Some History of Algebra

David LevineWoodinville High School

Early Algebra

http://www.archaeowiki.org/Image:Rhind_Mathematical_Papyrus.jpg

http://commons.wikimedia.org/wiki/File:Egyptian_A%27h-mos%C3%A8_or_Rhind_Papyrus_%281065x1330%29.png

The Rhind Papyrus

• Egypt: The Rind Papyrus (1650 B.C.) solved linear equations. The Cairo Papyrus (300 B.C.) solved simple quadratic equations

• Babylonia: Knew the quadratic formula by 1600 B.C.

• Greek algebra before 250 A.D. was based on geometry

• All of this early algebra was “rhetorical” – it used only words and no symbols

Indian and Arabic Mathematics

• India took Greek mathematics and developed early symbolic methods

• Arab mathematicians extended and spread Indian algebra to Europe

• The word "algebra" is named after the Arabic word "al-jabr" from the title of Persian mathematician Muhammad ibn Mūsā al-khwārizmī’’s 820 book. The word Al-Jabr means "reunion".

from the book al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala (The book of Summary Concerning Calculating by Transposition and Reduction)

Algebra Flourishes in Europe

• As Europe awoke from the dark ages, what we know as algebra began to develop

• It took hundreds of years for algebra’s modern symbols to evolve

Nicole Oresme (c. 1323-1382)• French economist, mathematician,

physicist, astronomer, philosopher, psychologist, musicologist, and theologian

• Advisor, chaplain, and chief secretary to King Charles V

• First to use fractional exponents• May have discovered the rules

(but he didn’t use modern notation)

mnnmmnnm xxxxx

http://sententiaedeo.blogspot.com/2010/07/galileos-giant-nicole-oresme.html

More Early Examples

Christoff Rudolff Coss (1525)addition of radicals

Simon Stevin (1585)multiplying decimals

François Viète (1540-1603)

• A lawyer by training, he was an amateur mathematician in the court of King Henry IV of France

• Known as “The Father of Algebra”

• Introduced the first systematic algebraic notation in 1591

• Used letters for constants and unknowns as in

In artem analyticam Isagoe (1591)

René Descartes (1596-1650)• Descartes used algebra

to describe points, lines, and circles geometrically using a coordinate system

y

x ExamplesPoint (-2, 1)Line y = x + 2Circle x2 + y2 = 100

3

1

Evariste Galois (1811-1832)

• A gifted mathematician from an early age• At age 16 his school wrote of him:

It is the passion for mathematics which dominates him, I think it would be best for him if his parents would allow him to study nothing but this, he is wasting his time here and does nothing but torment his teachers and overwhelm himself with punishments.

and described him as “singular, bizarre, original and closed”

• Opposed the Royalists during the French Revolution

• Was killed in a duel at age 21• Some say the duel was over a woman

(Stephanie-Felice du Motel), other say it was fought for the cause of the revolution

Galois’ Mathematics

• The night before he died, he tried to put together his theories for posterity and wrote in the margin:There is something to complete in this

demonstration. I do not have the time. • His work on Group Theory was scorned

during his lifetime but it became the basis of modern mathematics

• Group theory expresses the formal rules of algebra

Group• Expresses the formal rules of algebra• Examples: the integers under addition, and the

set {0, 1, 2, .. 11) under clock algebra

• Any group G has these four properties:Property Abstract Definition Integer Example

G = the set of integers Clock Algebra Example G = { 0, 1, 2, … 11 }

Closure For all a, b in G, the result of a ○ b is also in G

4 + 11 = 15, which is in G 4 + 11 = 3, which is in G

Associativity For all a, b and c in G, (a ○ b) ○ c = a ○ (b ○ c).

(3 + 5) + 8 = 8 + 8 = 16 3 + (5 + 8) = 3 + 13 = 16

(3 + 5) + 8 = 8 + 8 = 4 3 + (5 + 8) = 3 + 1 = 4

Identity Element

There exists an element I in G such that for all a in G, I ○ a = a and a ○ I = a.

