**600 B.C.**

**750 A.D.**

**1700 A.D.**

**37000 years ago**

**15th century**

**The History of Mathematics in a Nutshell**

5,000 BC

Language began to develop, and so did mathematics.

Greek Mathematics 600 B.C. to 450 A.D.

Probably began around 600 B.C., but that date is debated.

Thales and Pythagoras are considered the first Greek mathematicians

Continued to develop until about 400 A.D.

It was generally people of means who studied mathematics

Typically focused on geometric problems

Studied ratios rather than quantities like length

Their mathematics was probably influenced by the Babylonians

Hindu Mathematics 200 B.C. to 1250 A.D.

They used:

Decimal numeration system -- the system we still use today.

Base 10 place value system.

They made significant contributions to trigonometry

They studied linear equations with whole number solutions

They did not provide proofs, or methods, or derivations in their writings.

Arabian Mathematics 650 A.D. to 1200 A.D.

The House of Wisdom was founded in the 9th century to further the intellectual growth of the Arabic empire.

The early mathematicians adopted Euclid's form of proving propositions.

They used polynomials to solve certain algebraic problems.

They did not use, or consider, negative numbers.

Unlike the Greeks, they allowed for units of measurement.

Their main areas of study included trigonometry, geometry, algebra, and combinatorics.

Dark Ages 450 A.D. to 1120 A.D.

The political and social stability of the 10th century sparked the emergence of "cathedral schools" where mathematics was offered for study (but there was not lot of interest in the subject among students).

Many mathematical works were first translated from Arabic to a common language, and then again into Latin.

Universities were established in major European cities in the 13th and 14th centuries, but there was still little interest in mathematics.

Trade influenced the growth of mathematics.

Period of Transmission 950 A.D. to 1500 A.D.

Mayas developed a base-twenty place-value system.

Europeans develop methods of navigation and long-distance travel, bringing trigonometry and astrology to the forefront of mathematical importance and study.

Europeans set up schools and spread their methods of mathematics (as well as other aspects of their culture).

The rise of the merchant class called for more attention to algebra and computation.

Discoveries in geometry affected art, and the rise of linear perspective, a way of making artwork appear three dimensional.

Modern (first half) 1450 A.D. to 1700 A.D.

Algebra takes center stage.

Mathematical literature was still entirely rhetorical (words rather than symbols).

No standard notation had been developed yet.

Several attempts were made to solve cubic equations

Mathematicians were largely individually sponsored by patrons, and competed in the name of their patrons, and so kept much of their work secret.

No one could figure out how to solve the general quintic equation.

Theory of polynomials and roots evolved.

Coordinate Geometry was developed.

Link between geometry and algebra was established.

The Oldest Known Remnants of Mathematical Importance.

Found in Africa

As old as 37,000 years

It was used as a method of record keeping

It was also used for tax purposes

It was used to measure:

fields

volume of baskets

the number of workers needed for tasks

Greek Mathematicians

Thales -- 600 B.C. considered the first Greek mathematician

Pythagoras -- 500 B.C. known for the Pythagorean Theorem and for his work with incommensurable ratios.

Euclid-- 300 B.C. known for his writing "Elements", a compilation and proofs of Greek mathematics and conjecture.

Archimedes -- 250 B.C. wrote about areas and volumes of curved figures.

Apollonius -- 200 B.C. wrote about conic sections.

Diophantus -- 130 A.D. wrote about solving algebraic problems.

Pappus -- mid-4th century, wrote a collection and commentary of Greek mathematical works.

Hindu Mathematicians

Aryabta -- early 6th century A.D. first known Hindu mathematician

Brahmagupta and Bhaskara -- 7th century, first mathematicians to recognize negative numbers.

Bahskara -- 12th century, considered one of the most important Hindu mathematicians for his work in developing a method to provide whole number solutions to nx^2 + b = y^2.

Arabic Mathematicians

Muhammad Ibn Musa Al-Khwarizmi -- mid 9th century, explained the decimal place value system to his readers, and wrote other influential books.

Omar Khayyam -- 1048 A.D. worked on solving equations of degree 3.

Al-Kashi -- 14th century, worked on approximating the nth root of a number.

