of Attack
Mathematics: HOW’S THAT?
Author: Well, that’s a start, but making a section called “A Plan of Attack” isn’t the same as making a plan of attack. Philosophers call that a “use mention error.”
Mathematics: . . .
Author: Okay. There are several different routes we could take, and it’s best to keep them all in mind. If one route doesn’t work, we can fall back and try another. Any sentence we know that somehow involves is a possible weak point, so we need to focus on those. This thing tends to show up in sentences about circles, so let’s start with those. We know that the area of a circle with radius r is
So itself is just the area of a circle with radius 1. We’ve built up some artillery that lets us compute the area of certain curvy things, so that’s one possible path we could take.
Mathematics: WHAT? YOU TWO FIGURED OUT HOW TO COMPUTE THE ∫ THING?
Author: Well, sometimes. But not always.
Reader: Oh, that’s right, you weren’t there.
Author: Well it’s all on paper now. You’re welcome to walk back to Chapter 6 and check it out at your leisure. But come on, let’s focus. Other plans of attack!
Mathematics: OKAY, ALSO SHOWS UP IN THE DISTANCE AROUND A CIRCLE WITH RADIUS r:
Mathematics: SO SINCE r IS HALF THE DISTANCE ACROSS A CIRCLE, ITSELF IS JUST THE DISTANCE AROUND A CIRCLE WHOSE DISTANCE ACROSS IS 1.
Reader: At the end of the last chapter we found a way of computing curvy lengths, so that’s another possible route.
Mathematics: WHAT?
Reader: Oh, that’s right, you weren’t there for that either.
Mathematics: DID YOU TWO INVENT ANY MORE OF ME THAT I’VE YET TO BE INFORMED OF?
Reader: Nope, just Chapter 6.
Mathematics: GOOD. WELL, WE’VE GOT ALL THE CALCULUS WE’VE INVENTED SO FAR, LIKE THE HAMMERS.
Reader: And the anti-hammers.
Author: And the Nostalgia Device.
Mathematics: HEY, IN CHAPTER 5 WE USED THE NOSTALGIA DEVICE TO COMPUTE THAT NUMBER e. MAYBE IT WOULD WORK HERE TOO.
Reader: Good idea. How might we use the Nostalgia Device to compute ?
Author: Well when we computed e, we had this machine E(x) ≡ ex that spits out the number e when we feed it 1. The Nostalgia Device helped us talk about that machine using just addition and multiplication, like this:
Could we do that here?
Reader: We would need to think of a machine that spits out when we feed it some number.
Mathematics: SIMPLE ENOUGH. JUST DEFINE M(x) ≡ x, AND WE’VE GOT A MACHINE THAT SPITS OUT WHEN WE FEED IT .
Reader: I don’t know if that helps.
Author: Right. . . that’s true but it’s not really helpful. Using the Nostalgia Device on x just gives us x again. It’s true that 
, but that doesn’t really tell us what is.
Mathematics: THEN WHAT DO WE NEED?
Author: Well, in the example of ex, we didn’t just have a machine that spits out e when we feed it something. We had a machine that spits out e when we feed it something simple: the number 1. That gave us a description of e in terms of simpler things we already knew. We need to describe in terms of things that are as simple as possible.
Mathematics: OH. I DON’T KNOW HOW TO DO THAT.
Reader: Me neither.
Author: Me neither! But let’s not get discouraged. Look at all these weapons we’ve got to tackle the problem! (Author clears his throat.) We’ve got. . .
An Excessive Pile of Weaponry
(The characters look around the room, which is indeed littered with [title-of-this-section]. There are multiple hammers, equally many anti-hammers, an infinite magnifying glass, the Nostalgia Device, and two oddly shaped objects labeled “Lying” and “Correcting for the Lie.” A black sheet covers an object in the corner, next to a sign that says “Tricking the Mathematics: Out of Order.”)
Author: Look at all this stuff! There’s no way this problem can defeat us now.
Reader: Are you sure?
Author: No! But come on, let’s give it a shot. What do we need?
Mathematics: WELL, WE NEED A MACHINE THAT SPITS OUT WHEN WE FEED IT SOMETHING SIMPLE.
Reader: We kind of have the reverse of that.
Author: How do you mean?
Reader: Well, remember V and H? The things textbooks call sine and cosine?
Author: Of course.
Reader: Remember how V spits out 1 when we feed it 
?
Author: No.
