Mathematics: Its Magic and Mystery

~athematics ITS MAGIC AND MASTERY

AARON BAKST

NEW YORK

D. VAN NOSTRAND COMPANY, 250

FOURTH A.VENUE

INC.

Ch1athematics ITs

MAGIC AND MASTERY

In the manufacture of this book, the publishers have observed the recommendations of the War Production Board and any variation from previous printings of the same book is the result of this effort to conserve paper and other critical materials as an aid to war effort.

Copyright, 1941 by D.

VAN NOSTRAND COMPANY, INC.

ALL RIGHTS RESERVED

This book, or any part thereof, may not be reproduced in an} form without written permi.r.rion from the publisher.f.

First Published, September 1941 Reprinted November 1911, May 1942, January 1943 J a111tary 1945

PRINTED

!N

THE

U. S

A.

Pre/ace There was a time when mathematics was regarded as an intricate subject that could be mastered only by the elite of the scholarly world. Fortunately, this day is long passed. \Ve now know that anyone with a taste for figures and an interest in reading can find a vast enjoyment in the symmetry and harmony of basic mathematics and in the solving of mathematical problems. This book is designed to make mathematics interesting. The science is not treated formally; abstract conceptions and uninteresting and abstruse procedures are completely avoided; yet at the same time the book is sufficiently complete so as to give a broad picture of mathematical fundamentals. Mathematics, in order to be appreciated by those who do not have a flare for the intangible must be seen in the light of its versatility in the various fields of human endeavor. This is very much the central theme of this book. Although designed for amusement, the book is not restricted merely to novel and tricky and entertaining mathematical stunts. The prevailing trend in bOOMS on mathematical amusements published during the last three or four centuries has been in the direction of amusement in mathematics for the sake of the mathematics itself. It is not possible to enumerate all of the books published in this field but the most recent works of Ahrens, Ball, Fourray, Kraitchik, Lucas, Perelman and others rarely consider (excepting Perelman) the applications of mathematics. In this way, this book differs from the classical treatment of amusement in mathematics and it is hoped that it gives an answer to the reader's quest not only for amusement in mathematics but also for an easily read and clearly understandable discussion of mathematical processes. The material is simply developed. No proofs of any kind are used in the unfolding of the mathematical processes and properties. It is the firm belief of the author that v

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'ft1:athetna~ibs must be understood to be appreciated and that an underst:mding of the subject does not require involved theoretical Those who, after reading this book, care to conilnu~ their study of elementary mathematics thoroughly and exhaustibly wiiI find it profitable to refer to the books, "Mathematics for Self Study" by J. E. Thompson, D. Van Nostrand Co., Inc., New York, and "Integrated Mathematics" by John A. Swenson, Edwards Brothers, Ann Arbor, Michigan. In the compilation of the book the author referred freely to the vast literature on elementary mathematics and on recreations in mathematics. If there is any claim of originality, it is to originality of treatment and method. Otherwise, any similarity to any book published or unpublished is purely accidental and/or coincidental.

. , ::r ;', •. 4isc'u~si~ns, ~."

A.B. New York City May, 1941

Contents CHAPTER 1. NUMERALS

PAGE

& NUMERATION A Notch: Mathematics Is Born-Numeration Arrives, Accompanied by the Tax Collector-The Price-Tag Mystery-The Use of Numerals.

2.

1

9

SYSTEMS OF NUMERATION

The Strange Case of Dr. X.-Which Is the Easiest?This Trick Is Honest-The Two-System-How to Mystify Y our Friends-The Chinese Knew Their Digits-A Hole in 10,101-The Three-System-The Truth about Dr. X's Golf Scores-Other Systems-Homework Was Never Like This 1 + 1 = 1O-Remcmber This When That Collector Comes Around: 10 - 1 = 1 - Sometimes!-How to Multiply and Like It-Short Turns in Long Division-Some Addition and Multiplication Tables You Didn't Find in Grade School-Odd or Even?Fractions without Denominators. 3.

