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Math with Bad Drawings
Illuminating the Ideas That Shape Our Reality
By Ben Orlin
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In Math With Bad Drawings, Ben Orlin reveals to us what math actually is; its myriad uses, its strange symbols, and the wild leaps of logic and faith that define the usually impenetrable work of the mathematician.
Truth and knowledge come in multiple forms: colorful drawings, encouraging jokes, and the stories and insights of an empathetic teacher who believes that math should belong to everyone. Orlin shows us how to think like a mathematician by teaching us a brand-new game of tic-tac-toe, how to understand an economic crises by rolling a pair of dice, and the mathematical headache that ensues when attempting to build a spherical Death Star.
Every discussion in the book is illustrated with Orlin’s trademark “bad drawings,” which convey his message and insights with perfect pitch and clarity. With 24 chapters covering topics from the electoral college to human genetics to the reasons not to trust statistics, Math with Bad Drawings is a life-changing book for the math-estranged and math-enamored alike.
This is a book about math. That was the plan, anyway.
Somewhere, it took an unexpected left turn. Before long, I found myself without cell phone reception, navigating a series of underground tunnels. When I emerged into the light, the book was still about math, but it was about lots of other things, too: Why people buy lottery tickets. How a children’s book author swung a Swedish election. What defines a “Gothic” novel. Whether building a giant spherical space station was really the wisest move for Darth Vader and the Empire.
That’s math for you. It connects far-flung corners of life, like a secret system of Mario tubes.
If this description rings false to you, it’s perhaps because you’ve been to a place called “school.” If so, you have my condolences.
When I graduated from college in 2009, I thought I knew why mathematics was unpopular: It was, on the whole, badly taught. Math class took a beautiful, imaginative, logical art, shredded it into a bowl of confetti, and assigned students the impossible, mind-numbing task of piecing the original back together. No wonder they groaned. No wonder they failed. No wonder adults look back on their math experiences with a shudder and a gag reflex. The solution struck me as obvious: Math required better explanations, better explainers.
Then I became a teacher. Undertrained and oozing hubris, I needed a brutal first year in the classroom to teach me that, whatever I knew about math, I didn’t yet understand math education, or what the subject meant to my students.
One day that September, I found myself leading an awkward impromptu discussion of why we study geometry. Did grown-ups write two-column proofs? Did engineers work in “no calculator” environments? Did personal finance demand heavy use of the rhombus? None of the standard justifications rang true. In the end, my 9th graders settled on “We study math to prove to colleges and employers that we are smart and hardworking.” In this formulation, the math itself didn’t matter. Doing math was a weightlifting stunt, a pointless show of intellectual strength, a protracted exercise in résumé building. This depressed me, but it satisfied them, which depressed me even more.
The students weren’t wrong. Education has a competitive zero-sum aspect, in which math functions as a sorting mechanism. What they were missing—what I was failing to show them—was math’s deeper function.
Why does mathematics underlie everything in life? How does it manage to link disconnected realms—coins and genes, dice and stocks, books and baseball? The reason is that mathematics is a system of thinking, and every problem in the world benefits from thinking.
Since 2013, I’ve been writing about math and education—sometimes for publications like Slate, the Atlantic, and the Los Angeles Times, but mostly for my own blog, Math with Bad Drawings. People still ask me why I do the bad drawings. I find this odd. No one ever wonders why I choose to cook mediocre food, as if I’ve got a killer chicken l’orange that, on principle, I decline to serve. It’s the same with my art. Math with Bad Drawings is a less pathetic title than Math with the Best Drawings I Can Manage; Honestly, Guys, I’m Trying, but in my case they are equivalent.
I suppose my path began one day when I drew a dog on the board to illustrate a problem, and got the biggest laugh of my career. The students found my ineptitude shocking, hilarious, and, in the end, kind of charming. Math too often feels like a high-stakes competition; to see the alleged expert reveal himself as the worst in the room at something—anything—that can humanize him and, perhaps, by extension, the subject. My own humiliation has since become a key element in my pedagogy; you won’t find that in any teacher training program, but, hey, it works.