Reader: You never remember anything. But you said it yourself in Chapter 4. We defined V visually and figured out that
That’s kind of the reverse of the e situation.
Mathematics: EXPLAIN?
Reader: Well, instead of spitting out the number we want when we feed it something simple, like ex did, V spits out something simple when we feed it something we want! Or, something related to what we want. I mean, if we could figure out 
then we’d also have figured out .
Mathematics: OH! OKAY. SO, IF THERE WERE AN “OPPOSITE” OF V — WHATEVER THAT MEANS — THEN IT WOULD EAT 1 AND SPIT OUT , RIGHT?
Author: What’s the opposite of V?
Mathematics: I DON’T KNOW.
Author: I don’t know either.
Reader: Wait, does it matter if we’re not really familiar with the opposite of V? Can’t we just give it a name and see how it has to behave?
Author: Sure, I guess we could try.
Mathematics: LET ME NAME IT!
Author: Okay.
Mathematics: I’LL ABBREVIATE THE OPPOSITE MACHINE OF V AS. . . Λ. WE DON’T KNOW WHAT Λ LOOKS LIKE EXACTLY, BUT IT’S WHATEVER MACHINE DOES THIS:
NO MATTER WHAT NUMBER x IS. SO WHATEVER V DOES, Λ DOES THE OPPOSITE.
Reader: Exactly! So it’s got to be true that
Mathematics: SO IF WE COULD FIGURE OUT ANOTHER WAY TO COMPUTE Λ(1), WE’D BE DONE?
Author: I guess so. How are we supposed to do that?
Reader: We could use the Nostalgia Device.
Author: Nah, that won’t work. We don’t know any of the derivatives of Λ. We need to know all of the derivatives to use the Nostalgia Device.
Mathematics: PERHAPS WE COULD TRICK THE MATHEMATICS INTO TELLING US WHAT THE DERIVATIVES ARE.
Hammering with Reabbreviation
(Author and Reader simultaneously turn to Mathematics, with equally shocked expressions.)
Reader: Wait, the first time we ever met, you got all mad at us for using that phrase.
Author: Yeah, all of a sudden you don’t mind us saying “trick the Mathematics”?
Mathematics: WELL, I SAID IT WITH A LOWERCASE M, SO IT DOESN’T REFER TO ME. MY NAME STARTS WITH A CAPITAL M.
Reader: Those are both a capital M.
Author: No no no! Mathematics speaks in “small caps.” It’s a typography thing. Totally different emotional tone from just writing something in ALL CAPITAL LETTERS.
Reader: Wait. . . so the Mathematics character isn’t always yelling?
Author: What? Of course not! You didn’t think that this whole time, did you?
Reader: I don’t know. . . I may have. . .
Mathematics: ENOUGH, YOU TWO. LIKE I SAID, WE MIGHT BE ABLE TO TRICK THE MATHEMATICS BY DOING SOMETHING LIKE THIS: LET’S DEFINE THE AMBUSH MACHINE AS A(x) ≡ Λ(V(x)). SECRETLY, THIS IS JUST THE MACHINE A(x) = x, so 
. NOW WE CAN USE THE REABBREVIATION HAMMER, LIKE THIS:
Author: How does that help?
Reader: Well, there are two pieces on the far right. We know one of them, because 
.
Mathematics: EXACTLY. SO USE THAT TO REWRITE WHAT WE JUST DID. THAT GIVES:
Author: Okay, but what’s 
?
Reader: I don’t know.
Mathematics: COME ON, AUTHOR. REMEMBER, WE CAN ALWAYS CHANGE ABBREVIATIONS. WOULD IT HELP IF I REWROTE THE SAME THING LIKE THIS:
Author: No.
Reader: Oh! I see. Since we can always change abbreviations, those are all the same.
Mathematics: EXACTLY! THAT’S ALL I WAS TRYING TO SAY. WE CAN TRICK THE MATHEMATICS INTO TELLING US THE DERIVATIVE OF Λ BY DEFINING A(x) ≡ Λ(V(x)) AND THEN DIFFERENTIATING A IN TWO WAYS. ON THE ONE HAND, 
. ON THE OTHER HAND, WE CAN USE THE REABBREVIATION HAMMER LIKE WE DID ABOVE, WHERE WE FOUND
SO WE CAN THROW H(x) OVER TO THE OTHER SIDE TO ISOLATE THE DERIVATIVE OF Λ, LIKE THIS:
Author: Sorry to keep weighing down this discussion, but I still don’t see how this helps us at all. I can see how this type of argument can work in less tangled cases. Like, if you made an argument that showed something like
But what you wrote down in equation 6.34 is confusing and I don’t see how it helps. We want 
, and sure that’s the same as the original expression with V s in place of xs, because we can abbreviate things however we want, but when you change the V s to xs, what happens to that H(x) on the right side? If you’ve got an entire equation written with V as the “variable,” then I totally see how you can replace that with any other letter you want, but you’ve already got xs in there, and. . . I just don’t know what’s happening.
