MATH GIRLS by Hiroshi Yuki. Copyright Š 2007 Hiroshi Yuki.All rights reserved. Originally published in Japan by
Softbank Creative Corp., Tokyo.English translation Š 2011 Bento Books, Inc.
Translated by Tony Gonzalez with Joseph Reeder.Additional editing by Alexander O. Smith and Brian Chandler.
No portion of this book in excess of fair use considerations may be reproducedor transmitted in any form or by any means without written permission from
the copyright holders.
Published by
Bento Books, Inc.217 Tierra Grande CourtAustin, TX 78732-2458
USA
bentobooks.com
ISBN 978-0-9839513-1-5 (hardcover)ISBN 978-0-9839513-0-8 (trade paperback)
Library of Congress Control Number: 2011937467
Printed in the U.S.A.First edition, November 2011
To My Readers
This book contains math problems covering a wide range of difficulty.Some will be approachable by middle school students, while othersmay prove challenging even at the college level.
The characters often use words and diagrams to express theirthoughts, but in some places equations tell the tale. If you findyourself faced with math you donât understand, feel free to skip overit and continue on with the story. Tetra will be there to keep youcompany.
If you have some skill at mathematics, then please follow not onlythe story, but also the math. You might be surprised at what youdiscover.
âHiroshi Yuki
Contents
To My Readers 5
Prologue 1
1 Sequences and Patterns 31.1 Beneath a Cherry Tree 31.2 Outlier 61.3 Beginning the Pattern 8
2 Equations and Love Letters 112.1 Two Plus One Equals Three 112.2 Some Mental Arithmetic 122.3 The Letter 122.4 Ambush 132.5 Tetra 14
2.5.1 Defining Prime Numbers 152.5.2 Defining Absolute Values 18
2.6 Beneath an Umbrella 202.7 Burning the Midnight Oil 212.8 Mirukaâs Answer 242.9 Math by the Letters 26
2.9.1 Equations and Identities 262.9.2 The Forms of Sums and Products 30
2.10 Whoâs Behind the Math? 33
3 Shapes in the Shadows 353.1 Rotations 353.2 Projections 373.3 The Omega Waltz 43
4 Generating Functions 494.1 In the Library 49
4.1.1 Searching for Patterns 504.1.2 Sums of Geometric Progressions 514.1.3 Infinite Geometric Series 524.1.4 Creating Generating Functions 53
4.2 Capturing the Fibonacci Sequence 544.2.1 The Fibonacci Sequence 544.2.2 A Generating Function 564.2.3 Searching for a Closed Expression 574.2.4 On to the Infinite Series 594.2.5 The Solution 60
4.3 Reflections 64
5 Arithmetic and Geometric Means 675.1 The Second Note 675.2 Questions 685.3 Inequalities 705.4 Forging Ahead 775.5 A Glimpse Behind the Curtain 80
6 The Discrete World 856.1 Derivatives 856.2 Differences 896.3 Bringing Them Together 91
6.3.1 Linear Functions 916.3.2 Quadratic Functions 926.3.3 Cubic Functions 946.3.4 Exponential Functions 95
6.4 Traveling Between Two Worlds 97
7 Generalizations 1017.1 Back in the Library 101
7.1.1 With Miruka 1017.1.2 With Tetra 1047.1.3 Recurrence 105
7.2 Heading Home 1087.3 The Binomial Theorem 1097.4 Convolution 1167.5 Mirukaâs Solution 122
7.5.1 A Different Approach 1227.5.2 Confronting Generating Functions 1277.5.3 The Scarf 1307.5.4 The Final Hurdle 1307.5.5 The Leap 1337.5.6 A Circle with Zero Radius 137
8 Harmonic Numbers 1398.1 Treasure Hunt 139
8.1.1 Tetra 1398.1.2 Miruka 140
8.2 For Every Library There Exists a Dialogue. . . 1418.2.1 Partial Sums and Infinite Series 1418.2.2 Starting with the Obvious 1448.2.3 Propositions 1458.2.4 For All. . . 1488.2.5 There Exists. . . 150
8.3 An Infinite Upward Spiral 1538.4 An Ill-Tempered Zeta 1558.5 Overestimating Infinity 1568.6 Harmony in the Classrooom 1628.7 Two Worlds, Four Operators 1658.8 Keys to the Known, Doors to the Unknown 1728.9 If There Were Only Two Primes 173
8.9.1 Converging Geometric Progressions 1758.9.2 The Uniqueness of Prime Factorizations 1768.9.3 The Infinitude of Primes 176
8.10 Under the Stars 180
9 Taylor Series and the Basel Problem 185
9.1 Two Cards 1859.1.1 One for Tetra 1859.1.2 Polynomials of Infinite Degree 187
9.2 Learning on Your Own 1909.3 Getting Closer 191
9.3.1 Differentiation Rules 1919.3.2 Looping Derivatives 1939.3.3 The Taylor Series for sin x 1969.3.4 Taking Equations to the Limit 200
9.4 The Problem Within 2029.5 The Fundamental Theorem of Algebra 2059.6 Tetraâs Attempt 210
9.6.1 Trial and Error 2109.6.2 A Solution Emerges 2129.6.3 Defying Infinity 217
10 Partitions 22110.1 Partition Numbers 221
10.1.1 Ways to Pay 22110.1.2 Some Concrete Examples 222
10.2 The Fibonacci Sign 22810.3 Confessions 23010.4 Making Selections 23210.5 Three Solutions 235
10.5.1 Using Generating Functions 23610.5.2 Using an Upper Limit 24310.5.3 Using Brute Force 248
10.6 Lost and Found 25110.7 In Search of a Better Upper Bound 252
10.7.1 Starting with Generating Functions 25210.7.2 Changing Sums into products 25410.7.3 Taylor Series 25510.7.4 Harmonic Numbers 25910.7.5 Coming Home 26110.7.6 Tetraâs Notes 263
10.8 Goodbyes 264
Recommended Reading 273
Prologue
It is not enough to memorize.One must also remember.
Hideo Kobayashi
We met in high school.Iâll never forget them: The brilliant Miruka, forever stunning me
with her elegant solutions. The vivacious Tetra, with her earneststream of questions. It was mathematics that led me to them.
Mathematics is timeless.When I think back on those days, equations seem to pop into my
head and fresh ideas flow like a spring. Equations donât fade with thepassage of time. Even today they reveal to us the insights of giants:Euclid, Gauss, Euler.
Mathematics is ageless.Through equations, I can share the experiences of mathematicians
from ages past. They might have worked their proofs hundreds ofyears ago, but when I trace the path of their logic, the thrill thatfills me is mine.
Mathematics leads me into deep forests and reveals hidden trea-sures. Itâs a competition of intellect, a thrilling game where findingthe most powerful solution to a problem is the goal. It is drama. Itis battle.
2 Math Girls
But math was too hefty a weapon for me in those days. I had onlyjust gotten my hand around its hilt, and I wielded it clumsilyâlikeI handled life, and my feelings for Miruka and Tetra.
It is not enough to memorize. I must also remember.It all started my first year of high schoolâ
Chapter 1
Sequences and Patterns
One, two, three. One three.One, two, three. Two threes.
Yumiko OshimaThe Star of Cottonland
1.1 Beneath a Cherry Tree
The entrance ceremony on my first day of high school was held ona fine spring day in April. The principal gave a speech, filled withthe usual things people are supposed to say at times like these. Iremember maybe half.
...unfolding blossoms that you are. . . on this occasion of anew beginning. . . the proud history of this school. . . excel in yourstudies, as you excel in sports. . . learn while you are young. . .
I pretended to adjust my glasses to hide a yawn.On my way back to class after the ceremony, I slipped away
behind the school, and found myself strolling down a row of cherrytrees.
