mathematics 1 VOLUME Ab–Cy Barry Max Brandenberger, Jr., Editor in Chief Copyright © 2002 by Macmillan Reference USA, an imprint of the Gale Group All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photo- copying, recording, or by any information storage and retrieval system, with- out permission in writing from the Publisher. Macmillan Reference USA Macmillan Reference USA 300 Park Avenue South 27500 Drake Rd. New York, NY 10010 Farmington Hills, MI 48331-3535 Library of Congress Cataloging-in-Publication Data Mathematics / Barry Max Brandenberger, Jr., editor in chief. p. cm. Includes bibliographical references and index. ISBN 0-02-865561-3 (set : hardcover : alk. paper) - ISBN 0-02-865562-1 (v. 1 : alk. paper) ISBN 0-02-865563-X (v. 2 : alk. paper) - ISBN 0-02-865564-8 (v. 3 : alk. paper) - ISBN 0-02-865565-6 (v. 4 : alk. paper) 1. Mathematics-Juvenile literature. [1. Mathematics.] I. Brandenberger, Barry Max, 1971- QA40.5 .M38 2002 510-dc21 00-045593 Printed in the United States of America 1 2 3 4 5 6 7 8 9 10 Table of Contents VOLUME 1: Brain, Human Bush, Vannevar Preface List of Contributors C Calculators A Calculus Abacus Calendar, Numbers in the Absolute Zero Carpenter Accountant Carroll, Lewis Accuracy and Precision Cartographer Agnesi, Maria Gaëtana Census Agriculture Central Tendency, Measures of Air Traffic Controller Chaos Algebra Cierva Codorniu, Juan de la Algebra Tiles Circles, Measurement of Algorithms for Arithmetic City Planner Alternative Fuel and Energy City Planning Analog and Digital Comets, Predicting Angles, Measurement of Communication Methods Angles of Elevation and Depression Compact Disc, DVD, and MP3 Technology Apollonius of Perga Computer-Aided Design Archaeologist Computer Analyst Archimedes Computer Animation Architect Computer Graphic Artist Architecture Computer Information Systems Artists Computer Programmer Astronaut Computer Simulations Astronomer Computers and the Binary System Astronomy, Measurements in Computers, Evolution of Electronic Athletics, Technology in Computers, Future of Computers, Personal B Congruency, Equality, and Similarity Babbage, Charles Conic Sections Banneker, Benjamin Conservationist Bases Consistency Bernoulli Family Consumer Data Boole, George Cooking, Measurement of Bouncing Ball, Measurement of a Coordinate System, Polar Table of Contents Coordinate System, Three-Dimensional Flight, Measurements of Cosmos Form and Value Cryptology Fractals Cycling, Measurements of Fraction Operations Fractions PHOTO AND ILLUSTRATION CREDITS GLOSSARY Functions and Equations TOPIC OUTLINE G VOLUME ONE INDEX Galileo Galilei Games VOLUME 2: Gaming Gardner, Martin D Genome, Human Dance, Folk Geography Data Analyst Geometry Software, Dynamic Data Collxn and Interp Geometry, Spherical Dating Techniques Geometry, Tools of Decimals Germain, Sophie Descartes and his Coordinate System Global Positioning System Dimensional Relationships Golden Section Dimensions Grades, Highway Distance, Measuring Graphs Division by Zero Graphs and Effects of Parameter Dürer, Albrecht Changes E H Earthquakes, Measuring Heating and Air Conditioning Economic Indicators Einstein, Albert Hollerith, Herman Electronics Repair Technician Hopper, Grace Encryption Human Body End of the World, Predictions of Human Genome Project Endangered Species, Measuring Hypatia Escher, M. C. Estimation I Euclid and his Contributions IMAX Technology Euler, Leonhard Induction Exponential Growth and Decay Inequalities Infinity F Insurance agent Factorial Integers Factors Interest Fermat, Pierre de Interior Decorator Fermat’s Last Theorem Fibonacci, Leonardo Pisano Internet Field Properties Internet Data, Reliability of Financial Planner Inverses Table of Contents K Morgan, Julia Knuth, Donald Mount Everest, Measurement of Kovalevsky, Sofya Mount Rushmore, Measurement of Music Recording Technician L N Landscape Architect Nature Leonardo da Vinci Navigation Light Negative Discoveries Light Speed Nets Limit Newton, Sir Isaac Lines, Parallel and Perpendicular Number Line Lines, Skew Number Sets Locus Number System, Real Logarithms Numbers: Abundant, Deficient, Perfect, Lotteries, State and Amicable Lovelace, Ada Byron Numbers and Writing Photo and Illustration Credits Numbers, Complex Glossary Numbers, Forbidden and Superstitious Topic Outline Numbers, Irrational Volume Two Index Numbers, Massive Numbers, Rational Numbers, Real VOLUME 3: Numbers, Tyranny of Numbers, Whole Nutritionist M Mandelbrot, Benoit B. O Mapping, Mathematical Ozone Hole Maps and Mapmaking Marketer P Mass Media, Mathematics and the Pascal, Blaise Mathematical Devices, Early Patterns Mathematical Devices, Mechanical Percent Mathematics, Definition of Permutations and Combinations Mathematics, Impossible Pharmacist Mathematics, New Trends in Photocopier Mathematics Teacher Photographer Mathematics, Very Old Pi Matrices Poles, Magnetic and Geographic Measurement, English System of Polls and Polling Measurement, Metric System of Polyhedrons Measurements, Irregular Population Mathematics Mile, Nautical and Statute Population of Pets Millennium Bug Postulates, Theorems, and Proofs Minimum Surface Area Powers