Mathematics: OH, WELL THEN WE SHOULD TRY AND WRITE H IN TERMS OF V. IF WE COULD DO THAT, THEN EQUATION 6.34 WOULD BE WRITTEN COMPLETELY IN V LANGUAGE, AND THEN WE COULD JUST REABBREVIATE ALL THE VS TO xS, AND WE WOULD HAVE TRICKED ME INTO TELLING US THE DERIVATIVE OFΛ.
Author: Okay, I’m still not completely on board, but I think I see where you’re going. The formula for shortcut distances told us that V2 + H2 = 1,
which we could rewrite as 
. If we do that, then equation 6.34 becomes
Now there are only V s everywhere, so since we can abbreviate things however we want, it has to be true that
Author: But something still feels funny about this argument.
Reader: Well, you’re always saying that if we’re not sure about an argument, it helps to see if we can get the same result in a different way. We could try that.
Author: Alright. . . I guess we could run the same argument inside out.
Reader: How so?
Author: Well, in the above argument we used the sentence Λ(V(x)) = x to trick the mathematics into telling us Λ′(x). What if instead we used V(Λ(x)) = x, but otherwise did everything exactly the same? I mean, if we do that and get the same answer for Λ′(x), I guess I’d be a bit less worried. You got a minute?
Reader: A minute? If I’ve made it this far in the book, I’m probably an extremely patient person. Take your time.
Author: (Sigh3) You’re so great. Okay, I’ll try to be quick. This time let’s define A(x) ≡ V (Λ(x)), which is also just x because Λ and V undo each other, so the derivative of A is just 1 again. Now let’s use the same type of reasoning we did before:
(Reader’s patience in the face of Author’s unbounded verbosity warms Author’s heart, and Author is overcome with a feeling of 
(platonic+undefined) love, followed soon thereafter (read: now) by a deep feeling of embarrassment at having divulged the former feeling so publicly. After all, Author thought, one isn’t supposed to say such things in a textbook (or whatever this is). But I digress. . .)
The first equality is because A(x) = x, the third uses the hammer for reabbreviation, and the other two are just definitions. Now, on the far right we have the derivative of V, which is H. So we have
This H(Λ) piece looks funny, but it’s just H(stuff), and any stuff inside H can always be thought of as an angle. So we can just use the formula for shortcut distances to get H(Λ)2 + V(Λ)2 = 1. Then just isolate H(Λ) to get 
, which lets us write
Wait, that’s different from what we got last time.
Mathematics: I’M NOT SURE IT IS! WE STOPPED WRITING THE x’S SO THAT THINGS WOULD LOOK LESS SCARY, REMEMBER? LIKE WHEN YOU WROTE Λ INSTEAD OF Λ(x).
Author: Oh, right! So that V(Λ) piece is really V(Λ(x)), which by definition is just x. So we can rewrite what we just found like this:
which is exactly the same thing we got in equation 6.36!
Mathematics: CONVINCED?
Author: Slightly more than I was before.
Sharp Resists
Author: Okay, what now?
Mathematics: WELL, WE JUST CONVINCED OURSELVES THAT
WE DID THAT BECAUSE WE REALIZED THAT 
, SO IF WE COULD FIGURE OUT ANOTHER WAY TO COMPUTE Λ(1), THEN WE COULD COMPUTE !
Author: How does what we just did help us compute Λ(1)?
Mathematics: OH. . . I DON’T BELIEVE IT DOES.
Author: Ugh. . .
Mathematics: WE WERE PLANNING ON USING THE NOSTALGIA DEVICE, AND WE NEED ALL THE DERIVATIVES OF A MACHINE TO USE THAT. WE JUST FOUND THE FIRST ONE.
Author: Yeah, and it was a pain.
Reader: Can we use the fundamental hammer?
