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Everyday Calculus
Discovering the Hidden Math All around Us
By OSCAR E. FERNANDEZ PRINCETON UNIVERSITY PRESS
Copyright © 2014 Princeton University Press
All rights reserved.
ISBN: 978-1-4008-8566-4
CHAPTER 1
WAKE UP AND SMELL THE FUNCTIONS
It's Friday morning. The alarm clock next to me reads 6:55 a.m. In five minutes it'll wake me up, and I'll awake refreshed after sleeping roughly 7.5 hours. Echoing the followers of the ancient mathematician Pythagoras — whose dictum was "All is number"— I deliberately chose to sleep for 7.5 hours. But truth be told, I didn't have much of a choice. It turns out that a handful of numbers, including 7.5, rule over our lives every day. Allow me to explain.
A long time ago at a university far, far away I was walking up the stairs of my college dorm to my room. I lived on the second floor at the time, just down the hall from my friend Eric Johnson's room. EJ and I were in freshman physics together, and I often stopped by his room to discuss the class. This time, however, he wasn't there. I thought nothing of it and kept walking down the narrow hallway toward my room. Out of nowhere EJ appeared, holding a yellow Post-it note in his hand. "These numbers will change your life," he said in a stern voice as he handed me the note. Off in the corner was a sequence of numbers:
1.5 4.5 7.5
3 6
Like Hurley from the Lost television series encountering his mystical sequence of numbers for the first time, my gut told me that these numbers meant something, but I didn't know what. Not knowing how to respond, I just said, "Huh?"
EJ took the note from me and pointed to the number 1.5. "One and a half hours; then another one and a half makes three," he said. He explained that the average human sleep cycle is 90 minutes (1.5 hours) long. I started connecting the numbers in the shape of a "W." They were all a distance of 1.5 from each other — the length of the sleep cycle. This was starting to sound like a good explanation for why some days I'd wake up "feeling like a million bucks," while other days I was just "out of it" the entire morning. The notion that a simple sequence of numbers could affect me this much was fascinating.
In reality getting exactly 7.5 hours of sleep is very hard to do. What if you manage to sleep for only 7 hours, or 6.5? How awake will you feel then? We could answer these questions if we had the sleep cycle function. Let's create this based on the available data.
What's Trig Got to Do with Your Morning?
A typical sleep cycle begins with REM sleep — where dreaming generally occurs — and then progresses into non-REM sleep. Throughout the four stages of non-REM sleep our bodies repair themselves, with the last two stages — stages 3 and 4 — corresponding to deep sleep. As we emerge from deep sleep we climb back up the stages to REM sleep, with the full cycle lasting on average 1.5 hours. If we plotted the sleep stage S against the hours of sleep t, we'd obtain the diagram in Figure 1.1(a). The shape of this plot provides a clue as to what function we should use to describe the sleep stage. Since the graph repeats roughly every 1.5 hours, let's approximate it by a trigonometric function.
To find the function, let's begin by noting that S depends on how many hours t you've been sleeping. Mathematically, we say that your sleep stage S is a function of the number of hours t you've been asleep, and write S = f (t). We can now use what we know about sleep cycles to come up with a reasonable formula for f (t).
Since we know that our REM/non-REM stages cycle every 1.5 hours, this tells us that f (t) is a periodic function — a function whose values repeat after an interval of time T called the period — and that the period T = 1.5 hours. Let's assign the "awake" sleep stage to S = 0, and assign each subsequent stage to the next negative whole number; for example, sleep stage 1 will be assigned to S = -1, and so on. Assuming that t = 0 is when you fell asleep, the trigonometric function that results is
f(t) = 2 cos (4π/3 t) - 2,
where π ≈ 3.14.
Before we go off and claim that f (t) is a good mathematical model for our sleep cycle, it needs to pass a few basic tests. First, f (t) should tell us that we're awake (sleep stage 0) every 1.5 hours. Indeed, f (1.5)= 0 and so on for multiples of 1.5. Next, our model should reproduce the actual sleep cycle in Figure 1.1(a). Figure 1.1(b) shows the graph of f (t), and as we can see it does a good job of capturing not only the awake stages but also the deep sleep times (the troughs).
In my case, though I've done my best to get exactly 7.5 hours of sleep, chances are I've missed the mark by at least a few minutes. If I'm way off I'll wake up in stage 3 or 4 and feel groggy; so I'd like to know how close to a multiple of 1.5 hours I need to wake up so that I still feel relatively awake.
We can now answer this question with our f (t) function. For example, since stage 1 sleep is still relatively light sleeping, we can ask for all of the t values for which f (t) ≥ -1, or
2 cos (4π/3 t) - 2 ≥ -1.
