Mathematicians

In class, we have each been required to do one biography on a famous mathematician. From the presentations our peers give, we record a few facts they believe to be important. Here are our recordings:
__Aryabhata__
 * Quadratic functions always have two roots
 * Negative and irrational roots are possible

__Fibonacci__ __Aristotle__
 * Brought the Hindu-Arabic number system to the west
 * Fibonacci sequence
 * 1,1,2,3,5,8,13,…
 * x 2 + y 2 and x 2 – y 2 are even squares, but not x 4 – y 4
 * A lot of the knowledge and info that he taught was found in the notes that were left behind in The Lyceum
 * Of the people in his life, it was his father and Plato that seemed to capture his interest in the sciences
 * Science and philosophy were the driving forces behind his contributions to mathematics

__Pythagoras__
 * Did not discover the Pythagorean Theorem
 * Pythagoras believed that the soul was immortal and was reincarnated after death
 * The Pythagorean Brotherhood

__Brahmagupta__
 * Father of Zero
 * Introduced arithmetical rules for adding, subtracting, multiplying, and dividing with zero, negative, and positive numbers
 * Made huge advances in cyclic quadrilaterals

__Archimedes__
 * Archimedes’ Principle
 * Approximation of pi
 * Volume of a sphere is 2/3 the volume of a circumscribed cylinder

__Thales__
 * Founder of the School of Natural Philosophy in Miletus
 * Predicted the eclipse of 585 BCE
 * Credited for the development of the 5 theorems:
 * The circle is bisected by its diameter
 * Base angles of an isosceles triangle are equal
 * A pair of vertical angles formed by 2 intersecting straight lines are equal
 * A angle inscribed in a semi-circle is a right angle
 * 2 triangles are congruent if they have 2 angles and one side that are equal

__Euclid__
 * Greek mathematician best known for his work with geometry and the book he wrote: //The Elements//
 * Introduced the Euclidean Algorithm that was used to find the greatest common divisor of a set of numbers

__Augustus De Morgan__
 * The idea of the quantification of the predicate
 * De Morgan’s laws:
 * (A U B) c = A c n B c
 * (A n B) c = A c U B c
 * Developed De Morgan’s Rule for determining the convergence of a mathematical series

__Zeno of Elea__
 * Zeno’s paradoxes helped to pave the way for new branches of math and science, particularly calculus and the idea of an infinite series.
 * Improvement of the proof-by-contradiction method

__Henri Poincaré__
 * Millennium Prize Problem
 * Poincaré Conjecture
 * Acknowledged as a co-discoverer, with Albert Einstein and Hendrik Lorentz, of the special theory of relativity
 * Studied the n-body problem (determining the stability of the Solar System) which resulted in the birth of the chaos theory

__Hypatia of Alexandria__ She edited the work on the conics of Apollonius, which divided cones into different parts by a plane

She edited the third book of her fathers commentary on the //Almagest of Ptolemy//

Inventions
 * Plane astrolabe
 * Graduated brass hydrometer
 * Hydroscope

__Plato__ The founder of the Academy in Athens

Wrote 35 dialogues and 13 letters that have been used to teach a range of subjects such as philosophy, logic, rhetoric, and math

Helped to distinguish between pure and applied mathematics by widening the gap between “arithmetic” and “logistic”

__Apollonius of Perga__ Defined the ellipse, parabola, and hyperbola

Described the motion of the planets and the moon to be eccentric, and experience retrograde at times

Solved the Apollonius Problem

__Eudoxus of Cnidus__ Axiom of Edoxus Magnitudes are said to have a ratio to one another which is capable, when a multiple of either may exceed the other.

A theory of Proportion When a:b & c:d are equal; a/b = c/d if and only if, for all integers m and n, whenever ma < nb then mc < nd, and so on for > and =.

The Method of Exhaustion Method is finding an area of a shape through inscribing a polygon then adding more sides to it until the polygon converges to the area of the containing shape.

__Benoit Mandelbrot__ Showed how fractals can occur in many different places both in mathematics and in nature.

Found the fractal set of complex numbers known as the Mandelbrot Set

Although the idea of fractals existed before he coined the term, Mandelbrot had to come up with new mathematics and computer technology in order to show fractals graphically.

__Waclaw Sierpiński__ He gave the first example of an absolutely normal number

Revolutionized Polish mathematics and helped lay the foundations of the discipline of set-theoretic topology

__Nikolai Lobachevsky__ He was the first mathematician to look for another geometry that proved Euclid’s fifth postulate wrong.

Nikolai founded Non-Euclidean Geometry that stated, ‘For any infinite straight line and any point not on it, there are many other infinitely extending straight lines that pass through and which do not intersect.”

He founded the idea of approximating the roots of algebraic equations. Nikolai defined a function as a correspondence between two sets of real numbers.

__Evariste Galois__ Galois is generally considered to have been the first to develop the G**roup Theory**.

A step further: The **Galois Theory** provides a connection between field theory and group theory.

Galois moved to a more abstract perspective that was picked up by other mathematicians and eventually led to what is now known as **Abstract Algebra**.

__Diophantus of Alexandria__ The creation of Arithmetica

The profound influence on algebraic notation

Diophantine equations

__Muhammad Ibn Musa Al-Khwarizmi__ Al-Khwarizmi’s first book was very influential and provided the basis for the Algebra that we use today. The word “algebra” came from the book’s concept of al-jabr.

His second influential book introduced Hindu numerals along with many other “new arithmetic” concepts.

The translation of Al-Khwarizmi’s name led to the development of the word “algorithm”.