4 Math Girls
Iâm 15 now, I thought. 15, 16, 17, then graduation at 18. Onefourth power. One prime.
15 = 3 ¡ 516 = 2 ¡ 2 ¡ 2 ¡ 2 = 24 a fourth power
17 = 17 a prime
18 = 2 ¡ 3 ¡ 3 = 2 ¡ 32
Back in class everyone would be introducing themselves. I hatedintroductions. What was I supposed to say?
âHi. My hobbies include math and, uh. . . Well, mostly just math.Nice to meet you.â
Please.I had resigned myself to the idea that high school would turn out
much as middle school had. Three years of patiently sitting throughclasses. Three years with my equations in a quiet library.
I found myself by a particularly large cherry tree. A girl stood infront of it, admiring the blossoms. Another new student, I assumed,skipping class just like me. My eyes followed her gaze. Above us, thesky was colored in blurred pastels. The wind picked up, envelopingher in a cloud of cherry petals.
She looked at me. She was tall, with long black hair brushed backfrom metal frame glasses. Her lips were drawn.
âOne, one, two, three,â she said in precise, clipped tones.
1 1 2 3
She stopped and pointed in my direction, obviously waiting forthe next number. I glanced around.
âWho, me?âShe nodded silently, her index finger still extended. I was taken
off guard by the pop quiz, but the answer was easy enough.âThe next number is 5. Then 8, and after that 13, and then 21.
Next is, uh. . . âShe raised her palm to stop me and then issued another challenge.
âOne, four, twenty-seven, two hundred fifty-six.â
Sequences and Patterns 5
1 4 27 256
She pointed at me again. I immediately noticed the pattern.âI guess the next number is 3125. After that. . . uh, I canât do that
in my head.âHer expression darkened. â1â4â27â256â3125â46656,â she
said, her voice clear and confident. The girl closed her eyes andinclined her head toward the cherry blossoms above us. Her fingertwitched, tapping the air.
She was by far the strangest girl Iâd ever met, and I couldnât takemy eyes off her.
Her gaze met mine. âSix, fifteen, thirty-five, seventy-seven.â
6 15 35 77
Four numbers, again, but the pattern wasnât obvious. My headwent into overdrive. 6 and 15 are multiples of 3, but 35 isnât. 35and 77 are multiples of 7. . . I wished that I had some paper towrite on.
I glanced at the girl. She was still standing at attention beneaththe cherry tree. A cherry petal came to rest in her hair, but shedidnât brush it away. She didnât move at all. Her solemn expressionmade our encounter feel all the more like a test.
âGot it.âHer eyes sparkled and she showed a hint of a smile.â6â15â35â77. . . and then 133!âMy voice was louder than Iâd intended.She shook her head, sending the cherry petal fluttering softly to
the ground. âCheck your math,â she sighed, a finger touching theframe of her glasses.
âOh. Oh right. 11Ă 13 is 143, not 133.âShe continued with the next problem. âSix, two, eight, two, ten,
eighteen.â
6 Math Girls
6 2 8 2 10 18
Six numbers this time. I thought for a bit. The 18 at the endreally threw me, since I was expecting a 2. I knew there had to beanother pattern, though, and it hit me when I realized that all thenumbers were even.
âNext is 4â12â10â6.â I frowned. âKind of a trick question,though.â
âYou got it, didnât you?âShe approached, her hand extended for an unexpected handshake.
I took it, still unsure exactly what was going on. Her hand was softand warm in mine.
âMiruka,â she said. âNice to meet you.â
1.2 Outlier
I loved nighttime. Once my family was in bed, my time was my own.I would spread books before me, and explore their worlds. I wouldthink about math, delving deep into its forests. I would discoverfantastic creatures, tranquil lakes, trees that stretched up to the sky.
But that night, I thought about Miruka. I recalled our handshake.The softness of her hand, the way she smelled. She smelled like. . . likea girl. It was clear that she loved math; it was also clear that she wasstrangeâan outlier. Not many people introduced themselves with apop quiz. I wondered if I had passed.
I laid my glasses on my desk and closed my eyes, reflecting onour conversation.
The first problem, 1, 1, 2, 3, 5, 8, 13, ¡ ¡ ¡ , was the Fibonacci se-quence. It starts with 1, 1, and each number after that is the sum ofthe two before it:
1, 1, 1+ 1 = 2, 1+ 2 = 3, 2+ 3 = 5, 3+ 5 = 8, ¡ ¡ ¡
The next problem was 1, 4, 27, 256, 3125, 46656, ¡ ¡ ¡ . That was asequence like this:
Sequences and Patterns 7
11, 22, 33, 44, 55, 66, ¡ ¡ ¡
The general term for this would be nn. I could manage calculating44 and even 55 in my head, but 66? No way.
The problem after that was 6, 15, 35, 77, 143, ¡ ¡ ¡ . I got that bymultiplying:
2à 3, 3à 5, 5à 7, 7à 11, 11à 13, ¡ ¡ ¡
In other words <a prime number> Ă <the next prime number>.I couldnât believe Iâd messed up multiplying 11 Ă 13. Check yourmath, indeed.
The last problem was 6, 2, 8, 2, 10, 18, 4, 12, 10, 6, ¡ ¡ ¡ . In otherwords it was Ď, but with each digit doubled:
Ď = 3.141592653 ¡ ¡ ¡ pi
â 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, ¡ ¡ ¡ digits of Ď
â 6, 2, 8, 2, 10, 18, 4, 12, 10, 6, ¡ ¡ ¡ each digit doubled
What I didnât like about this problem was that you canât solveit unless you can remember the digits of Ď. You need to have thepattern already in your head. It relies on memorization.
Math isnât about dredging up half-remembered formulas. Itâsabout making new discoveries. Sure, there are some things thatrequire rote memorization: the names of people and places, words,the symbols of the elements. But math isnât like that. With a mathproblem, you have a set of rules. You have tools and materials, laidout on the table in front of you. Mathâs not about memorizing, itâsabout thinking. Or at least, thatâs what it is to me.
I noticed something interesting about that last problem. Mirukahad given me six numbers, instead of four like the rest. If she had justsaid, â6, 2, 8, 2,â then the series wouldnât necessarily be a doubling ofthe digits of Ď. Another, simpler pattern might be possible. Even ifshe had said, â6, 2, 8, 2, 10,â I could have answered with a series ofeven numbers separated by the number 2, like this:
6, 2, 8, 2, 10, 2, 12, 2, ¡ ¡ ¡
8 Math Girls
So she knew exactly what she was doing when she made thatseries longer than the others. And it hadnât surprised her when I gotit right. I could still see the smug curl of her mouth.
Miruka.I remembered the way she looked, standing there in the spring
light, her black hair a sharp contrast against the pink petals on thewind, slim fingers moving like a conductorâs. The warmth of herhand. Her fragrance.
That night, I couldnât think of much else.
1.3 Beginning the Pattern
By May, the novelty of being in a new school, going to new classrooms,and meeting new friends had largely faded. The days of same old,same old had begun.
I didnât participate in any after-school activities, preferring insteadto get away from school stuff as soon as I could. I wasnât particularlygood at sports, and just hanging out with friends had never appealedto me. Thatâs not to say I headed straight home. When classes weredone, I often went to the school library to do math, a habit I pickedup in middle school. No clubs for me; just reading, studying, andstaring out the library window.
But my favorite thing to do was tinker with the math I learnedin class. Sometimes I would start with definitions, seeing where theyled me. I jotted down concrete examples, played with variations ontheorems, thought of proofs. I would sit for hours, filling notebookafter notebook.
When youâre doing math, youâre the one holding the pencil, butthat doesnât mean you can write just anything. There are rules. Andwhere there are rules, thereâs a game to playâthe same game playedby all the great mathematicians of old. All you need is some freshpaper and your mind. I was hooked.