Nicole Oresme -- 1320 A.D. studied kinematics, and ratios, and presented graphical representations of changing quantities (mathematical modeling).

Leonardo of Pisa -- Published a "Book of Calculations" in 1202 A.D., which explained Hindu-Arabic numeration.

Luca Pacioli -- 1445 A.D. created practical mathematics for the merchants of Italy.

European Mathematicians of the Dark Ages

Modern Mathematicians

Robert Recorde -- 1510 A.D. wrote about mathematics in an entirely rhetorical manner.

Scipione del Ferro -- 1465 A.D., and Tartaglia -- 1500 A.D. were Italian mathematicians who made crucial breakthroughs in solving cubic equations.

Cardano -- 1501 A.D. worked with Tartaglia, and then betrayed him, revealing his secret methods and revealing his own expansion of those methods.

Rafael Bombelli -- 1526 A.D. expanded on Cardano's method to work with roots of negative numbers.

Francois Viete -- 1540 A.D. was a code-breaker for France, and was first to use letters to represent unknown values; he also promoted algebra.

Rene Decartes --1596 A.D. brought algebra to a mature state, developed the notation we use today, and took things out of context of dimension.

Pierre de Fermat -- 1601 A.D. helped develop coordinate geometry, and posed several number theory questions (which he also attempted to solve).

Modern (second half) 1700 A.D. to present

Many scholars were trying to unlock the secrets of the universe.

Bernoulli, Leibniz, Newton, L'Hospital, Cavalieri, Decartes, Fermat, and other scholars developed what we know as applied mathematics, and calculus, as a way of explaining the workings of the universe.

Anglican Bishop George Berkeley questioned the foundations of mathematics as sound, and this sparked controversy and also pushed many mathematicians to seek a firmer mathematical foundation.

Galileo Galilei -- 1564 A.D. was an astronomer who used mathematics to explain the motion of the universe.

Leonhard Euler -- 1707 A.D. was the greatest mathematician of his time, he explored many fields of study, and many branches of mathematics, refining works done by other mathematicians, and putting mathematics into detailed textbooks. He also invented e (the notation).

Pierre Laplace -- 1749 A.D. was an applied mathematician who wrote famous books on mechanics and probability.

Joseph-Louis Lagrange -- 1736 A.D. wrote a book on how and why things move.

Carl Gauss -- 1777 A.D. was a child prodigy who developed the Gaussian style of precise and concise proofs.

Augustin Cauchy -- 1789 A.D. was a professor of calculus who rebuilt calculus "the right way"; he introduced definitions and highlighted the Fundamental Theorem of Calculus.

Karl Weierstrauss -- 1815 A.D. was a teacher in Berlin, he completely transformed the basis of calculus through his clear and precise definitions.

Georg Cantor -- 1845 A.D. invented the notion of a set, which eventually became the foundation of all mathematics.

Niels Abel --1802 A.D. discovered there is no formula for the solution of the equation of degree 5.

Evariste Galois -- 1811 A.D. developed group theory hurriedly the night before he died in a duel.

Bernhard Riemann -- 1826 A.D. made discoveries that lead to the development of non-Euclidean geometry.

David Hilbert -- 1862 A.D. outlined what he considered the 23 most important (or going to be important to future generations) unsolved mathematical problems, and he was largely right.

Kurt Godel --1906 A.D. was the first to prove that some things cannot be proven.

Modern (second half) 1700 A.D. to present (cont)

The end of the French Revolution sparked a new drive for education in France.

The student-teacher structure pushed mathematicians toward rigor in their work.

The metric system was developed by the French Academy of Sciences.

There was a strong movement toward rigor and abstraction that brought about abstract algebra, and non-Euclidean geometry.

95% of mathematics was discovered after 1900.

The 20th and 21st centuries brought more focus on abstraction in mathematical study.

The invention of the computer brought with it a deep exploration of mathematical application, fractal geometry being one example.

By the late 20th century mathematics moved heavily toward applications.

Notable Mathematicians from 1700 A.D. to Present

Mathematics Today

Mathematics has found its way into technology such as cell phones, GPS, satellites, marketing, political strategy, CGI etc.

Mathematics is now more abstract than ever before, and also more applicable than ever before.