and Exponents Mitchell, Maria Predictions Möbius, August Ferdinand Primes, Puzzles of Table of Contents Probability and the Law of Large Numbers Space, Commercialization of Probability, Experimental Space Exploration Probability, Theoretical Space, Growing Old in Problem Solving, Multiple Approaches to Spaceflight, Mathematics of Proof Sports Data Puzzles, Number Standardized Tests Pythagoras Statistical Analysis Step Functions Q Stock Market Quadratic Formula and Equations Stone Mason Quilting Sun Superconductivity R Surveyor Symbols Radical Sign Symmetry Radio Disc Jockey Randomness T Rate of Change, Instantaneous Ratio, Rate, and Proportion Telescope Restaurant Manager Television Ratings Robinson, Julia Bowman Temperature, Measurement of Roebling, Emily Warren Tessellations Roller Coaster Designer Tessellations, Making Rounding Time, Measurement of Topology Photo and Illustration Credits Toxic Chemicals, Measuring Glossary Transformations Topic Outline Triangles Volume Three Index Trigonometry Turing, Alan VOLUME 4: U Undersea Exploration S Universe, Geometry of Scale Drawings and Models Scientific Method, Measurements and the V Scientific Notation Variation, Direct and Inverse Sequences and Series Vectors Significant Figures or Digits Virtual Reality Slide Rule Vision, Measurement of Slope Volume of Cone and Cylinder Solar System Geometry, History of Solar System Geometry, Modern W Understandings of Solid Waste, Measuring Weather Forecasting Models Somerville, Mary Fairfax Weather, Measuring Violent Sound Web Designer Table of Contents Z Topic Outline Zero Cumulative Index Photo and Illustration Credits Glossary Abacus A The abacus is the most ancient calculating device known. It has endured over time and is still in use in some countries. An abacus consists of a wooden frame, rods, and beads. Each rod represents a different place value—ones, tens, hundreds, thousands, and so on. Each bead represents a number, usu- ally 1 or 5, and can be moved along the rods. Addition and subtraction can easily be performed by moving beads along the wires of the abacus. The word abacus is Latin. It is taken from the Greek word abax, which means “flat surface.” The predecessors to the abacus—counting boards— were just that: flat surfaces. Often they were simply boards or tables on which pebbles or stones could be moved to show addition or subtraction. The earliest counting tables or boards may simply have been lines drawn in the sand. These evolved into actual tables with grooves in them to move the counters. Since counting boards were often made from materials that deteriorated over time, few of them have been found. The oldest counting board that has been found is called the Salamis Tablet. It was found on the island of Salamis, a Greek island, in 1899. It was used by the Babylonians around 300 B.C.E. Drawings of people using counting boards have been found dating back to the same time period. There is evidence that people were using abacuses in ancient Rome (753 B.C.E.–476 C.E.). A few hand abacuses from this time have been found. They are very small, fitting in the palm of your hand. They have slots with beads in them that can be moved back and forth in the slots similar to counters on a counting board. Since such a small number of these have been found, they probably were not widely used. However, they resemble the Chinese and Japanese abacuses, suggesting that the use of the abacus spread from Greece and Rome to China, and then to Japan and Russia. This Chinese abacus is representing the number 7,230,189. Note the two The Suanpan beads on each rod of the upper deck and five In China, the abacus is called a “suanpan.” Little is known about its early beads on each rod of the use, but rules on how to use it appeared in the thirteenth century. The suan- lower deck. Beads are pan consists of two decks, an upper and a lower, separated by a divider. The considered counted when moved towards the beam upper deck has two beads in each column, and the lower deck has five beads that separates the two in each column. Each of the two beads in the ones column in the top deck decks. 1 Abacus is worth 5, and each bead in the lower deck is worth 1. The column far- thest to the right is the ones column. The next column to the left is the tens column, and so on. The abacus can then be read across just as if you were place value in a reading a number. Each column can be thought of in terms of place value number system, the and the total of the beads in each column as the digit for that place value. power of the base assigned to each place; The beads are moved toward the middle beam to show different num- in base-10, the ones bers. For example, if three beads from the lower deck in the ones column place, the tens place, have been moved toward the middle, the abacus shows the number 3. If one the hundreds place, and so on bead from the upper deck and three beads from the lower deck in the ones column have been moved to the middle, this equals 8, since the bead from digit one of the sym- bols used in a number the upper deck is worth 5. system to represent the multiplier of each place To add numbers