Mathematics: OOH, MAYBE. WHAT MADE YOU THINK THAT?
Reader: Well, the fundamental hammer relates machines and their derivatives, so it might let us relate the information we just discovered about the derivative of Λ to information about Λ itself. I don’t know.
Author: Hey, yeah. The fundamental hammer says that
and we want Λ(1), so if we make b = 1 and use equation 6.38, we can write
Ugh. We just want Λ(1). It would be nice to get rid of that Λ(a) on the right.
Reader: Is there any a that makes Λ(a) = 0?
Mathematics: WELL, WE DON’T KNOW MUCH ABOUT Λ — WE JUST DEFINED IT TO BE WHATEVER MACHINE UNDOES V. WE KNOW V (0) = 0. BUT Λ UNDOES V, SO Λ(0) = 0, RIGHT?
Author: Makes sense.
Reader: Nice! So let’s choose a = 0 in equation 6.39 to get
Now we can combine this with the fact that 
from earlier to get
Author: Beautiful! Are we done?
Mathematics: I GUESS.
Reader: No we’re not! We still don’t have a specific number for ! That was the whole point.
Author: Well, how do we compute a specific number for ?
Reader: We’ve got to compute the integral in equation 6.41. How do we do that?
Author: I don’t know.
Mathematics: I DON’T KNOW EITHER.
Reader: You’ve got to be kidding. . .
Back to the Drawing Board
Reader: You mean to tell me that we’re no closer to figuring out than we were when we started!?
Author: I don’t know. We’re kind of “closer.” We’ve learned a lot more about how to attack the problem. . . but yeah, we’re still stuck. We don’t know how to compute the integral in equation 6.41.
Reader: We could just do it by brute force. Approximately.
Author: You mean by adding up the areas of a bunch of tiny rectangles?
Reader: Yeah!
Mathematics: THAT SOUNDS UNPLEASANT. I’D RATHER NOT.
Reader: Come on! This is taking so long!
Author: Alright, if you want. I’ll get us started.
Mathematics: ENJOY, YOU TWO. I’LL BE OVER HERE THINKING ABOUT THE PROBLEM BY MYSELF.
(Mathematics goes to the other side of the room.)
Author: Okay, well we could look at a bunch of points in the interval from 0 to 1, say
These are all the points 
, where n is some big number and k goes from 0 to n.
Reader: Okay, keep going!
Author: Well, the distance between any two of these points is the same. That is, 
.
Reader: Yeah. And then?
Author: Then we could approximate the integral in equation 6.41 by writing
Wow, this looks terrible.
Reader: Come on!
Author: Nah, it’s not worth it. How are we supposed to calculate this ourselves?
Reader: We could call Al and Sil! They said to call them if we ever had any number problems.
Author: I guess we could, but I feel guilty using Al and Sil to do things for us that we can’t do. Back when we were computing e, we invented two expressions that we could compute in principle. We just didn’t feel like doing the arithmetic. If we could get to a place like that, then I wouldn’t mind using Al and Sil.
Reader: Well why doesn’t this count?
Author: I guess it kind of does, but it’s a sum of a bunch of terms and each one has a square root in it. We don’t really know how to compute square roots.
Reader: Yes we do! We can just use the Nostalgia Device!
Author: Sure. . . I mean, we could compute any particular square root to arbitrary accuracy. But this is a sum of a huge number of square roots. I guess we could compute the whole sum to arbitrary accuracy by expanding each square root with the Nostalgia Device, cutting off each expression after a finite number of terms, and then adding up all the sums, but that would be a big ugly double sum. Even if we’d get the right answer in that case, it’s ugly. We might be able to find , but it doesn’t really feel like slaying.
Reader: Forget this!
(Reader leaves Author and heads over to Mathematics, who is busily working on something on the other side of the room.)
Reader: Hi, Math. Any progress?
(Mathematics continues working, talking to Reader without looking up from the table.)
Mathematics: I DON’T KNOW YET. I’VE BEEN PLAYING AROUND WITH OTHER MACHINES THAT SPIT OUT WHEN WE FEED THEM SOMETHING SIMPLE. YOU AND AUTHOR MAKE ANY PROGRESS?
Reader: Kind of. Not really. We got an expression for , but it involved square roots and it was really ugly, so Author said he didn’t really feel like we had solved the problem.
Mathematics: WHAT? WHY NOT?