Lots of days in the classroom, I strike out. Math feels to my students like a musty basement where meaningless symbols shuffle back and forth. The kids shrug, learn the choreography, and dance the tuneless dance.
But on livelier days, they see distant points of light and realize that the basement is a secret tunnel, linking everything they know to just about everything else. The students grapple and innovate, drawing connections, taking leaps, building that elusive virtue called “understanding.”
Unlike in the classroom, this book will sidestep the technical details.You’ll find few equations on these pages, and the spookiest ones are decorative anyway. (The hard core can seek elaboration in the endnotes.) Instead, I want to focus on what I see as the true heart of mathematics: the concepts. Each section of this book will tour a variety of landscapes, all sharing the underground network of a single big idea: How the rules of geometry constrain our design choices. How the methods of probability tap the liquor of eternity. How tiny increments yield quantum jumps. How statistics make legible the mad sprawl of reality.
Writing this book has brought me to places I didn’t expect. I hope that reading it does the same for you.
—BEN ORLIN, October 2017
HOW TO THINK LIKE A MATHEMATICIAN
To be honest, mathematicians don’t do much. They drink coffee, frown at chalkboards. Drink tea, frown at students’ exams. Drink beer, frown at proofs they wrote last year and can’t for the life of them understand anymore.
It’s a life of drinking, frowning, and, most of all, thinking.
You see, there are no physical objects in math: no chemicals to titrate, no particles to accelerate, no financial markets to destroy. Rather, the verbs of the mathematician all boil down to actions of thought. When we calculate, we turn one abstraction into another. When we give proofs, we build logical bridges between related ideas. When we write algorithms or computer programs, we enlist an electronic brain to think the thoughts that our meat brains are too slow or too busy to think for themselves.
Every year that I spend in the company of mathematics, I learn new styles of thought, new ways to use that nifty all-purpose tool inside the skull: How to master a game by fussing with its rules. How to save thoughts for later, by recording them in loopy Greek symbols. How to learn from my errors as if they were trusted professors. And how to stay resilient when the dragon of confusion comes nibbling at my toes.
In all these ways, mathematics is an action of the mind.
What about math’s vaunted “real-world” usefulness? How can spaceships and smartphones and godforsaken “targeted ads” emerge from this fantasy skyline of pure thought? Ah, patience, my friend. All that comes later. We’ve got to begin where all mathematics begins: That is to say, with a game…
WHAT IS MATHEMATICS?
Once at a picnic in Berkeley, I saw a group of mathematicians abandon their Frisbees to crowd around the last game I would have expected: tic-tac-toe.
As you may have discovered yourself, tic-tac-toe is terminally dull. (That’s a medical term.) Because there are so few possible moves, experienced players soon memorize the optimal strategy. Here’s how all my games play out:
If both players have mastered the rules, every game will end in a draw, forever and ever. The result is rote gameplay, with no room for creativity.
But at that picnic in Berkeley, the mathematicians weren’t playing ordinary tic-tac-toe. Their board looked like this, with each square of the original nine turned into a mini-board of its own:
As I watched, the basic rules became clear:
But it took a while longer for the most important rule of the game to emerge:
You don’t get to pick which of the nine mini-boards to play on. Instead, that’s determined by your opponent’s previous move. Whichever square she picks on the mini-board, you must play next in the corresponding square on the big board.
(And whichever square you pick on that mini-board will determine which mini-board she plays on next.)
This lends the game its strategic element. You can’t just focus on each mini-board in turn. You’ve got to consider where your move will send your opponent, and where her next move will send you—and so on, and so on.
(There’s only one exception to this rule: If your opponent sends you to a mini-board that’s already been won, then congratulations—you can go anywhere you like, on any of the other mini-boards.)
