the fascinating story behind our mathematicslawson/epis.pdf · thales was the grandfather of greek...

The Fascinating Story Behind OurMathematics

Jimmie Lawson

Louisiana State University

Story of Mathematics – p. 1

IntroductionIn every civilization that has developed writing wealso find evidence for some level of mathematicalknowledge.

Math was needed for everyday life: commercialtransactions and accounting, government taxes andrecords, measurement, inheritance.

Math was needed in developing branches ofknowledge: astronomy, timekeeping, calendars,construction, surveying, navigation

Math was interesting: In many ancient cultures(Egypt, Mesopotamia, India, China) mathematicsbecame an independent subject, practiced by scribesand others.

Old Babylon (2000-1700 B.C.)The Mesopotamian scribes represented numbers usinga place value system based on sixty.

For example, the number 742 would be written

I HH(600 + 60 + 60) II HH(10 + 10 + 1 + 1).

The Babylonians could do arithmetic, includingfractions, solve linear equations, extract square roots,and find large Pythagorean triples of integers:

(2291)2 + (2700)2 = (3541)2

They also studied astronomy (remember the 3wisemen), which needed elementary trigonometry.

I HH(600 + 60 + 60) II HH(10 + 10 + 1 + 1).

(2291)2 + (2700)2 = (3541)2

I HH(600 + 60 + 60) II HH(10 + 10 + 1 + 1).

(2291)2 + (2700)2 = (3541)2

I HH(600 + 60 + 60) II HH(10 + 10 + 1 + 1).

(2291)2 + (2700)2 = (3541)2

Babylonian Heritage

Our division of hours and degrees into 60 minutes andminutes into 60 seconds is based on Babylonianpractice and their base 60 numbering system.

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We could “babble on” about Babylon, but instead wemove on to speak about...

Babylonian Heritage

Our division of hours and degrees into 60 minutes andminutes into 60 seconds is based on Babylonianpractice and their base 60 numbering system.

6HHj HHH

��

We could “babble on” about Babylon, but instead wemove on to speak about...

Those Incredible GreeksGreek mathematical activity lasted for about 1000years (600 B.C.-400 A.D.). Of all ancient cultures, theGreeks were by far the most influential on ourmathematics and made major contributions,supremely in geometry, but also in number theory.

• Mathematical “truths” must be proved, not justobserved. Mathematics became deductive, notinductive.

• Mathematics begins with basic assumptions,called “axioms,” and definitions and builds onitself to add logical consequences or theoremsuntil a whole theory develops.

Greek Hall of FameThales was the grandfather of Greek mathematics. Hewas from the Greek colony of Ionia (western AsiaMinor, modern Turkey) and lived about 600 B.C.

He taught that proofs are the only reliable basis formathematical knowledge, not observation ormeasurement.

He also demonstrated some basic geometric theorems:

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α = 90◦

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α = 90◦

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α = β

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α = 90◦

The PythagoreansPythagoras and his school flourished about 500 B.C.in a Greek colony in southeast Italy. They weresomething of a secret society. Motto: Everything isnumber.

The Pythagorean school isknown for the Pythagoreantheorem (the world’s most fa-mous theorem).

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a2 + b2 = c2

They also studied musical ratios, irrational ratios(between a side and hypothenuse), and perfectnumbers (numbers that equal the sum of theirdivisors) to name a few.

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Best-Selling AuthorEuclid lived around 300 B.C. in Alexandria, Egypt. Inhis book, the Elements, he summarized 300 years ofGreek mathematics as a systematic and logical theory.Its style and content were enormously influential onlater mathematical tradition, the most influential andwidely read math book ever written.

The first part of the book starts with basic axioms anddefinitions and develops a comprehensive theory ofgeometry, plane and solid geometry.

Other topics covered include divisibility properties ofintegers and other basic number theory, regularpolyhedra, and an advanced theory of ratios.

The Sand ReckonerArchimedes (250 B.C.) lived in Sicily and was thegreatest mathematician of the ancient world. Some ofhis achievements were

• Combined mathematics and mechanics (principleof lever, various laws about physics of water),

• developed method for writing and using verylarge numbers (calculated how many grains ofsand would fill the universe),

• calculated very good approximation to π and avariety of areas and volumes,

• invented important new curves, such as the spiralof Archimedes.

Some “Late-Comers”Ptolemy (100 A.D.) was the greatest of Greekastronomers.

• He wrote the most important book on astronomyuntil Copernicus. It was widely known in Asiaunder the title “The Greatest.”

• Made basic contributions to spherical geometry.

Diophantus (300 A.D.) found solutions of equationsthat were integers or fractions for wide classes ofequations, for example, m2 + n2 = k2. These arecalled diophantine equations. He wrote a book abouthis results that greatly influenced later number theory.

India (500-1200 A.D.)The most famous invention of the Indianmathematicians/astronomers is their base 10numeration system. By 600 A.D. they had ninesymbols for the numbers 1 to 9, introduced placevalue, and had created a symbol, a dot or small circle,to denote an empty place (where we write 0). Theyalso developed methods for doing arithmetic with thissystem.