Reader: Because we couldn’t compute the answer ourselves. Well, we could, but he was being picky. It would have been a lot of numbers and approximations. But we were so close!
Mathematics: I CAN SYMPATHIZE. LOTS OF NUMBERS AND APPROXIMATIONS CAN BE UGLY. WHAT’S THE POINT OF ALL THIS IF WE DON’T GET SOMETHING WE LIKE?
Reader: But if we’re that picky, how are we ever going to solve this problem? I guess the answer itself had better be really simple or else you two won’t feel like we’ve really slayed . How on earth are we supposed to do that?
Mathematics: I DON’T KNOW. WHAT DID YOU SAY AUTHOR’S PROBLEM WAS?
Reader: Ultimately I guess he just didn’t like the square roots.
Mathematics: INTERESTING. . . I THINK I MIGHT KNOW HOW TO AVOID THEM.
Reader: Wait, are you serious?
Mathematics: MAYBE. WHERE DID THE SQUARE ROOT COME FROM IN THE FIRST PLACE?
Reader: When we were reabbreviating. We wanted to express the whole thing in terms of V. We wrote H2 + V2 = 1 and then isolated H to get 
. That’s when the square root came in.
Mathematics: SO IT’S V’S FAULT.
Reader: How do you mean?
Mathematics: WELL WHY WERE WE USING V IN THE FIRST PLACE?
Reader: Because we wanted a machine that spit out when we fed it something simple. We knew 
, which means 
. So if only we could figure out how to compute Λ(1), we’d know how to compute . What does that have to do with the square roots?
Mathematics: THE SQUARE ROOT CAME IN BECAUSE WE NEEDED IT IN ORDER TO EXPRESS V’S DERIVATIVE BY ONLY REFERRING TO V ITSELF. THE DERIVATIVE OF V IS H, NOT V ITSELF, AND EVENTUALLY THAT LED US TO THE UGLY SQUARE ROOTS.
Reader: I don’t see how this is getting us anywhere.
Mathematics: WE WANT A MACHINE THAT SPITS OUT WHEN WE FEED IT SOMETHING SIMPLE, RIGHT?
Reader: Right.
Mathematics: AND WE WANT TO AVOID THE OBNOXIOUS SQUARE ROOT THING HAPPENING AGAIN, SO WE DON’T WANT TO USE V OR H THEMSELVES.
Reader: Right.
Mathematics: WELL, I HAVE THIS CRAZY IDEA. REMEMBER ALL THOSE POINTLESS MACHINES LIKE “TANGENT” THAT AUTHOR WAS COMPLAINING ABOUT BACK IN CHAPTER 4?
Reader: Vaguely.
Mathematics: WHILE YOU TWO WERE OVER THERE, I WAS LOOKING BACK AT THAT PART OF THE BOOK. THIS MIGHT BE THE ONLY TIME THAT IT’S ACTUALLY USEFUL TO USE ONE OF THEM.
Reader: Don’t tell Author. He’ll probably get mad. What’s your idea?
Mathematics: OKAY, BACK IN CHAPTER 4, AUTHOR MENTIONED THIS MACHINE “TANGENT.” IT WAS DEFINED TO BE 
. NOW, I WAS LOOKING BACK AT CHAPTER 4, AND I SAW THAT ITS DERIVATIVE IS T′ = 1 + T2. SINCE ITS DERIVATIVE CAN BE WRITTEN JUST IN TERMS OF ITSELF WITHOUT USING SQUARE ROOTS, THIS SEEMS TO SOLVE BOTH OF THE PROBLEMS WE RAN INTO EARLIER.
Reader: Perfect. Let’s try the argument we made before.
Unearthing Something We Buried
Reader: Okay, so 
. When we feed it 
, it spits out. . . uh. . . 
. I don’t quite know what to do with that. Does T ever spit out anything less confusing?
Mathematics: WELL, IF WE FEED IT /4, THAT’S AN EIGHTH OF A FULL TURN, SO V AND H ARE THE SAME, AND 
. THAT’S FAIRLY SIMPLE.
Reader: Nice. And for the same reason as before, it’s not T that we care about. We care about its opposite, because we want a machine that spits out when we feed it something simple. So if there’s a machine ⊥(x) that can undo T, like this:
⊥(T(x)) = x
then we’d have 
, which means 
.
Mathematics: (Suddenly excited) THIS MAY ACTUALLY WORK. . .