The resulting scenarios look bizarre, with players missing easy two-and three-in-a-rows. It’s like watching a basketball star bypass an open layup and fling the ball into the crowd. But there’s a method to the madness. They’re thinking ahead, wary of setting up their foe on prime real estate. A clever attack on a mini-board leaves you vulnerable on the big board, and vice versa—it’s a tension woven into the game’s fabric.
From time to time, I play Ultimate Tic-Tac-Toe with my students; they enjoy the strategy, the chance to defeat a teacher, and, most of all, the absence of trigonometric functions. But every so often, one of them will ask a sheepish and natural question: “So, I like the game,” they’ll say, “but what does any of this have to do with math?”
I know how the world sees my chosen occupation: a dreary tyranny of inflexible rules and formulaic procedures, no more tension-filled than, say, insurance enrollment, or filling out your taxes. Here’s the sort of tedious task we associate with mathematics:
This problem can hold your attention for a few minutes, I guess, but before long the actual concepts slip from mind. “Perimeter” no longer means the length of a stroll around the rectangle’s border; it’s just “the two numbers, doubled and added.” “Area” doesn’t refer to the number of one-by-one squares it takes to cover the rectangle; it’s just “the two numbers, multiplied.” As if competing in a game of garden-variety tic-tac-toe, you fall into mindless computation. There’s no creativity, no challenge.
But, as with the revved-up game of Ultimate Tic-Tac-Toe, math has far greater potential than this busywork suggests. Math can be bold and exploratory, demanding a balance of patience and risk. Just trade in the rote problem above for one like this:
This problem embodies a natural tension, pitting two notions of size (area and perimeter) against one another. More than just applying a formula, solving it requires deeper insight into the nature of rectangles. (See the endnotes for spoilers.)
Or, how about this:
A little spicy, right?
In two quick steps, we’ve traveled from sleepwalking drudgery to a rather formidable little puzzle, one that stumped a whole school’s worth of bright-eyed 6th graders when I tossed it as a bonus on their final exam. (Again, see the endnotes for solutions.)
As this progression suggests, creativity requires freedom, but freedom alone is not enough. The pseudopuzzle “Draw two rectangles” provides loads of freedom but no more spark than a damp match. To spur real creativity, a puzzle needs constraints.
Take Ultimate Tic-Tac-Toe. Each turn, you’ve got only a few moves to choose from—perhaps three or four. That’s enough to get your imagination flowing, but not so many that you’re left drowning in a sea of incalculable possibilities. The game supplies just enough rules, just enough constraints, to spur our ingenuity.
That pretty well summarizes the pleasure of mathematics: creativity born from constraints. If regular tic-tac-toe is math as people know it, then Ultimate Tic-Tac-Toe is math as it ought to be.
You can make the case that all creative endeavors are about pushing against constraints. In the words of physicist Richard Feynman, “Creativity is imagination in a straitjacket.” Take the sonnet, whose tight formal restrictions—Follow this rhythm! Adhere to this length! Make sure these words rhyme! Okay… now express your love, lil’ Shakespeare!—don’t undercut the artistry but heighten it. Or look at sports. Humans strain to achieve goals (kick the ball in the net) while obeying rigid limitations (don’t use your hands). In the process, they create bicycle kicks and diving headers. If you ditch the rulebook, you lose the grace. Even the wacky, avant-garde, convention-defying arts—experimental film, expressionist painting, professional wrestling—draw their power from playing against the limitations of the chosen medium.
Creativity is what happens when a mind encounters an obstacle. It’s the human process of finding a way through, over, around, or beneath. No obstacle, no creativity.
But mathematics takes this concept one step further. In math, we don’t just follow rules. We invent them. We tweak them. We propose a possible constraint, play out its logical consequences, and then, if that way leads to oblivion—or worse, to boredom—we seek a new and more fruitful path.
For example, what happens if I challenge one little assumption about parallel lines?
Euclid laid out this rule regarding parallel lines in 300 BCE; he took it for granted, calling it a fundamental assumption (a “postulate”). This struck his successors as a bit funny. Do you really have to assume it? Shouldn’t it be provable? For two millennia, scholars poked and prodded at the rule, like a piece of food caught between their teeth. At last they realized: Oh! It is an assumption. You can assume otherwise. And if you do, traditional geometry collapses, revealing strange alternative geometries, where the words “parallel” and “line” come to mean something else entirely.