The quick way to find these intervals is to draw a horizontal line at sleep stage -1 on Figure 1(b) [??]. Then all of the t-values for which our graph is above this line will satisfy our inequality. We could use a ruler to obtain good estimates, but we can also find the exact intervals by solving the equation f (t) = -1:
[0, 0.25], [1.25, 1.75], [2.75, 3.25], [4.25, 4.75], [5.75, 6.25], [7.25, 7.75], etc.
We can see that the endpoints of each interval are 0.25 hour — or 15 minutes — away from a multiple of 1.5. Hence, our model shows that missing the 1.5 hour target by 15 minutes on either side won't noticeably impact our morning mood.
This analysis assumed that 90 minutes represented the average sleep cycle length, meaning that for some of us the length is closer to 80 minutes, while for others it's closer to 100. These variations are easy to incorporate into f (t): just change the period T. We could also replace the 15-minute buffer with any other amount of time. These free parameters can be specified for each individual, making our f (t) function very customizable.
I'm barely awake and already mathematics has made it into my day. Not only has it enabled us to solve the mystery of EJ's multiples of 1.5, but it's also revealed that we all wake up with a built-in trigonometric function that sets the tone for our morning.
How a Rational Function Defeated Thomas Edison, and Why Induction Powers the World
Like most people I wake up to an alarm, but unlike most people I set two alarms: one on my radio alarm clock plugged into the wall and one on my iPhone. I adopted this two-alarm system back in college when a power outage made me late for a final exam. We all know that our gadgets run on electricity, so the power outage must have interrupted the flow of electricity to my alarm clock at the time. But what is "electricity," and what causes it to flow?
On a normal day my alarm clock gets its electricity in the form of alternating current (AC). But this wasn't always the case. In 1882 a well-known inventor — Thomas Edison — established the first electric utility company; it operated using direct current (DC). Edison's business soon expanded, and DC current began to power the world. But in 1891 Edison's dreams of a DC empire were crushed, not by corporate interests, lobbyists, or environmentalists, but instead by a most unusual suspect: a rational function.
The story of this rational function begins with the French physicist André-Marie Ampère. In 1820 he discovered that two wires carrying electric currents can attract or repel each other, as if they were magnets. The hunt was on to figure out how the forces of electricity and magnetism were related.
The unexpected genius who contributed most to the effort was the English physicist Michael Faraday. Faraday, who had almost no formal education or mathematical training, was able to visualize the interactions between magnets. To everyone else the fact that the "north" pole of one magnet attracted the "south" pole of another — place them close to each other and they'll snap together — was just this, a fact. But to Faraday there was a cause for this. He believed that magnets had "lines of force" that emanated from their north poles and converged on their south poles. He called these lines of force a magnetic field.
To Faraday, Ampère's discovery hinted that magnetic fields and electric current were related. In 1831 he found out how. Faraday discovered that moving a magnet near a circuit creates an electric current in the circuit. Put another way, this law of induction states that a changing magnetic field produces a voltage in the circuit. We're familiar with voltages produced by batteries (like the one in my iPhone), where chemical reactions release energy that results in a voltage between the positive and negative terminals of the battery. But Faraday's discovery tells us that we don't need the chemical reactions; just wave a magnet near a circuit and voilà, you'll produce a voltage! This voltage will then push around the electrons in the circuit, causing a flow of electrons, or what we today call electricity or electric current.
So what does Edison have to do with all of this? Well, remember that Edison's plants operated on DC current, the same current produced by today's batteries. And just like these batteries operate at a fixed voltage (a 12-volt battery will never magically turn into a 15-volt battery), Edison's DC-current plants operated at a fixed voltage. This seemed a good idea at the time, but it turned out to be an epic failure. The reason: hidden mathematics.
Suppose that Edison's plants produce an amount V of electrical energy (i.e., voltage) and transmit the resulting electric current across a power line to a nineteenth-century home, where an appliance (perhaps a fancy new electric stove) sucks up the energy at the constant rate P. The radius r and length l of the power line are related to V by
r(V) = k [[square root of P0l]/V],
where k is a number that measures how easily the power line allows current to flow. This rational function is the nemesis Edison never saw coming.
For starters, the easiest way to distribute electricity is through hanging power lines. And there's an inherent incentive to make these as thin (small r) as possible, otherwise they would both cost more and weigh more — a potential danger to anyone walking under them. But our rational function tells us that to carry electricity over large distances (large l) we need large voltages (large V) if we want the power line radius r to be small (Figure 1.2). And this was precisely Edison's problem; his power plants operated at the low voltage of 110 volts. The result: customers needed to live at most 2 miles from the generating plant to receive electricity. Since start-up costs to build new power plants were too high, this approach soon became uneconomical for Edison. On top of this, in 1891 an AC current was generated and transported 108 miles at an exhibition in Germany. As they say in the sports business, Edison bet on the wrong horse.