__Carl Friedrich Gauss__ Successfully constructed the “17-gon” with a straight edge and compass

Wrote Disquisitiones Arithmeticae (great impact on mathematics)

Calculated of the orbit of the planet Ceres using the least squares approximation method

__Alexander Grothendieck__ Developed unifying themes in: algebraic geometry, number theory, topology and functional/complex analysis

Proved Riemann-Roch theorem algebraically

Introduced ‘Theory of Schemes’ which allowed him to solve two of Andre Weil’s number theory conjectures

__Niels Henrik Abel__ Overcame tragedy and opposition to make important mathematical discoveries

Proved the quintic equation had no solution

Provided a precedence for modern presentation of a proof

__Carl Gustav Jacob Jacobi__ Worked closely with Legendre Leading expert on elliptic functions

Discovered new ideas in number theory (Jacobi Symbol)

Found three new elliptic function formulas by substituting trigonometric functions into Legendre’s existing formulas

__Jacob Bernoulli__ Bernoulli developed the properties of //Bernoulli numbers//
 * These numbers arise in the series expansions of trigonometric functions, and are extremely important in number theory and analysis

Wrote the book //Ars Conjectandi// in 1713, discussing probability
 * //Ars Conjectandi// was the first statistics and probability book to be published
 * He looked at the relationship between theoretical probability and actual results (simulations or situations)
 * Jacob did not see each outcome as having equal rate of occurrence. He instead saw that outcomes could vary in rates of occurrence especially when discussing human life spans, health, etc.,
 * Unfortunately, his book was incomplete but laid the path for future statisticians to pick up where he left off.

Law of Large Numbers
 * Bernoulli added upon Cardano’s idea of the Law of Large Numbers
 * Bernoulli asserted that if a repeatable experiment had a theoretical probability (p) of turning out in a certain “favorable” way, then for any specified margin of error the ratio of favorable to total outcomes of some (large) number of repeated trials of that experiment would be within that margin of error.
 * By this principle, observational data can be used to estimate the probability of events in real-world situations.

__Johann Bernoulli__ Wrote Hydraulica, linking Newton’s concept of force to hydrodynamics (a turning point for hydrodynamics)

Because of Johann’s support, Liebniz’ calculus became the preferential from of mathematics in Europe (except England)

His second solution to the isoperimetric problem formed the foundation of calculus of variations

__Sophie Germain__ Made significant contributions to the study of underlying mathematical laws concerning vibrations of elastic surfaces

Contributed to the solving of Fermat’s Last Theorem, which would allow future mathematicians to progress in the proof’s development.

Opened doors for women in the fields of mathematics and science

__Augustine-Louis Cauchy__ Responsible for introducing the modern standard of rigor in calculus

He formally defined the terms limit, derivative, integral, continuity, and convergence in his book, //The Ecole Polytechnique Course in Analysis//.

__Gottfried Wilhelm Leibniz__ He developed notations for (integral sign) f(x) dx for calculus

He developed the binary system of arithmetic

His role in scientific societies (academies) being created in multiple areas

__Isaac Newton__ Wrote Opticks and discovered that white light is formed from a mix of colored light rays

Published Principia which included Newton’s Three Laws of Motion

Found that gravity was the force that acted upon objects, but could not explain where it came from

__Joseph Lagrange__ Published the ‘mecanique analytique’ which summarized all work on mechanics since Newton using differential equations

The metric system- the addition of a decimal base was largely due to him

Advances in the field of astronomy, including his essay on the Libration of the moon

__David Hilbert__ Hilbert developed many axioms, or rules/laws, for the field of mathematics

Hilbert established the basis for work on infinite-dimensional space, later named Hilbert space

Developed 23 interesting and important math problems that mathematicians are still solving today

__Bernhard Rieman__ Created Riemann Integral-found the limit of an integral of a function, over an interval using the Riemann Sum

Found the relationship between zero and the distribution of primes to discover the Riemann Zeta Function

Founder of Riemannian Geometry, which focused on the study of parallel lines on N-Dimensional shapes

__Pierre-Simon Laplace__

Laplace developed the nebular hypothesis which is the most widely accepted models of how the universe formed

Laplace concluded that any two planets must be in mutual equilibrium which led to the naming of the Laplace Coefficient

Laplace proposed the concept of black holes, saying that there were massive stars whose gravity was so immense that not even light could escape from that star’s surface

__Pierre de Fermat__

Fermat’s Last Theorem: x^n + y^n = z^n has no non-zero integer solutions for x, y and z when n>2, later proved by Andrew Whiles

Co-founded the theory of probability with Blaise Pascal

Light will always follow the shortest possible path

Laid the ground work for differential calculus years before Newton

__Gerolamo Cardano__ First to publish solutions for the cubic and quartic equations in his book Ars Magna

Developed a method to solve negative square roots which led to imaginary numbers

Took the first systematic look at probability in his “games on chance” book

__Blaise Pascal__ Pascal invented the Pascaline, one of the first mechanical computing devices

He wrote an important treatise on what is now Pascal’s Triangle

Pascal’s wager: “If God does not exist, one will lose nothing by believing in him, while if he does exist, one will lose everything by not believing.”

__Emilie du Chatelet__ Equation of energy: e~mv^2

Book about physical science called //Institution du physiques//

Translation of Newton’s //Principia Mathematica// into French with her own commentary

__Georg Cantor__ Cantor’s Theorem: The cardinality of the set of all subsets of any set is strictly greater than the cardinality of the set

Creator of set theory which was foundational for modern mathematics

Established the importance of one-to-one correspondence between sets

__Rene Descartes__ Wrote Geometrie, which laid the basic foundation for analytic geometry

Began using lowercase letters from the beginning of the alphabet to symbolize known quantities and lowercase letters from the end of the alphabet to symbolize unknown quantities

Used exponents on a variable to represent powers of that variable (allowed him to graph curves that contained multiple powers)