New rule, new game.
As it turns out, the same holds true of Ultimate Tic-Tac-Toe. Soon after I began sharing the game around, I learned that there’s a single technicality upon which everything hinges. It comes down to a question I confined to a parenthetical earlier: What happens if my opponent sends me to a mini-board that’s already been won?
These days, my answer is the one I gave above: Since that mini-board is already “closed,” you can go wherever you want.
But originally, I had a different answer: As long as there’s an empty space on that mini-board, you have to go there—even though it’s a wasted move.
This sounds minor—just a single thread in the tapestry of the game. But watch how it all unravels when we give this thread a pull.
I illustrated the nature of the original rule with an opening strategy that I dubbed (in a triumph of modesty) “The Orlin Gambit.” It goes like this:
When all is said and done, X has sacrificed the center mini-board in exchange for superior position on the other eight. I considered this stratagem pretty clever until readers pointed out its profound uncleverness. The Orlin Gambit wasn’t just a minor advantage. It could be extended into a guaranteed winning strategy. Rather than sacrifice one mini-board, you can sacrifice two, giving you two-in-a-rows on each of the other seven. From there, you can ensure victory in just a handful of moves.
Embarrassed, I updated my explanation to give the rule in its current version—a small but crucial tweak that restored Ultimate Tic-Tac-Toe to health.
New rule, new game.
This is exactly how mathematics proceeds. You throw down some rules and begin to play. When the game grows stale, you change it. You pose new constraints. You relax old ones. Each tweak creates new puzzles, fresh challenges. Most mathematics is less about solving someone else’s riddles than about devising your own, exploring which constraints produce interesting games and which produce drab ones. Eventually, this process of rule tweaking, of moving from game to game, comes to feel like a grand never-ending game in itself.
Mathematics is the logic game of inventing logic games.
The history of mathematics is the unfolding of this story, time and time again. Logic games are invented, solved, and reinvented. For example: what happens if I tweak this simple equation, changing the exponents from 2 into another number, like 3, or 5, or 797?
Oh! I’ve turned an ancient and elementary formula, easily satisfied by plugging in whole numbers (such as 3, 4, and 5), into perhaps the most vexing equation that humans have ever encountered: Fermat’s last theorem. It tormented scholars for 350 years, until the 1990s, when a clever Brit locked himself in an attic and emerged nearly a decade later, blinking at the sunlight, with a proof showing that whole number solutions can never work.
Or, what happens if I take two variables—say, x and y—and create a grid that lets me see how they relate?
Oh! I’ve invented graphing, and thereby revolutionized the visualization of mathematical ideas, because my name is Descartes and that’s why they pay me the big bucks.
Or, consider that squaring a number always gives a positive answer. Well, what if we invented an exception: a number that, when squared, is negative? What happens then?
Oh! We’ve discovered imaginary numbers, thereby enabling the exploration of electromagnetism and unlocking a mathematical truth called “the fundamental theorem of algebra,” all of which will sound pretty good when we add it to our résumés.
In each of these cases, mathematicians at first underestimated the transformative power of the rule change. In the first case, Fermat thought the theorem would yield to a simple proof; as centuries of his frustrated successors will attest, it didn’t. In the second case, Descartes’s graphing idea (now called “the Cartesian plane” in his honor) was originally confined to the appendix of a philosophical text; it was often dropped from later printings. And in the third case, imaginary numbers faced centuries of snubs and insults (“as subtle as they are useless,” said the great Italian mathematician Cardano) before being embraced as true and useful numbers. Even their name is pejorative, originating from a dis first uttered by none other than Descartes.
It’s easy to underestimate new ideas when they emerge not from somber reflection but from a kind of play. Who would guess that a little tweak to the rules (new exponent; new visualization; new number) could transform a game beyond imagination and recognition?