But the function r(V) has a split personality. Seen from a different perspective, it says that if we crank up the voltage V — by a lot — we can also increase the length l — by a bit less — and still reduce the wire radius r. In other words, we can transmit a very high voltage V across a very long distance l by using a very thin power line. Sounds great! But having accomplished this we'd still need a way to transform this high voltage into the low voltages that our appliances use. Unfortunately r(V) doesn't tell us how to do this. But one man already knew how: our English genius Michael Faraday.
Faraday used what we mathematicians would call "transitive reasoning," the deduction that if A causes B and B causes C, then A must also cause C. Specifically, since a changing magnetic field produces a current in a circuit (his law of induction), and currents flowing through circuits produce magnetic fields (Ampère's discovery), then it should be possible to use magnetic fields to transfer current from one circuit to another. Here's how he did it.
Picture Faraday — a clean-shaven tall man with his hair parted down the middle — with a magnet in his hand, waving it around a nearby circuit. Induction causes this changing magnetic field to produce a voltage Va in one circuit (Figure 1.3(a)). The alternating current produced would, by Ampère's discovery, produce another changing magnetic field. The result would be another voltage Vb in a nearby circuit (Figure 1.3(b)), producing current in that circuit.
As Faraday waves the magnet around, sometimes he does so closer to the loop and sometimes farther away; sometimes he waves it fast and other times slow. In other words, the voltage Va produced changes. Today, magnets are put inside objects like windmills that do the waving for us. As the blades rotate in the wind, the magnetic field produced inside the turbine also changes. In this case the changes are described by a trigonometric function (not by Faraday's crazy hand-waving). This alternating voltage causes the current to alternate too, putting the "alternating" in alternating current.
Great, we can now transfer current between circuits. But we still have the voltage problem: most household plugs run at low voltages (a fact left over from Edison's doings), yet our modern grids produce voltages as high as 765,000 volts; how do we reduce this to the standard range of 120–220 volts that most countries use?
Let's suppose that the original circuit's wiring has been coiled into Na turns, and that the nearby circuit's wiring has been coiled into Nb turns (Figure 1.4(a)). Then
Vb = [Nb/Na] Va.
This formula says that a high incoming voltage Va can be "stepped down" to a low outgoing voltage Vb by using a large number of turns Na for the incoming coiling relative to the outgoing coiling. This transfer of voltage is called mutual induction, and is at the heart of modern electricity transmission. In fact, if you step outside right now and look up at the power lines you'll likely see cylindrical buckets like the one in Figure 1.4(b). These transformers use mutual induction to step down the high voltages produced by modern electricity plants to lower, safer voltages for household use.
The two devices that got me going on this story — the iPhone and my clock radio — honor the legacies of both Edison and Faraday. My iPhone runs on DC current from its battery, and my clock radio draws its power from the AC current coming through the wall plug, itself produced dozens of miles away at the electricity plant by an alternating voltage. And somewhere in between, Faraday's mutual induction is at work stepping down the voltage so that we can power our devices.
But the real hero here is the rational function r(V). It spelled doom for Edison, but through a different interpretation suggested that we base our electric grid on voltages much higher than Edison's 110 volts. This idea of "listening" closely to mathematics to learn more about our world is a recurring theme of this book. We've already exposed two functions — the trigonometric f (t) and the rational r(V) — that follow you around everywhere you go. Let me wake up so that I can reveal even more hidden mathematics.
The Logarithms Hidden in the Air
It's now seven in the morning and my alarm clock finally goes off. It's set to play the radio when the alarm goes off, rather than that startling "BUZZ! BUZZ!" I can't stand. Back when I lived in Ann Arbor I would wake up to 91.7 FM, the local National Public Radio (NPR) station. But now that I live in Boston, 91.7 FM is pure static. What happened to the Ann Arbor station? Is my radio broken? Where's my NPR?!
The local NPR station for Boston is WBUR-FM, at 90.9 FM on the radio dial. Since I'm now far away from Ann Arbor my radio can't pick up the old 91.7 NPR station. We all intuitively know this; just drive far enough away from your home town and all your favorite radio stations will fade away. But wait a second, that's the same relationship that we saw in Figure 1.2 with the function r(V). Could there be another rational function lurking somewhere in the air waves?
(Continues...)
Excerpted from Everyday Calculus by OSCAR E. FERNANDEZ. Copyright © 2014 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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