SOME REMARKABLE PROPERTIES OF NUMBERS

31

Those 'Lucky' and 'Unlucky' Numbers-Number 12: The Ancients' Favorite-Versatile 365-Number 99, the Rapid Calculator-The Game of 999-The Thousandand-First Parlor Trick-Triple-Play 10,101-The Number 10,001-Tricky 9,999,99,999, and 999,999-A Puzzler with 111,111-Number Curiosities-The Oddity 123,456,789-Repeated-Digit Baffiers-Orderly 142,857-Cyclic Handy Men-Some Wit Testers. 4.

51

NUMBER GIANTS

Close-up of a Million-The Long Count-All Is Lost: That Housefly Is Here Again-Pity the Poor Billionaire: A Fifty-Nine-Mile Pile of Bills-You Owe $500-Now, Some Really Big Numbers-Archimedes Counts the Sands-A Cure for Egotists-Timetable for a Tour of the Stars.

5.

65

NUMBER PYGMIES

Some Gigantic Runts-Cosmic Small Fry-The Populous Bacterium-The Other Side of Zero"-Where "Split Seconds" Drag. .VIl

viii

Contents

CHAPTER 6. THERE'S SE'CRECY IN NUMBERS

PAGE

79

Trlx hoex erfa eeth mpel-Transposition: How to Scramble a -Message-More Scrambling: How to Make Cryptography More Cryptic-Befuddling the Foe, or How to Have Fun with the Fifth Column-How to Hide the Key and Keep Your Code from Talking-The Civil War's Secret Weapon: The Grille-Anatomy of a Grille-Behind the Grille: A Little Math-A Full-Dress Grille: The Shape's the Thing-Substitution: Alphabetical and Otherwise-Double Trouble for Code Kibitzers. 7.

THE ARITHMETIC OF MEASUREMENT

102

The Great- Pyramid Mystery-What a Little Juggling with Numbers Will Do-So's the National Debt-Nothing's Ever Right-The Turncoat Zero-Rounding Sums: Plus and Minus-Products and Quotients-To Stop Runaway Digits. 8.

SIMPLE CALCULATING DEVICES

113

This Little Pig Went to Market-A Homemade Calculating Machine-Daughters of Abacus: Suan-pan, Soroban, and Schoty. 9.

125 The Astounding Mr. Doe-How to Remember That Telephone Number-Arithmetical Rabbits-Multiplication Made Painless. 10. PROBLEMS & PUZZLES. 134 It's a System-Two-Timing Puzzles-Application of the Three-System-Arithmetic for Sherlock Holmes. 11. How THE NUMBER MAGICIAN DOES IT 150 "I Will Now Predict ..."-The Guessing Game-How To Make Your Own Tricks-If She's Coy about Her Age-Age and Date of Birth, Please. 12. ALGEBRA & ITs NUMBERS 158 It's Hotter than You Think-Both Ways at Once-A Rule for Signs-Multiplication Magic for the MillionSecrets of Some Curiosities. 13. THE ALGEBRA OF NUMBER GIANTS AND PYGMIES 171 The Fifth Operation: Making and Mastering GoogolsTaxi to the Moon-Jujitsu for Number Giants-Some Puzzling Results-I, 900,000,000,000,000,000,000,000,000,OOO,OOO-Up for Air: $50,000,000,000 Worth-Wood Is -Always Burning-Double Strength-Number Giants with a F~w Digits. RAPID CALCULATIONS

Contents

IX

CHAPTER PAGE 14. THE GRAMMAR OF ALGEBRA 186 The Heirs' Problem-Beware of the Math-Minded Mother-in-Law!-One for the Relief Administrator-A Rolling Pin Is How Heavy?-Million-Dollar SwapHelp Wanted: One Stag Line, Slightly Used-Men at Work: Fractions Ahead-Proceed at Own Risk-The Erring Speed Cop-An Accountant's Headache-Algebra and Common Sense. 15. ALGEBRA, Boss OF ARITHMETIC How the Stage Magician Does It-Performing with Three-Place Numbers-Making It Easier-Short-Order -Rapid Extraction of Roots-Less Work, Same Result.