I had assumed it was a game I would always play alone, even inhigh school. It turned out I was wrong.
Miruka shared homeroom with me, and she showed up in thelibrary a few days a week. The first time she walked up, I was sittingalone, working on a problem. She took the pencil out of my hand,
Sequences and Patterns 9
and started writing in my notebook. In my notebook! I wasnât sureif I should be offended or impressed. I decided on the latter.
Her math was hard to follow, but interesting. Exciting, even.âWhat was with the sequence quiz the other day?â I asked.âWhat other day?â She looked up, hand paused in mid-calculation.
A pleasant breeze drifted through the open window, carrying theindistinct sounds of baseball practice and the fragrance of sycamoretrees. âIâm drawing a blank here,â she said, tapping a pencilâmypencilâagainst her temple.
âWhen we first met. You know. . . under the cherry tree.ââOh, that. They just popped into my head. So what?ââI donât know. Just wondering.ââYou like quizzes?ââSure, I guess.ââYou guess, huh? Did you know there arenât any right answers to
those kinds of sequence problems?ââWhat do you mean?ââSay the problem was 1, 2, 3, 4, ¡ ¡ ¡ . Whatâs the answer?ââWell, 5 of course.ââNot necessarily. What if the numbers jumped after that, say to
10, 20, 30, 40, and then 100, 200, 300, 400? Itâs still a perfectly finesequence.â
âYeah, but, you canât just give four numbers and then say thenumbers jump. Thereâs no way to know that a 10 will come after the4.â
âWell, how many numbers do you need, then?â she asked, raisingan eyebrow. âIf the sequence goes on forever, at what point can youfigure out the rest?â
âAll right, all right. So thereâs always a chance that the patternwill suddenly change somewhere beyond what youâve seen. Still,saying that a 10 comes after 1, 2, 3, 4 makes for a pretty randomproblem.â
âBut thatâs the way the world works. You never know whatâsgoing to come next. Predictions fail. Check this out.â Miruka startedwriting in my notebook. âCan you give me a general term for thissequence?â
1, 2, 3, 4, 6, 9, 8, 12, 18, 27, ¡ ¡ ¡
10 Math Girls
âHmmm. . . Maybe,â I said.âIf you only saw the 1, 2, 3, 4, then youâd expect the next number
to be 5. But youâd be wrong. Rules wonât always reveal themselvesin a small sample.â
âOkay.ââAnd if you saw 1, 2, 3, 4, 6, 9, youâd expect the sequence to
keep increasing, right? But not here. The number after the 9 is an8. You see a pattern of increasing numbers, but then bam, one goesbackwards. Have you found the pattern yet?â
âWell, except for the 1 theyâre all multiples of 2 or 3. I canât figureout why that one number becomes smaller, though.â
âHereâs a solution to play with,â she said, writing:
2030, 2130, 2031, 2230, 2131, 2032, 2330, 2231, 2132, 2033, ¡ ¡ ¡
âLook at the exponents on the 2s and 3s. That should help yousee it.â
âWell, I know that anything raised to the zero power is 1, so yeah,when I do the calculations I get the right sequence,â I said, writingunderneath:
2030 = 1, 2130 = 2, 2031 = 3, ¡ ¡ ¡
âBut I still donât get it.ââThe exponents arenât enough, huh? How about this?â
2030,︸ ︡︡ ︸sum=0
2130, 2031,︸ ︡︡ ︸sum=1
2230, 2131, 2032,︸ ︡︡ ︸sum of exponents=2
2330, 2231, 2132, 2033,︸ ︡︡ ︸sum of exponents=3
¡ ¡ ¡
âOh, I see it now,â I said.âHey, speaking of multiples of 2 and 3ââ Miruka began, but a
shout from the library entrance interrupted her.âAre you coming, or what?ââOh, right. Practice today,â Miruka said.She returned my pencil and headed towards the girl standing
in the doorway. On her way out, she looked back. âRemind me totell you what the world would be like if there were only two primenumbers.â
Then she was gone and I was alone again.
Chapter 2
Equations and Love Letters
You fill my heart.
Moto HagioRagini
2.1 Two Plus One Equals Three
My second year of high school was pretty much the same as the first,except that the âIâ on my school badge became a âII.â Days flowedin an endless stream, each seeming just like the one that had comebefore, until a cloudy morning in late April.
Less than a month had passed since the start of the new schoolyear. I was walking through the school gates on my way to classwhen an unfamiliar girl called out to me.
âThis is for you,â she said, offering me a white envelope. Confused,I took it. She bowed curtly before scurrying off.
I peeked insideâit was a letter. But there was no time to read it.Stuffing the envelope into a pocket, I ran off to class.
It was the first time a girl had given me a letter since elementaryschool. Iâd caught a bad cold and was absent for a couple of days.One of my classmates came by my house to drop off homework, alongwith a note that said âGet better soon!â
The letter felt heavy in my pocket all during class.Just like Miruka said, you never know whatâs going to come next.
12 Math Girls
2.2 Some Mental Arithmetic
I had just finished eating lunch and was pulling out the girlâs letterwhen Miruka plopped down next to me, nibbling on a candy bar.
âPop quiz,â she announced. â1024. How many divisors?ââI gotta do it in my head?â I asked, cramming the letter back into
my pocket.âYep, and before I count to ten. 1, 2, 3ââI scrambled to think of numbers that would evenly divide 1024.
Definitely 1 and 2, but not 3âthat would leave a remainder. Iwas checking 4 when it struck me that 1024 is 210. I did a quickcalculation.
ââ9, 10. Timeâs up. How many?ââEleven. 1024 has eleven divisors.ââCorrect. How did you figure it out?â Miruka occupied herself by
licking chocolate off her fingers while awaiting my answer.âFrom the prime factorization of 1024, which gives you two to
the tenth power,â I said. âIf you write it out, you get this: â
1024 = 210 = 2à 2à 2à 2à 2à 2à 2à 2à 2à 2︸ ︡︡ ︸ten twos
âA divisor of 1024 has to divide it evenly,â I continued. âThatmeans it has to be in the form 2n, where n is some number from 0to 10. So a divisor of 1024 will be one of these numbers.â
Miruka nodded. âVery good. Okay, next problem. If you addedup all the divisors of 1024, what would the sumââ
âSorry,â I said, abruptly standing, âbut thereâs something I haveto do before class. Iâll see you later.â I turned away from an obviouslydisgruntled Miruka and left the room.
I was already thinking of ways to find the sum of the divisors of1024 as I headed for the roof.
2.3 The Letter
It was gloomy out. The usual lunchtime roof crowd was thin.I took the envelope from my pocket and removed the letter. It was
written on a sheet of white stationery in fountain pen with attractivehandwriting.
Equations and Love Letters 13
Hello.
My name is Tetra. We went to the same juniorhigh. Iâm one year behind you. Iâm writing to askyou for some advice about studying math.
Iâve been having problems with math for years.I heard that the classes get a lot harder in highschool, and Iâm looking for some way to get overmy âmath anxiety.â
I know itâs a lot to ask, but do you think you couldspare the time to talk? Iâll be in the lecture hallafter school.
âTetra
Tetra? As in mono-, di-, tri-?I was surprised to hear that we had gone to the same middle
schoolâI didnât remember her at all. That she was having problemswith math was less surprising; lots of students did, first-year studentsespecially.
I suppose this qualified as a bona fide letter. But somehow, itwasnât as exciting as Iâd hoped my first letter from a girl would be.
I read it four times.
2.4 Ambush
Classes had finished, and I was heading to the lecture hall whenMiruka ambushed me, appearing out of nowhere.
âWhat would the sum be?â she asked.â2047,â I answered without missing a beat.She frowned. âI gave you too much time to think about it.ââI guess. Look, IâmâââHeaded to the library?â Her eyes flashed.âNot today. I have to be somewhere.ââOh, yeah? Iâll give you some homework, then.âShe jotted something down on a piece of paper.