on the abacus, beads are moved toward the middle. To subtract numbers, beads are moved back to the edges of the frame. Look at the following simple calculation (12 (cid:2) 7 (cid:3) 19) using a suanpan. "12" "19" The abacus on the left shows the number 12. There is one bead in the lower deck in the tens place, so the digit in the tens column is 1. There are two beads in the lower deck in the ones place, so the digit in the ones col- umn is 2. This number is then read as 12. To add 7 to the abacus, simply move one bead in the ones column of the top deck (5) and two more beads in the ones column of the lower deck (2). Now the suanpan shows 9 in the ones column and 10 in the tens column equaling 19. The Soroban THE ABACUS VERSUS THE CALCULATOR The Japanese abacus is called the soroban. Although not used widely until the seventeenth century, the soroban is still used today. Japanese students In late 1946 a Japanese postal first learn the abacus in their teens, and sometimes attend special abacus official highly skilled in using schools. Contests have even taken place between users of the soroban and the soroban (Japanese abacus) the modern calculator. Most often, the soroban wins. A skilled person is usu- engaged in a contest with an ally able to calculate faster with a soroban than someone with a calculator. American soldier in adding, sub- tracting, and multiplying num- The soroban differs only slightly from the Chinese abacus. Instead of bers. The American used what two rows of beads in the upper deck, there is only one row. In the lower was then the most modern deck, instead of five rows of beads, there are only four. The beads equal the electromechanical calculator. In same amount as in the Chinese abacus, but with one less bead, there is no four of five contests, the Japan- carrying of numbers. For example, on the suanpan, the number 10 can be ese official with the soroban shown by moving the two beads in the upper deck of the ones column or was faster, being beaten only in only one bead in the upper deck of the tens column. On the soroban, 10 the multiplication problems. can only be shown in the tens column. The beads in the ones column only add up to 9 (one bead worth 5 and four beads each worth 1). 2 Absolute Zero The Schoty The Russian abacus is called a schoty. It came into use in the 1600s. Little is known about how it came to be. The schoty is different from other aba- cuses in that it is not divided into decks. Also, the beads on a schoty move on horizontal rather than vertical wires. Each wire has ten beads and each bead is worth 1 in the ones column, 10 in the tens column, and so on. The schoty also has a wire for quarters of a ruble, the Russian currency. The two middle beads in each row are dark colored. The schoty shows 0 when all of the beads are moved to the right. Beads are moved from left to right to show numbers. The schoty is still used in modern Russia. SEE ALSO Calcula- tors; Mathematical Devices, Early. Kelly J. Martinson Bibliography Pullan, J.M. The History of the Abacus.New York: Frederick A. Praeger, Inc., 1969. Absolute Value Absolute value is an operation in mathematics, written as bars on either side of the expression. For example, the absolute value of (cid:4)1 is written as (cid:2)(cid:4)1(cid:2). Absolute value can be thought of in three ways. First, the absolute value of any number is defined as the positive of that number. For example, (cid:2)8(cid:2) (cid:3) 8 and (cid:2)(cid:4)8(cid:2)(cid:3)8. Second, one absolute value equation can yield two solutions. For example, if we solve the equation (cid:2)x(cid:2) (cid:3) 2, not only does x (cid:3) 2 but also x (cid:3) (cid:4)2 because (cid:2)2(cid:2) (cid:3) 2 and (cid:2)(cid:4)2(cid:2) (cid:3) 2. Third, absolute value is defined as the distance, without regard to di- rection, that any number is from 0 on the real number line. Consider a real number a number formula for the distance on the real number line as (cid:2)k (cid:4) 0(cid:2), in which k is that has no imaginary part; a set composed of any real number. Then, for example, the distance that 11 is from 0 would all the rational and irra- be 11 (because (cid:2)11 (cid:4) 0(cid:2) (cid:3) 11). Likewise, the absolute value of 11 is equal tional numbers to 11. The distance for (cid:4)11 will also equal 11 (because (cid:2)(cid:4)11 (cid:4) 0(cid:2) (cid:3) (cid:2)(cid:4)11(cid:2) (cid:3) 11), and the absolute value of (cid:4)11 is 11. Thus, the absolute value of any real number is equal to the absolute value of its distance from 0 on the number line. Furthermore, if the absolute value is not used in the above formula (cid:2)k (cid:4) 0(cid:2), the result for any negative number will be a negative distance. Absolute value helps improve formulas in order to obtain realistic solutions. SEE ALSO Number Line; Numbers, Real. Michael Ota Absolute Zero In mathematics, there is no smallest number. It is always possible to find a number smaller than any number given. Zero is not the smallest number because any negative number is smaller than zero. The number line extends to infinity in both the positive and negative directions. However, when 3