747 = 7 × 100 + 4 × 10 + 7 × 1.

This numbering system was picked up by the Arabsand made its way west as the Hindu-Arabic numbers.

747 = 7 × 100 + 4 × 10 + 7 × 1.

The Birth of ZeroThe Hindus were the first to think of 0 not only as aplace holder (as in 703), but as a number in its ownright, with arithmetic properties such as 2 + 0 = 2 and7 × 0 = 0.

This required a whole new way of thinking: zero waspromoted from being nothing to being an actualnumber that one could work with.

Early important Indian mathematicians includedAryabhata, Brahmagupta, and Bhaskara, the latter twobeing among the first to work with negative quantities.

The Birth of ZeroThe Hindus were the first to think of 0 not only as aplace holder (as in 703), but as a number in its ownright, with arithmetic properties such as 2 + 0 = 2 and7 × 0 = 0.This required a whole new way of thinking: zero waspromoted from being nothing to being an actualnumber that one could work with.

Early TrigonometryGreek and Babylonian astronomers had introducedtrigonometry, but the Indians made importantcontributions. Greek trigonometric calculations hadrevolved around the chord of an angle and chordtables had been compiled.

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Chord of an angle

Early TrigonometryGreek and Babylonian astronomers had introducedtrigonometry, but the Indians made importantcontributions. Greek trigonometric calculations hadrevolved around the chord of an angle and chordtables had been compiled.

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Chord of an angle

The “Half-Chord” or SineThe Indian mathematicians realized the importance ofstarting with a central angle α, doubling it (2α), andthen taking half the chord of 2α, which they calledthe “half-chord.”

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Chord (α)

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Half-Chord(α)“Half-chord” was (mis)translated into Latin as“sinus,” from which we get “sine,”

sin(α) = (1/2)chord(2α).

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Half-Chord(α)

“Half-chord” was (mis)translated into Latin as“sinus,” from which we get “sine,”

sin(α) = (1/2)chord(2α).

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Chord (α)

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Half-Chord(α)“Half-chord” was (mis)translated into Latin as“sinus,” from which we get “sine,”

sin(α) = (1/2)chord(2α).Story of Mathematics – p. 14

Arab Math (800-1300 A.D.)• By 750 A.D. the Islamic Empire stretched all the

way from western India to parts of Spain. Itsimperial capital, Baghdad, quickly became thecultural center of the empire.

• A large library in Baghdad collected and housedGreek and Indian works together with Arabtranslations. Euclid’s Elements had a hugeimpact.

• The common Arabic language throughout theempire allowed them to collect and build on oneanother’s work.

Arabic ContributionsThe development of algebra was the most importantpart of Arabic mathematics. Arabic mathematicianslearned how to manipulate polynomials, solve variousalgebraic expressions, and much more.

All of this was done entirely with words, with nosymbols at all. Their methods of solution werefrequently geometric.

They also made important contributions to geometry,trigonometry and astronomy, and some work innumber theory and combinatorics (including earlyversions of “Pascal’s Triangle”).

Arabic ArithmeticAl-Khwarizmi (850 A.D.), from the area that is todayUzbekistan, wrote several influential books. One wasan explanation of the decimal place value system forwriting numbers and doing arithmetic, which camefrom India. It was the major source from whichEuropeans learned the new system.

Many translations into Latin began with the words“dixit Algorismi” (“so says Al-Khwarizmi”), so theword algorism came to mean the process of ofcomputing with Hindu-Arabic numbers. The moderncorresponding word is “algorithm,” a “recipe” forsolving some mathematical problem.

Arabic ArithmeticAl-Khwarizmi (850 A.D.), from the area that is todayUzbekistan, wrote several influential books. One wasan explanation of the decimal place value system forwriting numbers and doing arithmetic, which camefrom India. It was the major source from whichEuropeans learned the new system.

Many translations into Latin began with the words“dixit Algorismi” (“so says Al-Khwarizmi”), so theword algorism came to mean the process of ofcomputing with Hindu-Arabic numbers. The moderncorresponding word is “algorithm,” a “recipe” forsolving some mathematical problem.

More on Arabic AlgebraAl-Khwarizmi also wrote a basic book about algebracalled ”al-jabr w’all-muqabala,” which technicallymeant “bone-setting,” but more generally “restorationand compensation.” When this book was translatedinto Latin “al-jabr” became “algebra," the word westill use today.

The most famous of the Arabic mathematicians wasthe Persian Omar Khayyam, who worked on cubicequations. But his fame rested primarily on the factthat he was a talented poet. His poems much laterappeared in English in a famous translation by SirRichard Burton.

More on Arabic AlgebraAl-Khwarizmi also wrote a basic book about algebracalled ”al-jabr w’all-muqabala,” which technicallymeant “bone-setting,” but more generally “restorationand compensation.” When this book was translatedinto Latin “al-jabr” became “algebra," the word westill use today.

The most famous of the Arabic mathematicians wasthe Persian Omar Khayyam, who worked on cubicequations. But his fame rested primarily on the factthat he was a talented poet. His poems much laterappeared in English in a famous translation by SirRichard Burton.