14 Math Girls
Mirukaâs homework
Describe a method for summing the divisors of a givenpositive integer n.
âYou want me to give you a formula in terms of n?â I asked.âDonât strain yourself. Just the steps of a method will do.â
2.5 Tetra
The lecture hall was separate from the main school building, ashort walk across the school courtyard. The room was tiered, withthe podium at the bottom level so students could watch a lecturerperform experimentsâideal for physics and chemistry classes.
I found Tetra standing at the back with a nervous look on herface, a notebook and pencil case clutched tightly to her chest.
âOh, you came. Thank you so much,â she said. âUm, so, I wantedto ask you some questions, but I wasnât sure how, so I asked a friend,and she said that maybe this would be a good place to, uh, meet.â
Tetra and I sat down on a bench in the very back of the hall. Itook the letter she had given me that morning out of my pocket.
âI read your letter, but I have to be honest. I donât rememberyou from junior high.â
âNo, of course you donât. I wouldnât expect you to.ââSo how do you know about me? I didnât exactly stand out back
then.â Itâs hard to stand out when you spend all your free time inthe library.
âActually...you were kinda famous.ââIf you say so...â I held up her letter. âSo, about this. You said
you were having trouble with math?ââThatâs right. See, back in elementary school, I liked math just
fine. Doing the problems, working through thingsâall good. Butwhen I got to middle school, everything changed. I felt like I wasnâtreally getting a lot of what I was doing, you know? My math teachertold me it would only get harder in high school, so Iâd need to work
Equations and Love Letters 15
at it if I wanted to keep up. And I have been. But I want to do morethan regurgitate whatâs in the books. I want to understand.â
âYouâre worried about your grades?âTetra pressed her thumbnail against her lip. âNo, itâs not that.â
Her eyes darted about beneath her bangs. She reminded me of asmall, nervous animalâa kitten, maybe, or a squirrel. âWhen I knowwhatâs going to be on a test, Iâm fine. But when they start gettingcreative, I do worse. A lot worse.â
âYou follow what your teachers go over in class?ââMore or less.ââAnd you can do the homework?ââMostly. But somethingâs not sinking in.ââAll right,â I nodded. âTime to understand.â
2.5.1 Defining Prime Numbers
âLetâs try some specifics,â I began. âDo you know what a primenumber is?â
âI think so,â she said.âProve it. Give me a definition.ââWell, 5 and 7 are prime numbers...ââSure, but those are just examples. I want the definition.ââA prime number is, uh, a number that can only be divided by 1
and itself, right? One of my teachers made me memorize that.ââOkay. If we write that down, we get this: â
A positive integer p is a prime number if it canbe evenly divided only by 1 and p.
I showed my notebook to Tetra. âSo this is your definition?ââYeah, that looks right.ââClose, but not quite.ââBut 5 is a prime number, and it can only be divided evenly by 1
and 5.ââIt works for 5. But if p was 1, according to this definition 1
would be a prime number too, since it would only be divisible by 1and p. But the list of primes starts with 2, like this: â
2, 3, 5, 7, 11, 13, 17, 19, ¡ ¡ ¡
16 Math Girls
âOh right, 1 isnât a prime,â Tetra replied. âI remember learningthat now.â
âSo your definition isnât perfect, but there are a few ways to fixit. You might add a qualifier at the end, like this: â
A positive integer p is a prime number if it canbe evenly divided only by 1 and p. However, 1 isnot a prime.
âAn even better way would be to put the qualifier up front: â
An integer p that is greater than 1 is a primenumber if it can be evenly divided only by 1 andp.
âYou could also give the condition as a mathematical expression: â
An integer p > 1 is a prime number if it can beevenly divided only by 1 and p.
âThose definitions make sense,â Tetra looked up from my notebook.âAnd I know that 1 isnât a prime, but I donât get it. I mean, whosays 1 canât be a prime? What difference does it make? There has tobe a reason.â
âA reason?â I raised an eyebrow.âYeah, isnât there some kind of theory or something behind all
this?âThis was interestingâI didnât meet a lot of people who understood
the importance of being convinced.âThat was a stupid question, wasnât it.â Tetra said.âNo. No, itâs a great question. The primes donât include 1 because
of the uniqueness of prime factorizations.ââThe uniqueness of what? You lost me.ââItâs a property of numbers that says that a positive integer n
will only have one prime factorization. So for 24, the only primefactorization is 2 Ă 2 Ă 2 Ă 3. You could write it 2 Ă 2 Ă 3 Ă 2 or3 Ă 2 Ă 2 Ă 2 if you wanted, but theyâre all considered the same,because the only difference is in the order of the factors. In fact,itâs so important to keep prime factorizations unique that 1 isnâtincluded in the primes, just to protect this uniqueness.â
Equations and Love Letters 17
Now it was Tetraâs turn to raise an eyebrow. âYou mean you candefine something one way just to keep it from breaking somethingelse?â
âKind of a harsh way of putting it, but yeah.â I tapped my pencilon the notebook. âItâs more like this: Mathematicians are always onthe lookout for useful concepts to help build the world of mathematics.When they find something really good, they give it a name. Thatâswhat a definition is. So you could define the primes to include 1 if youwanted to. But thereâs a difference between a possible definition anda useful one. Using your definition of primes that included 1 wouldmean you couldnât use the uniqueness of prime factorizations, so itwouldnât be very useful. That making sense now? The uniqueness, Imean.â
âI think so.ââYou think so, huh? Look, itâs up to you to make sure you
understand something.ââIf I donât know if I understand, how can I make sure?ââWith examples. An example isnât a definition, but coming up
with a good example is a great way to test one out.â I wrote out aproblem in my notebook and handed it to Tetra:
Give an example showing that including 1 as aprime number would invalidate the uniqueness ofprime factorizations.
âOkay,â Tetra replied. âIf 1 was prime, then you could factorize24 in lots of ways. Like this: â
2Ă 2Ă 2Ă 3
1Ă 2Ă 2Ă 2Ă 3
1Ă 1Ă 2Ă 2Ă 2Ă 3...
âPerfect example,â I said. âSee? Examples are the key to under-standing.â A look of relief washed over Tetraâs face. âHowever,â Icontinued, âinstead of saying âlots,â it would be better to say âmultiple,âor âat least two.â Saying it that way is more. . . â
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âPrecise.â Tetra finished.âExactly. âLotsâ isnât very precise, because thereâs no way to know
how many it takes to become âlots.â ââAll these wordsâdefinition, example, prime factorization, unique-
ness. You donât know how much this helps. I didnât realize howimportant language was in math.â
âThatâs a great point. Language is extremely important in math-ematics. Math uses language in a very precise way to make surethereâs no confusion. And equations are the most precise language ofall.â
âEquations are a language?ââNot just any language. The language of mathematicsâand your
next lesson.â I glanced around the lecture hall. âItâll be easier if I usethe blackboard. Câmon.â
I headed down the stairs at the center of the lecture hall. I hadonly taken a few steps when I heard a yelp, and Tetra came crashinginto me, nearly sending both of us sprawling.
âSorry!â Tetra said. âI tripped. On the step. Sorry!ââItâs cool,â I said.This is going to be more work than I thought.
2.5.2 Defining Absolute Values
We reached the blackboard without further incident, and I picked upa piece of chalk before turning to Tetra. âDo you know what absolutevalues are?â
âYeah, I think so. The absolute value of 5 is 5, and the absolutevalue of â5 is 5 too. You just take away any negatives, right?â
âWell, sort of. Letâs try a definition. Tell me if this looks good toyou.â I wrote on the board.