209

16. ALGEBRA LOOKS AT INSTALMENT BUYING Debt on the Instalment Plan-How to Learn the Price of Money-Advertising Is an Art-What Makes the Instalment Wheels Go 'Round and 'Round-The HowMuch Ladder-How to Buy a House on Instalment-The Rich Get Rich, and the Poor Try to Borrow:

224

17 . CHAIN-LETTER ALGEBRA The Silk-Stocking-Bargain Bubble-Dream of an Opium Eater-The Family-Minded Fly-The Arithmetical Plague of Australia-The How-Many-Times Ladder-A Sequence for the Rabbits-A Clever Rat-The Banker's Mathematics.

246

18. STREAMLINING EVERYDAY COMPUTATION Napier's Escape from Drudgery-Napier's Clue: Addition Is Easier-Number's Common Denominator: The Lowly Logarithm-A Mathematical Rogue's Gallery: Every Number Has Its Fingerprint-What You Pay the Middle-Men-Logarithm Declares a Dividend-Sing a Song of Sixpence-The Roar of a Lion and the Twinkle of a Star-The Algebra of Starlight: 437,000 Full Moons = 1 Bright Midnight-The Trade-in Value of Your Car: a = ACI - r)n.

264

19. THE BANKERS' NUMBER-JACK OF ALL TRADES 291 Simple Life-How to Make Money Breed-The Shortest Insurance Policy Ever Written: 'e'-How Fast Your Money Can Grow-How Fast Your Money Can GoThe Mathematics of Slow Death-How Fast Is Slow Death?-The Curves of Growth and Death-The Bankers' Number and Aviation-How Hot is Your Coffee?-The Bankers' Number Gets Around.

x

Contents

CHAPTER PAGE 20. How TO HAVE FUN WITH LADY LUCK 329 Fickle as the Flip of a Coin-How Good Is a Hunch?The 'What' in 'What a Coincidence'-Ten in a Row: Algebra. Hits the Jackpot-Mugwump Math: Head or Tail?-An Ancient Lullaby: Baby Needs Shoes-What the 'Odds' Are-How Long Will You Live?-What Price Life?-Drop a Needle, Pick Up a Probability. 21. THE THINKING MACHINES Hollywood Goes Mathematical-That Good-Movie Formula-Memo for Burglars-Why Robbers Blow Up Safes -Wisdom by the Turn of a Crank-An Infinity of Nonsense-A Tip for Radio Baritones.

354

22. POSTOFFICE MATHEMATICS 365 How to Keep out of the Dead-Letter Department: An Introduction to Relativity-'Somewhere' in Math: A Game of Chess-Where Einstein Began-Don't Let Relativity Disturb You: 2 Is Still 2-If All the World Were a Pancake-This Expanding Universe: Three-Dimensional Worlds-Geometric Comet: Y = 6. 23. NEW WORLDS FOR OLD Totalitarian Utopia: World without Freedom-The Inhibited Insect-The Perils of Flatland-An Arithmetic Oddity in a Geometric Dress-Unsquaring the SquareSurveying in Flatland-"Seeing" Three-Dimensional Pictures-Formula for Creating a World: Algebra + Imagination.