3 + 0 = 3 3 + 0 = 3

Inverse Element

For each a in G, there exists an element b in G such that a ○ b = I and b ○ a = I

3 + (–3) = 0 3 + 9 = 0

A Ring is a Group

• A ring is a group with both addition and multiplication operations

• Addition is commutative and has an inverse operation (there are negative number)

• Multiplication isn’t necessarily commutative and it doesn’t have an inverse (there’s no division)

• Multiplication distributes over addition• Examples: rationals, reals under these limits

3 (4 + 5) = 12 + 8, but 12 + 8 ≠ 8 + 12

A Field is a Ring

• A ring is a field where multiplication is commutative and has an inverse (division is allowed)

• Examples: rationals, reals, both with division• The rules of algebra we learn in school are mostly for the

field of real numbers• In college math, the term algebra usually refers to the

study of the structure of groups, rings, and fields. Group theory allows proofs of complicated questions such as Fermat’s last theorem:

Group Theory in Chemistry

• A point group is a set of symmetry operations forming a mathematical group, for which at least one point remains fixed under all operations of the group.

• A crystallographic point group is a point group which is compatible with translational symmetry in three dimensions.

• There are a total of 32 crystallographic point groups, 30 of which are relevant to chemistry.

(http://en.wikipedia.org/wiki/Molecular_symmetry)

Some Common Point GroupsPoint group Symmetry elements Simple description Illustrative species

C1 E No symmetry, chiral CFClBrH, lysergic acid

Cs E σh Planar, no other symmetrythionyl chloride,

hypochlorous acid

Ci E i Inversion centeranti-1,2-dichloro-1,2-

dibromoethane

C∞v E 2C∞ σv linearhydrogen chloride,

dicarbon monoxide

D∞h E 2C∞ ∞σi i 2S∞ ∞C2 linear with inversion centerdihydrogen, azide, anion, carbon dioxide

C2 E C2 "open book geometry," chiral hydrogen peroxide

C3 E C3 propeller, chiral triphenylphosphine

C2h E C2 i σh Planar with inversion center trans-1,2-dichloroethylene

C3h E C3 C32 σh S3 S3

5   Boric acid

C2v E C2 σv(xz) σv'(yz) angular (H2O) or see-saw (SF4)water, sulfur, tetra-fluoride, sulfuryl fluoride

C3v E 2C3 3σv trigonal pyramidalammonia, phosphorus

oxychloride

C4v E 2C4 C2 2σv 2σd square pyramidal xenon oxytetrafluoride

Fermat’s Last Theorem

Galois’ work paved the way to proving Fermat’s Last Theorem. In 1637, Pierre de Fermat conjectured:

If an integer n is greater than 2, then there are no integers a, b, and c that solve

an + bn = cn

Example: If n = 2 then 32 + 42 = 52

Example: There are no integers a, b, c that

solve a3 + b3 = c3

Pierre de Fermatc. 1604-1665French lawyer and amateur mathematician

Fermat’s proved his theorem for n = 4, others including Sophie Germain (1776–1831) proved it for other values of n.

In 1993, Andrew Wiles (1953–present) very dramatically proved Fermat’s Last Theorem when n is any integer.

Fermat’s Last TheoremMATH RIOTS PROVE FUN INCALCULABLEby Eric Zorn • News Item (June 23, 1993) -- Mathematicians worldwide were excited and pleased

today by the announcement that Princeton University professor Andrew Wiles had finally proved Fermat's Last Theorem, a 365-year-old problem said to be the most famous in the field.

• Yes, admittedly, there was rioting and vandalism last week during the celebration. A few bookstores had windows smashed and shelves stripped, and vacant lots glowed with burning piles of old dissertations. But overall we can feel relief that it was nothing -- nothing -- compared to the outbreak of exuberant thuggery that occurred in 1984 after Louis DeBranges finally proved the Bieberbach Conjecture.

• "Math hooligans are the worst," said a Chicago Police Department spokesman. "But the city learned from the Bieberbach riots. We were ready for them this time."

• When word hit Wednesday that Fermat's Last Theorem had fallen, a massive show of force from law enforcement at universities all around the country headed off a repeat of the festive looting sprees that have become the traditional accompaniment to triumphant breakthroughs in higher mathematics.

• Mounted police throughout Hyde Park kept crowds of delirious wizards at the University of Chicago from tipping over cars on the midway as they first did in 1976 when Wolfgang Haken and Kenneth Appel cracked the long-vexing Four-Color Problem. Incidents of textbook-throwing and citizens being pulled from their cars and humiliated with difficult story problems last week were described by the university's math department chairman Bob Zimmer as "isolated."

Algebra Can Prove Anything?

• Victorian Hubris: By the end of the 19th century, mathematicians began to believe that by combining logic and algebra, they could prove any logical algebraic question

• Algebra is limited: Kurt Gödel proved in 1931 that some algebraic questions can’t be answered