Definition of |x|, the absolute value of x:
|x| =
{x if x > 0
âx if x < 0
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âI remember having trouble with this. If getting the absolutevalue of x means taking away the minuses, why does a minus showup in the definition?â
âWell, âtaking away the minusesâ is kinda vague in a mathematicalsense. Not that I donât know what you mean. Youâre on the righttrack.â
âWould it be better to say change minuses to pluses?ââNo, thatâs still pretty vague. Letâs say we want the absolute value
of âx: â|â x|
âWell,â Tetra said, âwouldnât you take the minus away, leavingx?â
|â x| = x
âNot quite. What if x = â3? How would you calculate that?âTetra picked up her own piece of chalk. âLetâs see. . . â
|â x| = |â (â3)| because x = â3
= |3| because â(â3) = 3
= 3 because |3| = 3
âRight,â I said. âIf you use your definition, |â x| = x, then whenx = â3 you would have to say that | â x| = â3. But in this case|â x| = 3. Or put another way, |â x| = âx.â
Tetra stared at the board. âI see. Since thereâs no sign in frontof the x, I never thought of x being a negative number like â3. Butthatâs the whole point of using a letter like x, isnât itâit lets youdefine something without giving all sorts of specific examples.â
âThatâs right,â I said. âJust saying âtake off the minusesâ isnâtgood enough. You have to be strict with yourselfâyou have to thinkabout all the possibilities. No cutting corners.â
Tetra nodded slowly. âGuess Iâm gonna have to get used to that.âShe slumped into a chair and started fiddling with the corner of hernotebook. âI was just wasting time in junior high,â she said.
I waited quietly for her to continue.âNot that I didnât study. But I wasnât looking at the definitions and
equations the right way. I wasnât strict enough. I was too. . . sloppy.âTetra let out a long sigh.
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âThatâs in the past,â I said.âHuh?ââJust do things right from here on out.âShe sat up, eyes wide.âYouâre right. I canât change the past, but I can change myself.âI smiled. âGlad I could help with the breakthrough, but we should
probably call it a day. Itâs getting dark out. Weâll pick this up nexttime.â
âNext time?ââIâm usually in the library after school. If you have any more
questions, you know where to find me.â
2.6 Beneath an Umbrella
Outside the lecture hall, Tetra stopped and looked up at the sky.Clouds hung low and grey, and it had started to rain.
âFigures,â she grumbled.âNo umbrella?â I asked.âI was running late this morning. Guess I forgot it. I even watched
the weather and everything!â She shrugged. âWell, itâs not raininghard. Iâll be okay if I run.â
âYouâll be soaked by the time you get to the train station. Câmon,weâre going the same way. My umbrellaâs big enough for both of us.â
She smiled. âThanks!âIâd never shared an umbrella with a girl before. It was a bit
awkward at first, but I matched her pace and managed to keep fromtripping or jabbing her with an elbow. We walked slowly throughthe soft spring shower. The road was quiet, the bustle of the townlost in the rain.
It was cool to have someone younger who looked up to me, and Ifound myself surprised by just how much I had enjoyed talking withher. She was so easy to readâwhen she understood something, andwhen she didnât.
âSo howâd you know?â she asked.âKnow what?ââHowâd you know what was giving me trouble?â
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âOh, that. Well, a lot of the stuff we talked about todayâprimenumbers, absolute values, all thatâthatâs stuff I wondered about too,back in the day. When Iâm studying math and I donât understandsomething, it bugs me. Iâll think about it for weeks, Iâll read about it,and then suddenly Iâll just get it. And the feeling when it happens...After youâve felt that a few times you canât help but like math. Andthen you start getting better at it, andâoh, we should turn here.â
âThatâs not the way to the station.ââItâs a lot quicker if you cut through this neighborhood.ââOh...ââYeah, itâs a great shortcut.âTetra slowed her pace to a crawl and I found myself having a
hard time matching her speed the rest of the way.The rain was still falling when we reached the station.âI think Iâm gonna hit the book store,â I said. âGuess Iâll see you
tomorrow.â I started to leave. âOh, here,â I offered her my umbrella.âWhy donât you take this.â
âYouâre going? Oh. Okay, well. . . thanks for all the help. It reallymeans a lot to me.â She bowed deeply.
I nodded and darted for the bookstore.Tetra called after me. âAnd thanks for the umbrella!â
2.7 Burning the Midnight Oil
That night I sat in my room, recalling my conversation with Tetra.She had been so sincere, so enthusiastic. She definitely had potential.I hoped that she would learn to enjoy math.
When I talked to Tetra, I slipped into teacher mode. Talking withMiruka was a very different thing. With Miruka, I had to scrambleto keep up. If anything, she was the one teaching me. I rememberedthe homework she had given me; without a doubt the first time Iâdgotten homework from another student.
Mirukaâs homework
Describe a method for summing the divisors of a givenpositive integer n.
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I knew that I could always solve the problem by finding all thedivisors of n and adding them together, but that felt like a cheat. Iwondered if I couldnât find a better way, and the prime factorizationof n looked like a good place to start.
I thought back on the problem we worked on at lunch, for 1024 =
210. Maybe there was some way to generalize this, like writing n asa power of a prime:
n = pm for prime number p, positive integer m
If n = 1024, then for this equation Iâd have p = 2 and m = 10. IfI wanted to list all the divisors of n like I did for 1024, then it wouldbe something like this:
1,p,p2,p3, ¡ ¡ ¡ ,pm
So for n = pm, I could find the sum of the divisors by addingthem up:
(sum of divisors of n) = 1+ p+ p2 + p3 + ¡ ¡ ¡+ pm
That would be the answer for a positive integer n that could bewritten in the form n = pm, at least. I pushed on to see if I couldnâtgeneralize it for other numbers too. It shouldnât be too hard; all Ineeded to do was generalize prime factor decomposition.
One way to write the prime factor decomposition for a positiveinteger n would be to take p,q, r, ¡ ¡ ¡ as primes and a,b, c, ¡ ¡ ¡ aspositive integers and write it like this:
n = pa à qb à rc à ¡ ¡ ¡ à whoa!
Hang on. This wonât work using just letters. If I went througha,b, c, ¡ ¡ ¡ theyâd eventually run into p,q, r, ¡ ¡ ¡ and that would reallyconfuse things.
I wanted to write an expression that looked something like 23 Ă31 Ă 74 à ¡ ¡ ¡ Ă 133, the product of a bunch of terms in the formprimeinteger. So I could write the primes as p0,p1,p2, ¡ ¡ ¡ ,pm andthe exponents as a0,a1,a2, ¡ ¡ ¡ ,am. Adding all of those subscriptsmight make things look a bit more complicated, but at least it would
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let me generalize. It would also let me use m+1 to mean the numberof prime factors in the prime factor decomposition of n. I startedrewriting.
Now given a positive integer n, I could generalize its prime factordecomposition:
n = pa00 Ă p
a11 Ă p
a22 à ¡ ¡ ¡ à pamm ,
where p0,p1,p2, ¡ ¡ ¡ ,pm are primes and a0,a1,a2, ¡ ¡ ¡ ,am are posi-tive integers. When n was in this form, then a divisor of n wouldlook like this:
pb00 Ă p
b11 Ă p
b22 à ¡ ¡ ¡ à pbmm ,
where b0,b1,b2, ¡ ¡ ¡ ,bm was an integer:
b0 = one of 0, 1, 2, 3, ¡ ¡ ¡ ,a0b1 = one of 0, 1, 2, 3, ¡ ¡ ¡ ,a1b2 = one of 0, 1, 2, 3, ¡ ¡ ¡ ,a2
...
bm = one of 0, 1, 2, 3, ¡ ¡ ¡ ,am
I looked back at what I had written, surprised at how messy itwas to write it out precisely. All I wanted to say was that, to writea divisor, you just leave the prime factors as they are, and movethrough the exponents 0, 1, 2, ¡ ¡ ¡ for each one. But generalizing thistook an alphabet soupâs worth of symbols.