377

24. PASSPORT FOR GEOMETRIC FIGURES 392 Moving Day: It's all Done by Math-Life on a MerryGo-Round-Some Points on the Way to Infinity-Babylonian Heritage: 360 Pieces of Circle-How to Run Around in Distorted Circles-Nature's Favorite: The Ellipse-Meet the Circle's Fat Friends: Sphere, Hypersphere, and Ellipsoid-This Curve May Kill You: The Parabola-How to Get Your Geometric Passport-Every Passport Has Its Picture. 25. MAN'S SERVANT-THE TRIANGLE 420 Measure Magic-The Triangle: Simple, Eternal, and Mysterious-The Bases of Comparison-Similar Triangles and Their Properties-Two Especially Helpful Triangles -The Triangle as a Superyardstick-How to Measure Distant Heights-An Instrument for Measuring Any Height-Measuring Heights with a Mirror-Measuring

Contents CHAPTER

Xl PAGE

Distances between Inaccessible Objects-Measuring between Two Inaccessible Points-Another Method for Inaccessible Points-What the Surveyor Does When He Surveys. 26.

THE TRIANGLE-MAN'S MASTER

446

The Triangle Key to Measllrement-How to Measure Angles-Angles and Their Ratios-The Ratio That Does Everything. 27.

CIRCLES, ANGLES, AND AN AGE-OLD PROBLEM

456

The Circle: Sphinx to the Mathematician, Old Saw to the Carpenter-The Elusive Pi-Taking the Girth of a Circle -Trig Without Tables-This Lopsided World: Even Your Best Friend Is Two-Faced-Squaring the Circle. 28.

THE MATHEMATICS OF SEEING

469

Can You Trust Your Eyes?-Actual and Apparent Sizes -The Circle Family: Meet Cousin Chord-Don't Eclipse the Moon with a Match-How Good Is Your Eyesight? -The Glass Eye: Sees All, Knows All-The Historical Eye: Don't Look Now, Boys. 29.

THE LOST HORIZON

479

If the Earth Were Flat-How Far Is Faraway?-Seeing Is Not So Simple-How Far Can You See?-Distance Observation in Sea Warfare-Reaching Out for the Horizon -Celestial Illusion: Sunrise and Sunset-The Soaring Horizon: Another Illusion. 30.

THE SHAPE OF THINGS

493

Foundation for a Figure-Tailoring with Straight Lines and Angles-Three Straight Lines and Three AnglesAdding Another Line and Angle-Squashing the Rectangle-There's Method in Their Math-Math's Trade Secret Applied to Other Fields-How to Wrap a Circle -Jumping Off into Infinity. 31.

THE SIZE OF THINGS

520

Birth of a World-Measurements in Flatland-Algebra to the Rescue of Geometry-The Area of a Circle-The Area of Any Figure. 32.

ESCAPE FROM FLATLAND

Measuring in Three Dimensions-What's in a Figure?The Refugee Returns to His Land of Flight: Flatland Again-Comes the Revolution: Cones and PyramidsJust How Big Is the Earth?-Speaking of Volumes: It's a Small World-Name Your Figure.

538

xu

Contents

CHAPTER PAGE 33. How ALGEBRA SERVES GEOMETRY 564 200 Men and an Egg-The Lesson of the Shrinking Dime -Tin-Can Economy-Square vs. Rectangle-Squares, Circles, and Suds-Sawing Out the Biggest Log-Cutting Corners from a Triangle, or How to Get the Most Out of a Garret-How to Make Money in the Box Business -Nature Study: Why Is a Sphere?-Fashion Your Own Funnel. 34. CORK-ScREW GEOMETRY .

585

But the Earth Isn't Flat ...-Shortest Distances in Three Dimensions-A Tip for the Spider-How to Know Your Way arouna a Prism-Journey across a Pyramid-Points about a Glass-It All Depends on Your Direction-The Screw: Industry Spirals Up Its Stairway-Coming around the Cone: Helix, the Spiral Screw-The Shortest Route on Earth. 35. MATHEMATICS, INTERPRETER OF THE UNIVERSE Railroading among the Stars-Portrait of a TimetableTiming Straight-Line Motion-The Algebra of Speed -Speed on the Curves-Why Don't We Fly Off into Space?-What Makes the Universe Hang Together?The Democracy of a Tumble-A Multiplication Table for Physical Relations-What Came First, the Chicken or the Egg?-Flatfoot to the Universe: Gravitation.

