The Mathematical Palette

670 Pages·2004·7.766 MB·English

by  Ronald Staszkow, Robert Bradshaw

5564_Staszkow_Endsheets1 1/14/04 1:27 PM Page 2
A TIME-LINE OF MATHEMATICS
(Dates are approximate)
s
Humans possess number sense and sense ofform Egyptian Hieroglyphic system ofnumeration (3000 B.C.) Rhind (1650 B.C.) and Moscow Papyrus (1850 B.C.) show evidence ofalgebra Chinese books Arithmetic of9 Sections andBook ofChangesshow understanding ofgeometry and permutations (1100 B.C.) Mayan system ofnumeration (300 B.C.) Chinese system ofnumeration,their counting boarduse negative numbers (200 B.C.)Origin ofHindu-Arabic Numerals (150 B.C.) Hindus study math in relation to astronomy Sûyra Siddhãnta — trigonometry (400) Brahmagupta — algebra,negative numbers (625) Symbol for zero (0) — Gwalior,India (870)Abû’l Wefâ — trigonometry tables (920)Omar Khayyan — algebra,cubic equations (1100)Bhãskara — permutations,combinations (1150)Suan-pan abacus widely used in China (1200)Ch’in Chiu-Shao — higher degree equations (1247)Nasîr Ed-dîn al-Tûsî — trig.text (1250)
HINDU/ARABIAN PERIOD (200 B.C. TO A.D. 1250)
PREHISTORIC PERIOD
EGYPTIAN/BABYLONIAN PERIOD (3000 B.C. TO A.D. 260)
TIMES
(TO 3000 B.C.)
GREEK PERIOD (600 B.C. TO A.D. 450)
Humans have ability to tally and count nian sexagesimal system ofnumeration (2300 B.C.) (cid:1)(cid:1)pBabylonians use 258 (2000 B.C.) Babylonians and Egyptians develop arithmetic and practical geometryThales — deductive geometry (600 B.C.)Pythagorean school (540 B.C.)Ionic Greek numeration system (450 B.C.)Platonic Schools (380 B.C.)Eudoxus — method ofexhaustion (370 B.C.)Aristotle — logic,syllogisms (340 B.C.)Euclid — axiomatic geometry (300 B.C.)pArchimedes — areas,volumes,(225 B.C.)Apollonius — conic sections (225 B.C.)Eratosthenes — sieve for primes (200 B.C.)Hipparchus — trigonometry (140 B.C.)Menelaus — spherical triangles (100 B.C.) Ptolemy — the Almagest,trig (125)Diophantus — algebra (250) Pappus — centroids,volumes (300) Hypatia — 1st woman ofmathematics (410)Zeno ofElea — paradoxes ofmotion (450) d Arab scholars preserve mathematics ofthe GreeksWang Hs’iao-t’ung — cubic equations (625) Al-Khowarizmi — the term algebra(825) Many translations ofArabic texts into LatinFibonacci — use ofHindu-Arabic numbers (1202) St.Thomas Aquinas — scholastic logic (1250)
o n
byl u a
Ba nd
Hi
5564_Staszkow_Endsheets1 1/14/04 1:27 PM Page 3
ki
Chu Shih-Chieh — binomial coefficients (1303)Ockham — formal logic (1320)Oresme — fractional exponents (1360) Hindu-Arabic numerals take present form (1479)(cid:2)(cid:3)Chuquet — notation for ,offractions (1484)Calandri — long division process (1491) (cid:1)Rudoff— square root symbol () (1525) Diez Feile — writes 1st math book in Mexico (1556)(cid:1)Recorde — symbol for the equal sign () (1557)(cid:7)Clavius — the dot () for multiplication (1583)Napier — logarithms (1614)Kepler and Briggs — table oflogarithms (1624)Descartes — analytic geometry (1637) Fermat — probability,analytic geometry,number theory,calculus (1629–1654)Pascal — projective geometry,calculating machine,probability,binomial coefficients (1642–1654)Seki Ko(cid:2)wa — determinants magic squares (1685)Newton — calculus (1687) Jakob Bernoulli — theory ofprobability (1713)DeMoivre — actuary math,complex numbers (1720) Maria Agnesi — popular math text book (1748)(cid:8)Euler — uses the symbols i,e,and (1750)First Bank ofthe U.S.