I don’t think the mathematicians at that picnic had this in mind as they huddled over a game of Ultimate Tic-Tac-Toe. But they didn’t need to. Whether we’re cognizant of it or not, the logic game of inventing logic games exerts a pull on us all.
WHAT DOES MATH LOOK LIKE TO STUDENTS?
Alas, this will be a short, bleak chapter. I’d apologize for that, but I’ll be too busy apologizing for other things, like the often soul-pureeing experience of math education.
You know what I’m talking about. To many students, “doing math” means executing the prescribed sequence of pencil maneuvers. Mathematical symbols don’t symbolize; they just dance across the page in baffling choreography. Math is a tale told by an abacus, full of “sin” and “θ,” signifying nothing.
Allow me to offer two brief apologies:
First, I apologize to my own students for any times I’ve made you feel this way about math. I’ve tried not to; then again, I’ve also tried to stay on top of email, cut back on ice cream, and avoid four-month lapses between haircuts. Please enter my plea as “merely human.”
Second, I apologize to mathematics, for any violence you have suffered at my hands. In my defense, you’re an intangible tower of quantitative concepts bound together by abstract logic, so I doubt I’ve left any lasting scars. But I’m not too proud to say I’m sorry.
That’s it for this chapter. I promise the next one will have more explosions, like any good sequel should.
WHAT DOES MATH LOOK LIKE TO MATHEMATICIANS?
It’s very simple. Math looks like language.
A funny language, I’ll admit. It’s dense, terse, and painstaking to read. While I zip through five chapters of a Twilight novel, you might not even turn the page in your math textbook. This language is well suited to telling certain stories (e.g., the relations between curves and equations), and ill suited to others (e.g., the relations between girls and vampires). As such, it’s got a peculiar lexicon, full of words that no other tongue includes. For example, even if I could translate into plain English, it wouldn’t make sense to someone unfamiliar with Fourier analysis, any more than Twilight would make sense to someone unfamiliar with teenage hormones.
But math is an ordinary language in at least one way. To achieve comprehension, mathematicians employ strategies familiar to most readers. They form mental images. They paraphrase in their heads. They skim past distracting technicalities. They draw connections between what they’re reading and what they already know. And—strange as it may seem—they engage their emotions, finding pleasure, humor, and squeamish discomfort in their reading material.
Now, this brief chapter can’t teach fluent math any more than it could teach fluent Russian. And just as literary scholars might debate a couplet by Gerard Manley Hopkins or the ambiguous phrasing of an email, so mathematicians will disagree on specifics. Each brings a unique perspective, shaped by a lifetime of experience and associations.
That said, I hope to offer a few nonliteral translations, a few glimpses into the strategies by which a mathematician might read some actual mathematics. Consider it Squiggle Theory 101.
A common question I get from students: “Does it matter whether I multiply by 11 or 13 first?” The answer (“no”) is less interesting than what the question reveals: that in my students’ eyes, multiplication is an action, a thing you do. So one of the hardest lessons I teach them is this: Sometimes, don’t.
You don’t have to read 7 × 11 × 13 as a command. You can just call it a number and leave it be.
Every number has lots of aliases and stage names. You could also call this number 1002 – 1, or 499 × 2 + 3, or 5005/5, or Jessica, the Number That Will Save Planet Earth, or plain old 1001. But if 1001 is how this number is known to its friends, then 7 × 11 × 13 isn’t some quirky and arbitrary moniker. Rather, it’s the official name you’d find on the birth certificate.
7 × 11 × 13 is the prime factorization, and it speaks volumes.
Some key background knowledge: Addition is kind of boring. To wit, writing 1001 as the sum of two numbers is a truly dull pastime: you can do it as 1000 + 1, or 999 + 2, or 998 + 3, or 997 + 4… and so on, and so on, until you slip into a boredom coma. These decompositions don’t tell us anything special about 1001, because all numbers can be broken up in pretty much the same way. (For example, 18 can be written as 17 + 1, or 16 + 2, or 15 + 3…) Visually, this is like breaking a number up into two piles. No offense, but piles are dumb.
