Transcript General history of algebra

Some History of Algebra
David Levine
Woodinville High School
The Rhind Papyrus
Early Algebra
• 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
http://commons.wikimedia.org/wiki/File:Egyptian_A%27hmos%C3%A8_or_Rhind_Papyrus_%281065x1330%29.png
http://www.archaeowiki.org/Image:Rhind_Mathematical_Papyrus.jpg
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 alkhwārizmī’’s 820 book. The word
Al-Jabr means "reunion".
from the book alKitāb al-muḫtaṣar fī
ḥisāb al-ğabr wa-lmuqābala (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
x x x
m n
n m
x 
m n
x
mn
(but he didn’t use modern notation)
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
Examples
Point (-2, 1)
Line
y =  13 x + 2
Circle x2 + y2 = 100
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
Closure
Abstract Definition
For all a, b in G, the result of
a ○ b is also in G
Associativity For all a, b and c in G,
(a ○ b) ○ c = a ○ (b ○ c).
Identity
There exists an element I in G
Element
such that for all a in G,
I ○ a = a and a ○ I = a.
Inverse
For each a in G, there exists an
Element
element b in G such that
a ○ b = I and b ○ a = I
Integer Example
G = the set of integers
4 + 11 = 15, which is in G
Clock Algebra Example
G = { 0, 1, 2, … 11 }
4 + 11 = 3, which is in G
(3 + 5) + 8 = 8 + 8 = 16
3 + (5 + 8) = 3 + 13 = 16
3+0=3
(3 + 5) + 8 = 8 + 8 = 4
3 + (5 + 8) = 3 + 1 = 4
3+0=3
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 Groups
Point group
Symmetry elements
Simple description
Illustrative species
C1
E
No symmetry, chiral
CFClBrH, lysergic acid
Planar, no other symmetry
thionyl chloride,
hypochlorous acid
Inversion center
anti-1,2-dichloro-1,2dibromoethane
linear
hydrogen chloride,
dicarbon monoxide
Cs
Ci
C∞v
E σh
Ei
E 2C∞ σv
D∞h
E 2C∞ ∞σi i 2S∞ ∞C2
linear with inversion center
dihydrogen, 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 S35
Boric acid
E C2 σv(xz) σv'(yz)
angular (H2O) or see-saw (SF4)
water, sulfur, tetra-fluoride,
sulfuryl fluoride
C2v
C3v
E 2C3 3σv
trigonal pyramidal
ammonia, 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
Pierre de Fermat
c. 1604-1665
French lawyer and
amateur
mathematician
solve a3 + b3 = c3
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
Theorem
MATH RIOTS PROVE FUN INCALCULABLE
by 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