Mathematics Vol 1 (Ab-Cy)

207 Pages·2002·8.46 MB·English

by  Barry Max Brandenberger

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