With things generalized to this extent, I figured the rest wouldbe easy. To find the sum of the divisors I just had to add all of theseup.
(sum of divisors of n) = 1+ p0 + p20 + p30 + ¡ ¡ ¡+ p
a00
+ 1+ p1 + p21 + p31 + ¡ ¡ ¡+ p
a11
+ 1+ p2 + p22 + p32 + ¡ ¡ ¡+ p
a22
+ ¡ ¡ ¡+ 1+ pm + p2m + p3m + ¡ ¡ ¡+ pamm
I paused, realizing that what I had written was wrong. This wasnâtthe sum of all the divisors, it was the sum of just those divisors that
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can be described as a power of a prime factor. I had said the form ofa divisor was this:
pb00 Ă p
b11 Ă p
b22 à ¡ ¡ ¡ à pbmm
So I had to find all the combinations of powers of prime factors,multiply those together, and add them up. I found this easier towrite as an equation than to put into words, so thatâs what I did.
My answer to Mirukaâs homework
Write the prime factorization of the positive integer nas follows:
n = p0a0 à p1a1 à p2a2 à ¡ ¡ ¡ à pmam ,
where p0,p1,p2, ¡ ¡ ¡ ,pm are prime numbers anda0,a1,a2, ¡ ¡ ¡ ,am are positive integers. Then the sumof the divisors of n is as follows:
(sum of divisors of n) = (1+ p0 + p20 + p30 + ¡ ¡ ¡+ p
a00 )
à (1+ p1 + p21 + p31 + ¡ ¡ ¡+ p
a11 )
à (1+ p2 + p22 + p32 + ¡ ¡ ¡+ p
a22 )
à ¡ ¡ ¡à (1+ pm + p2m + p3m + ¡ ¡ ¡+ pamm )
I went to bed wondering if there wasnât a cleaner way to writethis. . . and whether I was even right at all.
2.8 Mirukaâs Answer
âWell, itâs right,â Miruka said the next day, âbut itâs kind of a mess.âBlunt as ever, I thought, but what I said was, âIs there some
way to make it simpler?â
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âYes,â she immediately replied. âFirst, you can use this for thelong sums.â Miruka started writing in my notebook as she talked.âAssuming that 1â x 6= 0. . . â
1+ x+ x2 + x3 + ¡ ¡ ¡+ xn =1â xn+1
1â x
âOh, of course. The formula for the sum of a geometric progres-sion.â
Miruka jotted down the proof. Show off.
1â xn+1 = 1â xn+1 equal sides
(1â x)(1+ x+ x2 + x3 + ¡ ¡ ¡+ xn) = 1â xn+1 factor left side
1+ x+ x2 + x3 + ¡ ¡ ¡+ xn =1â xn+1
1â xdivide by 1â x
âYou can use that to turn all your sums of powers into fractions,âshe continued. âYou should also use Î to tidy up the multiplication.âShe wrote the symbol large on the page.
âThatâs a capital Ď, right?ââRight. This one doesnât have anything to do with circles, though.
Î works like ÎŁ does, but for multiplication. ÎŁ is a capital Greek âSâfor âsum,â and Î is a capital Greek âPâ for âproduct.â If you wantedto write out a definition for it...â
Definition of the Sigma operator
mâk=0
f(k) = f(0) + f(1) + f(2) + f(3) + ¡ ¡ ¡+ f(m)
Definition of the Pi operator
mâk=0
f(k) = f(0)à f(1)à f(2)à f(3)à ¡ ¡ ¡ à f(m)
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âNow, check out how much cleaner things are when you use Î .â
Mirukaâs answer
Write the prime factorization of the positive integer nas follows:
n =
mâk=0
pakk ,
where pk is a prime number and ak is a positive integer.Then the sum of the divisors of n is as follows:
(sum of divisors of n) =mâk=0
1â pak+1k
1â pk.
âOkay,â I said. âLots of letters in there, but it is shorter. By theway, you going to the library today?â
âNope.â Miruka shook her head. âIâm off to practice with Ay-Ay.She says she has a new piece ready.â
2.9 Math by the Letters
I was working on some equations when Tetra came up to me with asmile and an open notebook.
âLook what I did! I copied all the definitions out of my mathbook from last year, and made my own example for every one!â
âAll in one night? Thatâs dedication.ââOh no, I love doing stuff like this. And I thought of something
when I was going through my old textbook. Maybe the differencebetween simple and advanced math is that, in advanced math, youuse letters in the equations.â
2.9.1 Equations and Identities
I nodded. âRight, and since you brought up using letters in equations,letâs talk a little bit about equations and identities. Youâve seenequations like this: â
xâ 1 = 0
Equations and Love Letters 27
âSure. x = 1.ââHow about this one?â
2(xâ 1) = 2xâ 2
She frowned. âI think I can solve that if I clean it up a little: â
2(xâ 1) = 2xâ 2 the problem
2xâ 2 = 2xâ 2 expand the left side
2xâ 2xâ 2+ 2 = 0 move the right side terms to the left
0 = 0 simplify the left side
âHuh? I ended up with 0 = 0.ââThatâs right, because 2(xâ 1) = 2xâ 2 isnât an equation, itâs an
identity. See how when you expanded the 2(xâ 1) on the left side,you got 2xâ 2 on the other? Theyâre identical, right? To be precise,this is an identity in x. That means that no matter what x is, thestatement will be true.â
âSo identities are different from equations?ââUh huh. An equation is a statement thatâs true when you replace
the xs with a certain number. An identity is a statement thatâs truewhen you replace the xs with any number. When youâre doing aproblem that deals with an equation, youâre probably trying to findthe value of x that makes the statement true. When youâre doinga problem that deals with an identity, youâre probably trying toshow that any value of x will work. Do that, and youâve proven theidentity.â
âI get it. I guess Iâve always known about identities, I just neverthought of them as being so different.â
âMost people donât. But you use them all the time. Almost allthe formulas you learn outside of math are actually identities.â
âHow can you tell the difference?ââYou have to look at the context and ask yourself what the person
who wrote it intended it to be.ââIâm not sure I follow.â
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âWell, for example, if you want to change the form of a statement,you use identities. Here, look at this: â
(x+ 1)(xâ 1) = (x+ 1) ¡ xâ (x+ 1) ¡ 1= x ¡ x+ 1 ¡ xâ (x+ 1) ¡ 1= x ¡ x+ 1 ¡ xâ x ¡ 1â 1 ¡ 1= x2 + xâ xâ 1
= x2 â 1
âSee the equals signs in each line? Theyâre forming a chain ofidentities. You can follow the chain, checking everything out step bystep, until you end up with this: â
(x+ 1)(xâ 1) = x2 â 1
âOkay.ââChains of identities like this give you a slow-motion replay of
how a statement transforms from one thing into another. Donât freakout because thereâs a bunch of statements. Just follow them along,one at a time. Now take a look at this: â
x2 â 5x+ 6 = (xâ 2)(xâ 3)
= 0
âThe first equals sign there is creating an identity. Itâs tellingyou that no matter what you stick into this x here, x2 â 5x + 6 =
(xâ 2)(xâ 3). But that second equals sign is creating an equation.Itâs saying you donât have to solve x2â 5x+ 6 = 0 for x, you can justsolve its identity, (xâ 2)(xâ 3) = 0, instead.â
âNot bad for two lines.âI nodded. âThereâs one more kind of equality besides equations
and identities that you should know about: definitions. When youhave a really complex statement, definitions let you name part ofit to simplify things. You use an equals sign for this, but it doesnâtmean you have to solve for anything, like with an equation, or proveanything, like with an identity. You just use them in whatever wayâsconvenient.â
Equations and Love Letters 29
âCan you give me an example?ââWell, say youâre adding together two numbers, alpha and beta.