609

36. THE FIRING SQUAD & MATHEMATICS What Happens When You Pull the Trigger?-How Strong Is a Bullet?-You May Not Hit the Target, but You'll Get a Kick out of This-The Path of a BulletBig Bertha's Secret-Aerial Artillery: Bigger than Bertha-The Algebra of a Fired Shell-Bertha's Shell and Johnny's Top-Out of the Firing Pan into the Fire-The Gun's Angle of Elevation-How to Determine the Firing Angle-If You Want to Hit Your Target, Don't Aim at It.

641

37. OF MATH & MAGIC 678 Computing Is Believing-How a Locomotive Takes a Drink-Gaining Weight on a Merry-Go-Round-The Cheapest Way Around the World: Go East, Old ManThe 'Human Cannon Ball'-Want to Break a Record? Don't Go to Berlin!-The 'Devil's Ride' or Looping the Loop-Stepping on Y our Brakes-The Hammer and Anvil Circus Act-The Engineer's Dilemma-Gravitation and Flirtation-Getting the Moon Rocket Started.

Contents

X111

PAGE

705

ApPENDIX

1. Signs and Symbols Used in This Book.

I. Algebra

2. Addition of Signed Numbers-3. Multiplication and Division of Signed Numbers-4. Multiplication of Polynominals-S. Division of Polynominals by a Monomial6. Some Formulas for the Multiplication of Polynomials -7. Fundamental Property of Fractions-S. Addition and Subtraction of Fractions-9. Multiplication of Fractions10. Division of Fractions-II. Proportion-I2. Some Results of a Proportion-13. The Arithmetic Mean-l4. Raising to a Power-IS. Extraction of Roots-I6. The 'Sign of the Root-17. Special Properties of Powers and Roots-IS. Operations with Powers and Roots-I9. Extraction of Square Roots of Numbers-20. Arithmetic Progressions-21. Geometric Progressions-22. Logarithms-23. Raising of a Binomial to a Power-24. Equations of the First Degree in One Unknown-25. Rules for the Solution of Equations of the First Degree in One Unknown-26. Systems of Equations of the First Degree in Two Unknowns-27. Quadratic Equations-2S. Formulas for the Roots of Quadratic Equations-29. The Test for the Roots of a Quadratic Equation-30. The Properties of the Roots of a Quadratic Equation.

II. Geometry 31. Lines and Angles-32. Triangles-33. Parallel Lines 34. Quadrangles-35. The Altitude, the Angle Bisector and the Median of a Triangle-36. Four Remarkable Points in a Triangle-37. Formulas Associated with Triangles-3S. Formulas for the Areas of Polygons39. The Circle-40. Two Circles-41. The Circumference and Area of a Circle-42. Regular Polygons-43. Polyhedra-44. Formulas for the Areas of the Surfaces and the Volumes of Polyhedra.

III. Trigonometry 4S. Trigonometric Ratios of an Acute Angle-46. Some Special Values of'Trigonometric Ratios-47. Formulas for the Solution of Right Triangles-4S. Fundamental Formulas of Trigonometry-49. Solution of Triangles.

XIV

Contents PAGE

IV. Tables

A. Table of Logarithms of Numbers-B. Table of Squares of Numbers-C. Table of Square Roots of NumbersD. Tables of Trigonometric Ratios. V. Approximate Formulas for Simplified Computation ANSWERS TO PROBLEMS

771

INDEX

783

".

