A Brief History of Algebra The Greeks: Euclid, Pythagora, Archimedes Indian and arab mathematicians Italian mathematics in the Renaissance The Fundamental Theorem of Algebra Hilbert s problems 1
Pythagoras, 500 BC Pythagorean triplets: known to babylonians, mayans and hindus. a 2 + b 2 = c 2 Examples: (120, 119, 169), (3456, 3367, 4825), (13500, 12709, 18541). These appear in the Plimpton 322 tablet, Hammurapi dinasty. Clearly they must have had the algorithm to construct them. The triplets appear on manuals for construction of temples and sacred buildings. Irrational numbers: the proof 2 is not rational. 2
Euclid, 300 BC The 13 books of the elements describe geometrical facts about triangle circles and other planar and spatial figures. It is the first axiomatic exposition of mathematics (nowadays the only approach used). Parallel postulate: Given a line l and a point P not on l, then there exists exactly one line through P that does not meet l. Euclid felt unconfortable in using this axiom and tried to prove as much as possible without using it. Non-euclidean geometry: Gauss, Bolyai, Lobatchevski. 3
Archimedes 287-212 BC Diophantus 250 BC Archimedes was the first to use the symbol π and gave the approximation: 3 + 10 71 < π < 3 + 1 7 Archimedes invented the fluxion method to compute the volume of certain solids. This is closely related to integration as we know it and contains the seminal idea of limit. Diophantus was the first to examine the integral and rational solutions of equations of type: x n + y n = z n, x 2 Ny 2 = ±1, Last Fermat s Theorem Pell s equation 4
Indian and Arab mathematics, V I XII century AD Bramagupta (VI century) made astonishing astronomical predictions on the distant stars. Bhaskara (1114-1813 AD) discovered the Cakravala, and algorithm to solve the Pell s equation (the complete treatment was done by Lagrange 1736-1813). The arabs carried through the centuries the books by Euclid, which were never translated into latin! Al-Kwaritzmi (780-850 AD) wrote the formula for the solution of the second degree equation (known much before his time) in his book Al-jabr. Any formula is proved algebraically and/or geometrically. 5
Beginnings Algebra in Europe Fibonacci (Filio Bonacci, 1180-1240) published three important works: 1. Liber Abbaci (1202) 2. Flos (1225) 3. Liber Quadratorum Types of questions addressed: Find a number X such that X 2 +5 and X 2 5 are both squares. The Fibonacci Numbers: 1,1,2,3,5,8,... Formula: X N = X N 1 + X N 2, X 0 = X 1 = 1 X N = (1 + 5/2) N + (1 5/2) N Problem: How many sheeps can we have after 5 years if we start with one sheep and each sheep produces one every year and from the second year on is productive? 6
Fra Luca Pacioli (1445-1514) wrote Summa de aritmetica, geometrica proportioni e proportionabilita in italian. He started using notation for expressions like: 40 320 RV 40mR320 R meant radical, RV meant apply radical to the whole expression and so on. 7
The Bologna s School: cubic and biquadratic equations Scipione Del Ferro (Bologna 1465-1526) solved the equation: X 3 + px = q but did not publish his results! Antonio Maria Fior (Del Ferro s pupil) and Niccolo Tartaglia issued public challenges to each other: the loser had to pay dinner for 30 people. Gerolamo Cardano (Milan 1501-1576) published the book Ars Magna with the solution of third and second degree polynomial equations. These formulas are now known as Cardano s formulas. Cardano gave the first definition of imaginary number. The complex numbers were later extensively studied by Gauss (1777-1885). 8
The Fundamental Theorem of Algebra Given any polynomial P with complex coefficients, P(z) = z N +a 1 z N 1 +a 2 z N 2 +... a N 1 z+a N there are unique (up to a permutation) complex numbers r 1... r N, not necessarily distinct, such that: P(z) = (z r 1 )(z r 2 )...(z r N ). The methods used by Gauss to prove this theorem form the foundation of the theory of functions of a complex variable. No algebraic proof is available! 9
Real version of the fundamental theorem of algebra: Given any polynomial P with real coefficients, P(z) = z N +a 1 z N 1 +a 2 z N 2 +... a N 1 z+a N then: P(z) = (z r 1 )(z r 2 )...(z r k ) (z 2 + 2s 1 z + t 1 )...(z 2 + 2s m z + t m ) where r i, s j, t l are real numbers and either k or m can be zero. No formulas are available for degree greater than 4. This remarkable discover is due to Galois (1811-1832) who founded Galois theory (at age 17!), Abel (1802-1829), Ruffini (1765-1822). These are the so called no go theorems. 10
Hilbert problems The birth of modern algebra (around 1900) is regarded to take place with Hilbert basis theorem: If A = C[x 1... x n ] is the algebra of polynomials in the variables x 1... x n with coefficients in C, then every ideal I A is finitely generated. During the Hilbert s presentation of the proof of this result Gordan exclaimed: This is not mathematics, it is theology! Hilbert presented in 1900 a list of 23 problems for mathematicians to work on. The second problem is to give to mathematics a sound foundation so that every mathematical statement can be resolved to be either true or false. To everybody great surprise Goedel (1906-1978) proved that in mathematics there 11
are statements that are true but cannot be demonstrated to be so. This is the famous Goedel incompleteness Theorem. Some of the Hilbert problems are keeping mathematicians busy today like the 8 t h problem also known as the Riemann hypothesis: The real part of any non-trivial zero of the Riemann zeta function is.