(1791)Babbage — steam powered calculator (1823)Lobachevsky — non-Euclidean geometry (1829) Georg Riemann — non-Euclidean geometry,calculus(1850–1854)(cid:2)Weierstrass — absolute value symbol (),foundations ofcalculus (1841–1874)Hollerith — electrical tabulating device (1880) Markov — probability theory,Markov Chains (1907)Federal Reserve Act — U.S.banking system (1913)Ronald Fisher — statistics sampling techniques (1920)Thomas Watson forms IBM (1924) Mauchley and Eckert — ENIAC computer (1945)Gertrude Cox — statistical design (1950)Mandelbrot — fractal geometry (1967)Hewlett Packard — programmable calculator (1974)Team ofMichael Cameron,George Woltman,Scott Kurows(cid:3)13466917et al.discovered a new record prime 21 (2001)Dr.Kanada ofUniversity ofTokyo calculates Pi to 1.241 trillion digits (2002)
OF TRANSMISSION
(1000–1500)
EARLY MODERN PERIOD (1450 TO 1800) MODERN PERIOD (1800 TO PRESENT)
)) ) )))))) )))))))s)))) ))))r)))))))) ) ))))
95 4 975125 1505903u4699 9034e21375040 0 8813
2932 46 485254555758 62636465656666ul68707172 74757681mb8384848486888991 92 95969799
Florence,Italy outlaws Hindu-Arabic numerals (1Burley — formal logic (1 Regiomontanus — trigonometry (1 (cid:2)(cid:3)Widmann — addition (),subtraction () signs (1(cid:4)(cid:5)Apianus — notation for ,offractions (1Cardano — cubic equations (1Rheticus — right triangle trigonometry (1Bombelli — imaginary numbers (1Stevin — decimal fractions (1 (cid:4)Oughtred — multiplication sign (),slide rule (1Cavalieri — indivisibles in calculus (1Desargues — projective geometry (1(cid:6)Wallis — negative exponents,infinity symbol () (1(cid:5)Rahn — division sign () (1Isomura — magic circles (1Barrow — calculus,differentiation (1Leibniz — symbolic logic,calculating machine,calc(1666–1pWilliam Jones — symbol for pi () (1First math book printed in America (1Greenwood — writes 1st American math book (1 Achenwall — the word statistik(1Gabriel Cramer — systems ofequations (1Bayes — statistics,origin ofpolls (1Laplace — probability theory (1uss — statistics,normal curve,the term complex nu(1809–1Carl Jacobi — theory ofdeterminants (1Ada Byron — computer programming (1George Boole — logic,Boolean algebra (1Möbius — topology (1John Venn — logic,Venn diagrams (1Burroughs — practical adding machine (1Whitehead,Russell — Principia Mathematica (1 Lukasiewicz,Post,Wittgenstein — truth tables (1 Kilby,Texas Instruments — integrated circuit (1Gilbert Hyatt — computer microprocessor chip (1Edward Roberts — first personal computer (1ProofofFermat’s Last Theorem by Andrew Wiles (1
a
G
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5564_Staszkow_FM_i-xviii 1/14/04 1:28 PM Page iii
The
Mathematical
Palette
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5564_Staszkow_FM_i-xviii 1/14/04 1:28 PM Page v
TThhee
MMaatthheemmaattiiccaall
PPaalleettttee
T H I R D E D I T I O N
RONALD STASZKOW
Ohlone College
ROBERT BRADSHAW
Ohlone College
Australia •Canada •Mexico •Singapore •Spain
United Kingdom •United States
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CONTENTS
Preface xi
Numbers—Old and New
chapter
1
OVERVIEW 2
A SHORT HISTORY OF NUMBERS AND NUMERALS 2
PROJECTS 4
1.1 Ancient Systems ofNumeration 6
1.2 The Hindu-Arabic System and Fractions 17
1.3 Numeration Systems with Other Bases 23
1.4 The Numbers ofTechnology 31
1.5 Types ofNumbers 39
Summary 48
Review 50
Test 51
Logical Thinking
chapter
2
OVERVIEW 54
A SHORT HISTORY OF LOGIC 54
PROJECTS 55
2.1 Logic,Statements,and Definitions 56
2.2 Inductive and Deductive Reasoning 64
2.3 Symbolic Logic and Truth Tables 74
2.4 Logic and Flowcharts 85
2.5 Logic and Puzzles 93
Summary 103
Review 105
Test 106
vii