Numerals & Numeration

A Notch: Mathematics Is Born Somewhere back in the early days of humankind a hairy hand hesitatingly notched a tree to record a kill, or the suns of his journeying. Later, some Stone Age Einstein may have formulated the theory of the notch to keep track of his flock-each scratch (or notch) upon a piece of wood corresponding to one animal and the total scratches therefore equaling the total flock. In some such unprofessorial way man took his first plunge into the mathematical world and came up with a revolutionary concepthow to count. F or when man learned how to count, he acquired a scientific tool with which he could break up the universe into its component units and thus master the size and shape of things. Some of us, however, are still stuck in the notch. Peasants in many countries have made little improvement upon the ancient shepherd's theory. And this is also true of modern city dwellers, although few realize it. Suppose you receive a bill and wish to compare the items enumerated with those you ordered and received. To be certain that each item is accounted for, you make a check mark beside it on the bill to indicate that the charge is correct. If there are several purchases of the same kind, you may place two or more check marks. In "modern, scientific" office work, too, especially in bookkeeping when individual objects are counted, it is a common

2

Mathematics-Its Magic

&

Mastery

practice to rec~rd similar individual items by writing a short vertical stroke (the notch again) for each. Thus:

I II /I I

II II III I I

denotes denotes denotes denotes denotes

1 object 2 objects 3 objects 4 objects 5 objects

To avoid confusion, when a fifth object is counted, it is often recorded (or, in technical terms, tabulated) by a cross stroke, thus: ffH. This method of crossing out facilitates further counting. The method of writing the fifth stroke horizontally, however, is not universally accepted. Recording may be done by vertical strokes until nine objects are counted (I I I I I I I I I), and, when a tenth is added, by another vertical stroke or by a horizontal stroke U/// H-l/-l). When all the objects have been checked off it is easy to rewrite in numerical form the results of the work-sheet tabulation. Thus, if we have Chairs Tables Beds

we can write

//// -ffI+ /f/IL If/-/- III

H-ff //// fI+f II

ffH Ii-hi IIII 23 chairs 17 tables 14 beds

Notice that every -HH represents five objects. procedure of the rewriting is as follows:

Thus, the

4 times 5 = 20, and 20 plus 3 = 23 3 times 5 = 15, and 15 plus 2 = 17 2 times 5 = 10, and 10 plus 4 = 14

The same objects might have been recorded as follows: Chairs Tables Beds

i////,{!// i-l/////// I I I //////i// IIIIIII -fIN //1/ / /I I I

However, the tabulation by four vertical strokes and one horizontal is preferable, because the fewer strokes, the less the chance of a mistake.

Numerals & Numeration

3

Numeration Arrives, Accompanied by the Tax Collector The art of recording numbers, or numeration, was invented to meet the needs of a society more complex than our ancient herdsman's. The numbers now in use, however, are-if the age of civilization is considered-of comparatively recent origin. These numerals (0, 1,2, 3,4,5,6, 7, 8, and 9) have been in common use for not more than a thousand years. They are known as Hindu-Arabic figures, because they originated in India and were introduced to Europe by the Arabs. Before their introduction, sums were done in various ways, all lengthy. For example, in Russia under the Tatar occupation (during the thirteenth and fourteenth centuries) receipts for taxes collected by Tatar officials were of this form:

$$$ ®®@ ®@@

DDDDDDD XXXX Ijlll/II!

J /II/Ill

IIIIII This recorded the payments of the iasak ( the Tatar word for taxes) of 3,674 rubles and 46 kopecks (1 ruble equals 100 kopecks). Thus:

$

denoted

1,000 RUBLES

@

denot-ed

100 RU BLES

denoted

10 RUBLES

denoted

I RU BLE

o

X

/II! I! LfI ITT/Tll

I

denoted

10 KOPECKS

denoted

I KO PECK.

4

Mathematics-Its Magic

Mastery

&

The border drawn around the numerals on the receipt was prescribed by law on the optimistic theory that it would tend to prevent additions or other finagling. The symbols for the numerals enabled the people, who by and large could neither read nor write, to keep records of their payments. This way of writing numbers was used (in a variety of forms) by many ancient peoples. F or example, the Egyptians wrote their numbers this way: denoted 10

n f

de-noted

I

7hus((((1iill~~5 Mill de-noted 45,62.3 The Babylonians enumerated as follows (they counted by 60):

~

-