You could name themâin other words define them asââsâ like this: â
s = ι+ β an example definition
Tetraâs hand shot up. âQuestion!ââThis isnât class, Tetra. You donât have to raise your hand.âShe lowered her hand. âBut Iâm confused. Why did you name it
âsâ?ââIt doesnât really matter what you name it. You can use s, t,
whatever you want. Then once youâve said, okay, from now on s =Îą + β, you can just write s instead of having to write Îą + β everytime. Learn to define things, and youâll be able to write math thatâseasier to read and understand.â
âSo what are Îą and β then?ââWell, they could be letters that you defined somewhere else.
When you define something like s = Îą + β, that usually meansyouâre using the letter on the left side of the equals sign to namethe expression on the right. So here, youâd be using s as the name ofsomething that you made out of Îą and β.â
âAnd you can name them anything you want, right?ââBasically, yeah. Except that you shouldnât use a name that youâve
already used to define something else. Like, if you defined s = ι+ β
in one place, and then turned around and redefined it as s = ιβ,youâd start to lose your audience.â
âYeah, I can see that.ââThere are also some generally accepted definitions, like using Ď
to mean the ratio of the circumference of a circle to its diameter, ori to represent the imaginary unit, so it would be kind of weird to usethose names for something else. Anyway, if youâre reading through amath problem and you see a new letter popping up, donât panic, justthink to yourself, âoh, this must be a definition.â If youâre readingmath and it says something like âdefine s as Îą+βâ or âlet s be Îą+βâyouâre looking at a definition.â
âGot it.âI put down my pencil. âHereâs an idea. Next time youâre going
through your book, try looking for mathematical statements with
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letters in them and asking yourself if theyâre equations, identities,definitions, or something else altogether.â
Tetra nodded enthusiastically.âYou know,â I told her, âevery mathematical statement you find
in your textbook was written to express a thought. Just remember,âI said, pausing for effect, âthereâs always somebody behind the math,sending us a message.â
2.9.2 The Forms of Sums and Products
âOh, one more important thing,â I told her. âYou should always payattention to the overall form of a mathematical expression.â
âWhatâs that mean?ââTake a look at this statement, for example. An equation, right?â
(xâ Îą)(xâ β) = 0
âThe expression on the left side of the equals sign is telling youto multiply. In other words, itâs in multiplicative form. The thingsthat are being multiplied together are called factors.â
(xâ Îą)︸ ︡︡ ︸factor
(xâ β)︸ ︡︡ ︸factor
= 0
âThe same âfactorâ in prime factorization?â Tetra asked.âSure. Factorizing something means breaking it down into a
multiplicative form. Prime factorization means breaking it down intoa multiplicative form where all the factors are prime numbers. Oh,and most people leave out the multiplication sign when multiplyingthings. So all of these are the same equation, just written differentways: â
(xâ Îą)Ă (xâ β) = 0 using a Ă sign
(xâ Îą) ¡ (xâ β) = 0 using a ¡ sign(xâ Îą)(xâ β) = 0 using nothing
âOkay,â she said.âNow,â I added, âfor (xâÎą)(xâβ) = 0 we know that at least one
of the two factors has to equal zero. We can say that because itâs inmultiplicative form.â
Equations and Love Letters 31
âSo...if we multiply two things together and the result is zero, oneof the factors has to be zero. That makes sense.â
âWell, itâs better to say that at least one of them has to be zero.Because they both might be, right?â
âOkay, at least one of them. This is that precise mathematicallanguage thing we talked about yesterday, isnât it.â
âRight. So anyway, since we know that at least one of the factorsis zero, do you see how this equation is true when x â Îą = 0 orxâ β = 0? Another way to say this is that x = Îą,β is a solution tothis multiplicative form equation.â
âI follow.ââOkay. So letâs see what happens when we expand (xâÎą)(xâβ): â
(xâ Îą)(xâ β) = x2 â Îąxâ βx+ ιβ
âBy the way,â I asked her, âdo you think this is an equation?ââNope!â Tetra replied quickly. âItâs an identity!ââNot bad. Okay, âexpanding somethingâ means changing products
into sums. On the left side there are two factors being multipliedtogether, and on the right side there are four terms being addedtogether.â
âSorry, terms?ââYeah, when you add things together, you call them terms. Here,
let me show you a diagram with everything labeled: â
expandâââââ
(xâ Îą)︸ ︡︡ ︸factor
(xâ β)︸ ︡︡ ︸factor
= (x2)︸︡︡︸term
+(âÎąx)︸ ︡︡ ︸term
+(âβx)︸ ︡︡ ︸term
+(ιβ)︸ ︡︡ ︸term
ââââââfactorize
âWe can still do some cleanup on this expression,â I continued.âItâs a bit of a mess as it is: â
x2 â Îąxâ βx+ ιβ
âWell,â Tetra said, âwe could take the things that have an x inthem, like âÎąx and âβx...â
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âTry to call them âterms,â not âthings,â okay?â I said. âAlso, termslike âÎąx and âβx that only have one x should be called âfirst degreeterms of x,â or simply âfirst degree terms.â â
âOkay...â Tetra scratched her head. âHow âbout we bring togetherthe first degree terms of x. Like this: â
x2 + (âÎąâ β)x︸ ︡︡ ︸first degree terms
+ιβ
âExactly. Thatâs a good explanation of what to do with terms,but normally you would go one more step and bring the minus signto the outside of the parentheses: â
x2 â (Îą+ β)x+ ιβ
âYouâve probably heard of that as âcombining like terms.â âShe frowned. âHeard of it, yes. Thought about it, no.ââA quick quiz, then. Is this an identity or an equality?â
(xâ Îą)(xâ β) = x2 â (Îą+ β)x+ ιβ
âAll weâve done is expand and combine like terms, right? So thisshould be true for any value of x. . . which makes it an identity!â
âVery good! Moving on, then. We started out talking about thisequation, which is in multiplicative form: â
(xâ Îą)(xâ β) = 0 equation in multiplicative form
âUsing the identity that we just created, we can rewrite theequation. This is called an equation in additive form: â
x2 â (Îą+ β)x+ ιβ = 0 equation in additive form
âThese equations are in different forms, but theyâre the sameequation. All weâve done is use an identity to change the form of theleft side.â
âGot it.ââWhen we looked at the multiplicative form, we said that the
solution to the equation was x = ι,β. That means the solution to
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the equation in additive form must also be x = Îą,β. After all, theyârethe same equation.â
(xâ Îą)(xâ β) = 0 equation in multiplicative form
m same equations, same solutions
x2 â (Îą+ β)x+ ιβ = 0 equation in additive form
âYou can use this to solve some simple second degree equationsjust by looking at them. Here, take a look at these two. Pretty similar,arenât they?â
x2 â (Îą+ β)x+ ιβ = 0 (solution: x = Îą,β)
x2 â 5x+ 6 = 0
âWell...â She paused for a moment. âOh, I see! You can just thinkof the 5 as being Îą+ β, and the 6 as ιβ.â
âExactly. So to solve x2 â 5x+ 6 = 0, all you have to do is thinkof two numbers that equal 5 when you add them, or 6 when youmultiply them. That would be x = 2, 3, right?â
âMakes sense.ââMathematical expressions come in all sorts of forms. Multiplica-
tive and additive are just two of the possibilities. Remember thatsolving equations like ă additive form ă = 0 can be tough, but prob-lems like ămultiplicative form ă = 0 are super simple.â
âHuh, itâs like putting equations in multiplicative form is a wayof solving them, isnât it,â Tetra said. âYou know, I think Iâm gettingthe hang of this.â
2.10 Whoâs Behind the Math?
âI wish my teachers taught me as well as you do,â Tetra said.I grinned. âItâs a lot easier one-on-one. If I lose you, you can
slow me down and ask questions. You could always try that in classsometimes.â
Tetra pondered that for a moment. âWhat if Iâm studying some-thing and thereâs no one around to ask?â she asked.