Mathematics Moves WestThe Italian Renaissance included a burst ofmathematical activity as Greek and Arabicmathematics became known. The Arabicmathematicians knew how to solve linear andquadratic equations, but made limited progress onsolving cubic (degree 3) equations. The crucialbreakthrough was made in Italy in a series ofdiscoveries and an accompanying soap opera ofevents.

(1) First Scipione del Ferro (1465-1526) and thenTartaglia (1500-1557) discovered a method of solvingcertain classes of cubic equations, but kept it secret towin public competitions.

Mathematics Moves WestThe Italian Renaissance included a burst ofmathematical activity as Greek and Arabicmathematics became known. The Arabicmathematicians knew how to solve linear andquadratic equations, but made limited progress onsolving cubic (degree 3) equations. The crucialbreakthrough was made in Italy in a series ofdiscoveries and an accompanying soap opera ofevents.

(1) First Scipione del Ferro (1465-1526) and thenTartaglia (1500-1557) discovered a method of solvingcertain classes of cubic equations, but kept it secret towin public competitions.

Act II(2) Girolamo Cardano, a physician by trade,convinced Tartaglia to share the secret of the cubicwith him, promising never to reveal it. But once heknew Tartaglia’s method ...

(3) Cardano was able to generalize it to a method ofsolving any cubic equation. Feeling it was now mainlyhis own work, he no longer felt bound to secrecy, andwrote a book, called Ars Magna (“The Great Art”), inwhich he told all. Tartaglia was furious!!

But there was a “little” problem.

Act IIISometimes Cardano’s methods involved square rootsof negative numbers (imaginary or complex numbers),even when the answer was real.

(4)The missing step was provided by Rafael Bombelli(1526-1572), who developed a method of dealing withsuch expressions that we might call the beginning of atheory of complex numbers, numbers of the forma + bi.

(5) Lodovico Ferrari (1522-1565), a student ofCardano’s, showed how to extend Cardano’stechniques to solve all equations of degree 4.

Algebra Comes to FranceMajor progress took place in algebra in the first halfof the 17th century, primarily at the hands of twotalented Frenchmen, René Descartes (1596-1650), aphilospher, and Pierre de Fermat (1601-1665), alawyer, judge, and amateur mathematician.

(1) Algebraic notation (+,−,×, =, x2) was inventedand solidified into something close to ours. Descartessuggested using letters near the end of the alphabet forunknowns or variables and those near the front forconstants.

(2) No one could figure out how to solve fifth degreeequations, but a general theory of polynomials andtheir roots evolved.

Analytic Geometry(3) Fermat and Descartes linked algebra and geometryusing what we now call “coordinate geometry” or“analytic geometry” (thus we call the coordinate planethe “cartesian plane).” Both Fermat and Descartesused it to study curves by their equations and showedthe power of algebra to solve difficult geometricproblems.

-50 0 50-20

Analytic Geometry(3) Fermat and Descartes linked algebra and geometryusing what we now call “coordinate geometry” or“analytic geometry” (thus we call the coordinate planethe “cartesian plane).” Both Fermat and Descartesused it to study curves by their equations and showedthe power of algebra to solve difficult geometricproblems.

-50 0 50-20

Fermat’s Last Theorem(4) Fermat, after reading the works of Diophantus,introduced a whole new class of algebraic problems innumber theory. His most famous “theorem”(unproved) was that

xn + yn 6= zn

for all positive integers x, y, z and n ≥ 3.

This assertion was not proved true until 1994 byAndrew Wiles of Princeton.

Fermat’s Last Theorem(4) Fermat, after reading the works of Diophantus,introduced a whole new class of algebraic problems innumber theory. His most famous “theorem”(unproved) was that

xn + yn 6= zn

for all positive integers x, y, z and n ≥ 3.

This assertion was not proved true until 1994 byAndrew Wiles of Princeton.

Modern MathematicsToday we have only had time to sketch the history ofour math up to the time of the rise of modernmathematics (1600-2004). With the discovery of thecalculus by Newton and Leibnitz in the 2nd half of the17th century, a deep and wide mathematical streamhas developed that has resulted one of the greatest andmost profound human intellectual achievements and istoday practiced worldwide.

Suggested Reading: A Parrot’s Theorem by DenisGuedj. A novel about a French family who find aparrot and receive a shipment of books that plungesthem into both a mystery and a search into “thefascinating story of mathematics.”

Modern MathematicsToday we have only had time to sketch the history ofour math up to the time of the rise of modernmathematics (1600-2004). With the discovery of thecalculus by Newton and Leibnitz in the 2nd half of the17th century, a deep and wide mathematical streamhas developed that has resulted one of the greatest andmost profound human intellectual achievements and istoday practiced worldwide.

Suggested Reading: A Parrot’s Theorem by DenisGuedj. A novel about a French family who find aparrot and receive a shipment of books that plungesthem into both a mystery and a search into “thefascinating story of mathematics.” Story of Mathematics – p. 25

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