Reader: Alright, let’s try to do exactly what we did with Λ, and maybe it’ll pay off this time. First, let’s define A(x) ≡ ⊥(T(x)), so secretly A(x) = x, and its derivative is 
. Then we can use the hammer for reabbreviation to write
Mathematics: NOW WE CAN USE THE FACT THAT
FROM CHAPTER 4, TOGETHER WITH EQUATION 6.42, TO WRITE
Reader: Nice. Everything is written in terms of T, so we can reabbreviate to get
In the earlier argument with V, what did we do next?
(Reader looks back.)
Right, we used the fundamental hammer on the derivative. Now we can use the fundamental hammer on the derivative of ⊥ to write
Mathematics: OOH, I HAVE A GOOD FEELING ABOUT THIS. WE WANT ⊥(1), SO LET’S MAKE b = 1. WE DON’T WANT TO BOTHER WITH THAT OTHER TERM ⊥(a), SO LET’S CHOOSE AN a FOR WHICH ⊥(a) = 0.
Reader: Well, T(0) = 0, so ⊥(0) = 0, too. But then by the far right of equation 6.43, the above integral with a = 0 and b = 1 is just ⊥(1).
Mathematics: FANTASTIC. AND WE ALREADY KNOW THAT 
, SO WE CAN WRITE
Unleashing the Nostalgia Device
Mathematics: WHAT NOW?
Reader: Do we know how to compute this integral?
Mathematics: I DON’T.
Reader: I don’t either.
Mathematics: IF WE DID ALL THIS FOR NOTHING, I MAY HAVE TO QUIT THE BOOK.
Reader: Me too. This is painful.
(Author walks over to Reader and Mathematics.)
Author: Hi there, you two. Any progress?
Reader: We were so close again. . . and we just got stuck.
Author: What did you do?
Mathematics: GO SEE FOR YOURSELF.
(Author skims the above conversation.)
Author: (Subdued) Oh no. . .
(Author sits in silence for a moment.)
Reader: (To Mathematics) Uh-oh, I think he’s mad about us using that machine he kept complaining about back in Chap—
Mathematics: IT WAS READER’S IDEA!!!
Author: No no, it’s not that. It’s just. . . if you two quit the book. . . either of you. . . I don’t think I could get through this. After everything we’ve been through. . . I can’t go back to doing this alone. . .
Reader: Oh.
Mathematics: OH.
Author: Please don’t leave. . . Or, you can. If you want. . . It’s okay if you do. . . I know it’s rough. It’s a lot of gory details from time to time, especially right now. But, I mean, the only alternative would be to hide something from you. I’m not sure I could finish the book if I had to do that. Even though it would be less work, it would be so much harder. I don’t want to have to lie to you. So, you can leave if you want. But while you’re here. . . let’s try to get through this together. . . okay?
Reader: Okay.
Mathematics: OKAY.
(A moment of (awkward+comfortable) silence elapses.)
Mathematics: I’M NOT SURE I COULD LEAVE IF I WANTED TO. . .
Author: Hah. Anyways, that’s a nice result you two just discovered.
Reader: So what? We’re stuck!
Author: Maybe you’re not. While you two were talking, I was playing around with the Nostalgia Device on the version of the problem with the square roots. It was really ugly, but it gave me an idea. Check it out. You two just showed that
Now what if we just expand 
with the Nostalgia Device?
Mathematics: THAT’LL NEVER WORK. WE’D NEED TO FIGURE OUT ALL OF ITS DERIVATIVES.
Reader: Well, the zeroth derivative was just the machine itself, right?
Author: Right! So M(0)(0) ≡ M (0) = 1.
Reader: And the first derivative is
so M′(0) = 0.
Mathematics: THE SECOND ONE IS JUST THE DERIVATIVE OF THAT:
M″(x) = (−1)(−2)(2x)(1 + x2)−3 + (−2) · (1 + x2)−2
So M″(0) = −2.
Author: This is getting really ugly. Can we cheat?
Reader: How?
Author: Well, 
is just the machine 
with x2 plugged in.
Reader: So?
Author: Couldn’t we just use the Nostalgia Device on 
and then plug in x2 afterward? It seems like it would be so much simpler.
Reader: Would that work?
Author: Not sure. But it seems like it has to work.
Reader: I guess it’s worth a shot.
Mathematics: WAIT, WE’RE STARTING OVER?