Multiplication: now that’s where the party’s at. And to join the festivities, you need to deploy our first math-reading strategy: forming mental images.
As the picture on the previous page shows, multiplication is all about grids and arrays. 1001 can be seen as a giant structure of blocks, 7 by 11 by 13. But that’s just getting started.
You can visualize this as 13 layers of 77 each. Or, if you tilt your head sideways, it’s 11 layers of 91 each. Or, tilting your head a different sideways, seven layers of 143 each. All of these ways to decompose 1001 are immediately evident from the prime factorization… but virtually impossible to discern from the name 1001 without laborious guesswork.
The prime factorization is the DNA of a number. From it, you can read all the factors and factorizations, the numbers that divide our original and the numbers that don’t. If math is cooking class, then 7 × 11 × 13 isn’t the pancake recipe. It’s the pancake itself.
To casual fans, π is a mystic rune, a symbol of mathematical sorcery. They ponder its irrationality, memorize thousands of its digits, and commemorate Pi Day on March 14 by combining the most glorious of mankind’s arts (dessert pies) with the least glorious (puns). To the general public, π is an object of obsession, awe, even something approaching worship.
And to mathematicians, it’s roughly 3.
That infinite spool of decimal places that so captivates laypeople? Well, mathematicians aren’t that bothered. They know math is about more than precision; it’s about quick estimates and smart approximations. When building intuition, it helps to streamline and simplify. Intelligent imprecision is our next crucial math-reading strategy.
Take the formula A = πr2, which many students have heard so often the mere phrase “area of a circle” triggers them to scream, “Pi r
- "The book is a more polished, extensive discussion of the concepts that pepper Orlin's blog, featuring his trademark caustic wit, a refreshingly breezy conversational tone, and of course, lots and lots of bad drawings. It's a great, entertaining read for neophytes and math fans alike because Orlin excels at finding novel ways to connect the math to real-world problems-or in the case of the Death Star, to problems in fictional worlds."—Ars Technica
- "Ben Orlin is terribly bad at drawing. Luckily he's also fantastically clever and charming. His talents have added up to the most glorious, warm, and witty illustrated guide to the irresistible appeal of mathematics."—Hannah Fry, mathematician, University College London and BBC presenter
- "Brilliant, wide ranging, and irreverent, Math with Bad Drawings adds ha ha to aha. It'll make you smile - plus it might just make you smarter and wiser."—Steven Strogatz, Professor of Mathematics, Cornell University, author of The Joy of x
- "MATH WITH BAD DRAWINGS is a gloriously goofy word-number-and-cartoon fest that drags math out of the classroom and into the sunlight where it belongs. Great for your friend who thinks they hate math - actually, great for everyone!"—Jordan Ellenberg, author of How Not To Be Wrong
- "Ben Orlin has hit the seemingly unattainable sweet spot. He has written a book that is funny and serious, that is entertaining and informative, and that would interest a reader with or without a background in mathematics. Math with Bad Drawings would be a wonderful book for people who love math, used to love math, want to love math, want to know what math is good for, or just want to know what math really is."—Math Horizons
- "Orlin's ability to masterfully convey interesting and complex mathematical ideas through the whimsy of drawings (that, contrary to the suggestion of the title, are actually not that bad) is unparalleled. This is a great work showing the beauty of mathematics as it relates to our world. This is a must read for anyone who ever thought math isn't fun, or doesn't apply to the world we live in!"—John Urschel, mathematician named to Forbes® "30 Under 30" list of outstanding young scientists and former NFL player
- "Illuminating, inspiring, and hilarious, Math with Bad Drawings is everything you wanted to learn in class but never thought to ask. A joyful romp through mathematics and all its wisdom."—Bianca Bosker, author of the New York Times-bestselling Cork Dork
- On Sale
- Sep 18, 2018
- Page Count
- 376 pages
- Black Dog & Leventhal