âIf I donât get something after a careful reading, I mark the pageand move on. After a while, Iâll come back to that page and read it
34 Math Girls
one more time. If I still donât get it, I move on again. Sometimes Iâllswitch to a different bookâbut I keep going back to the part I didnâtunderstand. Once, I came across an expansion of an equation that Ijust couldnât follow. After agonizing over it for four days, I decidedthere was no way it could be right, so I contacted the publisher.Turns out it was a misprint.â
âNice catch!â Tetra shook her head. âGuess it pays to keep at it.ââWell, math takes time. I mean, thereâs so much history to it.
When youâre reading math, youâre trying to relive the work of count-less mathematicians. Trace through the development of a formula,and you might be following centuries of work. With depth like that,itâs not enough just to read. You have to become a mathematicianyourself.â
âSounds like a tall order.ââWell, itâs not like you have to get a PhD, but when youâre reading
math, you do have to make an effort to get into it. Donât just read it,write it out. Thatâs the only way to be sure you really understand.â
Tetra gave a slight nod. âIt kind of got to me, what you saidabout equations being a language, and that thereâs somebody onthe other side of the equation trying to send a message.â She lookedoff into the distance, her words coming faster. âMaybe itâs only myteacher, or the author of the textbook, but I can imagine it beinga mathematician from hundreds of years ago too. It kind of makesmath real, if you know what I mean.â
She looked away and smiled. âI think I might love youââ Shelooked back at me, then her eyes went wide as she realized what shehad just said.
I cocked an eyebrow.ââTeaching me!â She blurted, a few seconds too late.I looked around nervously.Tetraâs face was burning red. âMath,â she whispered. âI love you
teaching me math.â
Index
Aabsolute inequality, 71absolute value, 18, 52additive form, 32arbitrarily close, 89argument, 41arithmetic mean, 76average, 72
BBasel problem, 209, 212, 216,
259binomial theorem, 109, 234
CCatalan numbers, 138, 202,
233closed expression, 54coefficient, 53combination, 111, 234combining like terms, 32common ratio, 52complex number, 38condition, 73
continuous, 88convolution, 138, 174cosine function, 193
Ddefinition, 17, 28
of absolute value, 18of prime numbers, 15
derivative, 87, 133, 165difference, 90
of squares, 49difference equation, 96differential operator, 88differentiation, 190, 191, 261
of trig functions, 195rules, 192, 194
discrete, 89, 165divergance, 159, 162divisors, 12double angle formulas, 36
Ee, 95element (â), 148
286 Math Girls
equationdifference, 96, 165, 167quadratic, 63, 121second degree, 33, 60simultaneous, 61system of, 62versus identity, 26
Euler, Leonhard, 180, 217, 234exists (â), 151exponent
fractional, 78negative, 166
Ffactor, 30factorization, 30, 60, 207, 211,
218falling factorial, 93, 112, 134,
165Fibonacci numbers, 244Fibonacci sequence, 6, 55, 56,
59, 202, 233first degree term, 32for all (â), 148formula, 76
for a geometric series, 161,175, 241, 257
functionexponential, 95, 164generating, 53, 117higher order, 88
fundamental theorem of alge-bra, 206
Ggeneral term, 7, 54, 117, 233,
236
for the Fibonacci se-quence, 62
generalization, 22, 37, 51, 108,109, 188
generating function, 53, 55, 57,117, 232, 236, 253
geometric mean, 76geometric progression, 52, 175geometric series, 52, 178given, 75
Ii, 38, 145identity, 71
versus equation, 26, 186imaginary number, 38, 78, 145inequality, 157
absolute, 71of arithmetic and geomet-
ric means, 76, 77infinity, 141instantaneous rate of change,
87integers, 71
Llimit, 52, 142, 218logarithm, 165, 254
Mmathematical induction, 244max-min table, 261mean
arithmetic, 76geometric, 76
memorization, 7, 76multiplicative form, 30
Index 287
Nnumber
harmonic, 163, 168, 260imaginary, 78, 145integer, 71irrational, 96natural, 148negative, 73prime, 15, 176real, 70, 71square, 70
Ooperator
difference, 90, 167differential, 88, 165product, 80summation, 80
Ppartition number, 221, 236,
253upper limit, 243
Î , 25, 80Ď, 7, 217polynomial, 187
infinite dimensional, 187power laws, 57power series, 53, 117, 131, 187,
191, 201precision
in language, 146prime factorization, 22, 26, 30product
formal, 174of a sum, 49of infinite series, 175
of infinite sums, 177of sums, 119
proof by contradiction, 176proposition, 145
Qquadratic equation, 63quadratic formula, 121
Rreal number, 70, 71recurrence relation, 57, 105,
233rotation, 35
Ssequence, 53
infinite, 132problems, 9
seriesgeometric, 52harmonic, 162, 179, 204infinite, 52, 59, 142, 200
ÎŁ, 25, 80, 107, 114, 254sine function, 188, 190, 210,
218square number, 70square root, 73, 78subscripts, 22, 144sum
finite, 144of a geometric progres-
sion, 25of powers, 53of products, 118partial, 52, 142, 200
288 Math Girls
TTaylor series, 196, 198, 200,
214, 255
Uuniqueness of prime factoriza-
tions, 16, 176, 178unit circle, 40upper limit, 243, 253
Vvariable
naming, 29, 69, 71, 145,188
Zzeta function, 155, 162, 204,
209
Other works by Hiroshi Yuki
(in Japanese)
¡ The Essence of C Programming, Softbank, 1993 (revised1996)
¡ C Programming Lessons, Introduction, Softbank, 1994(Second edition, 1998)
¡ C Programming Lessons, Grammar, Softbank, 1995
¡ An Introduction to CGI with Perl, Basics, SoftbankPublishing, 1998
¡ An Introduction to CGI with Perl, Applications, SoftbankPublishing, 1998
¡ Java Programming Lessons (Vols. I & II), SoftbankPublishing, 1999 (revised 2003)
¡ Perl Programming Lessons, Basics, Softbank Publishing,2001
¡ Learning Design Patterns with Java, Softbank Publishing,2001 (revised and expanded, 2004)
¡ Learning Design Patterns with Java, MultithreadingEdition, Softbank Publishing, 2002
¡ Hiroshi Yukiâs Perl Quizzes, Softbank Publishing, 2002
¡ Introduction to Cryptography Technology, SoftbankPublishing, 2003
¡ Hiroshi Yukiâs Introduction to Wikis, Impress, 2004
¡ Math for Programmers, Softbank Publishing, 2005
¡ Java Programming Lessons, Revised and Expanded (Vols.I & II), Softbank Creative, 2005
¡ Learning Design Patterns with Java, MultithreadingEdition, Revised Second Edition, Softbank Creative, 2006
¡ Revised C Programming Lessons, Introduction, SoftbankCreative, 2006
¡ Revised C Programming Lessons, Grammar, SoftbankCreative, 2006
¡ Revised Perl Programming Lessons, Basics, SoftbankCreative, 2006
¡ Introduction to Refactoring with Java, Softbank Creative,2007
¡ Math Girls / Fermatâs Last Theorem, Softbank Creative,2008
¡ Revised Introduction to Cryptography Technology, SoftbankCreative, 2008
¡ Math Girls Comic (Vols. I & II), Media Factory, 2009
¡ Math Girls / GĂśdelâs Incompleteness Theorems, SoftbankCreative, 2009
¡ Math Girls / Randomized Algorithms, Softbank Creative,2011