Author: No, don’t worry. But let’s try to find the derivatives of a slightly different machine. Here, watch. Define
The zeroth derivative is just the machine itself, so m(0)(0) ≡ m(0) = 1.
Mathematics: THE FIRST DERIVATIVE OF m IS
Reader: The second derivative is
Wow, this is a lot easier!
Author: Right? And we can see that the nth derivative will be
Hmm. . . what’s that?
Mathematics: WELL, YOU’VE GOT n NEGATIVE THINGS, RIGHT?
Author: Yeah.
Mathematics: THAT’S n COPIES OF (−1). SO MOVE ALL OF THOSE TO THE FRONT. THEN THE REST IS JUST n!, RIGHT? THAT IS, m(n)(0) = (−1)nn!
Reader: And that’s all we need to use the Nostalgia Device on m(s), so
Now what?
Author: Well, M(x) = m(x2), so just toss x2 in there:
Why were we doing this?
Mathematics: WE WERE TRYING TO WRITE THE EXPRESSION READER AND I CAME UP WITH IN A WAY THAT LET US GET UNSTUCK. WE HAD WRITTEN
BUT NOW THAT WE USED THE NOSTALGIA DEVICE ON IN THAT INDIRECT SORT OF WAY, EQUATION 6.46 LETS US WRITE
Reader: That may be the scariest thing I’ve ever seen.
Author: Yeah, I don’t think we know what to do with this.
Mathematics: SURE WE DO!
Infinite Shattering with Anti-Hammers
Mathematics: IT’S NOT AS SCARY AS IT LOOKS. HERE, I’LL WRITE IT THIS WAY:
Reader: Oh, wow. That’s a lot better.
Author: I guess now we can use our anti-hammer for addition to break apart the integral.
Reader: Does that work for infinite sums?
Author: I don’t know, but let’s hope it works for this one! It’s either that or give up, so let’s keep moving. Once we break the integral apart, we just need to think of an anti-derivative for each piece. Check it out:
which is just an abbreviation for
Sharp Yields
Reader: So is just four times an infinite back-and-forth sum of the handstands of all the odd numbers?
Author: I guess so.
Mathematics: HEY, AND IF n IS A WHOLE NUMBER, THEN 2n IS ALWAYS EVEN AND 2n + 1 IS ALWAYS ODD, SO WE COULD WRITE THE SAME THING THIS WAY:
Author: Perfect — there are no square roots or anything. We described just using arithmetic! We could absolutely compute this ourselves, in principle.
Reader: Does that mean we can call Al and Sil?
Mathematics: CERTAINLY.
(Mathematics borrows Author’s phone again and dials a number.)
Mathematics: AL, LISTEN, ARE YOU BUSY?. . . YOU ARE BUSY?. . . WHAT ARE YOU DOING?. . . OH! THAT’S WONDERFUL. I’M SO GLAD YOU TWO FINALLY. . . LISTEN, I NEED TO BE QUICK. THERE ISN’T MUCH TIME. WE KNOW YOU CAN’T HANDLE INFINITE JOBS, BUT WOULD YOU MIND COMPUTING. . .
(Mathematics whispers equation 6.52 into the phone.)
WHEN N IS 100,000?
Author: How long is this gonna take?
Mathematics: HE SAYS IT’S ABOUT 3.14160.
Author: Wow! That was quick. Do you mind doing it again? Ask him for the first million terms.
Mathematics: (Time passes) HE SAYS IT’S ABOUT 3.14159.
Reader: Nice, it looks like the first three decimal places have stabilized, and the other two aren’t changing much.
Author: And hey, remember back in Chapter 4 we guessed that should be somewhere in the neighborhood of 3? I guess we were right. There’s some more evidence that all those arguments we just made were probably on the right track.
Mathematics: I CAN’T BELIEVE WE’RE FINISHED.
Author: I know. . . That was by far the most difficult thing we’ve done so far. But we did it! We slayed . In the end, it just turned out to be
Author: See how much was being hidden from us when we were all just told that the area of a circle is πr2, and—
Reader: Hey, that reminds me. You said in Chapter 4 that after we figure out how to compute this thing, we would start calling it π.
Author: Did I?
(Author flips back to Chapter 4.)
Author: Huh. You’re right. I guess I did say that.
Mathematics: WHAT ON EARTH IS π?
Reader: Well, in the textbooks, π is—
Author: Forget that. Let’s keep calling it . We earned this.
Reader: Works for me.