# Great Ideas in Mathematics

Mathematics is one of the oldest human pursuits, and a part of every educational program in recorded history. Plato valued geometry and arithmetic, applied math and theoretical math. "Let no one ignorant of geometry enter my door," was inscribed on his Academy of Learning. He held that computational skills were necessary to both the merchant and the soldier, "who must learn the art of numbers or he will not know how to array his troops." And for the philosopher, he said that, "arithmetic has a very great and elevating effect, compelling the mind to reason about abstract number."

Such an ancient discipline, still vigorous today, necessarily contains some very deep and influential ideas. The Great Ideas, the Best of the Best, include these five topics:

 Analytic Geometry    A merger of two ideas with enduring consequences. Modeling    Connecting the concrete world with the abstract ideas of the mind. Calculus    Mathematics' contribution to the Renaissance, calculus supports modern science and technology. Probability and Statistics    Being precise about uncertainty. Experiment, Conjecture, and Proof    The essence of the mathematical enterprise.

Many important and beautiful ideas are found in mathematics, some of which fall under the heading of Theorem and Proof. Other ideas are more general: procedures, trends, approaches to problems. There are also ideas that have proved useful in other fields: an application of mathematics can have lasting effects too. A partial list of other important ideas is below. Each makes for interesting reading and exploration.

Theorems and Proofs General Topics - Math & History
• The area of a circle
• The volume of a sphere
• The volume of a pyramid
• The Pythagorean Theorem
• The irrationality of the square root of 2
• There are exactly five regular polyhedra
• The Euclidean algorithm
• Any proof from Euclid's Elements
• The fundamental theorem of arithmetic
• The fundamental theorem of algebra
• Ratios of the Fibonacci sequence approach a limit
• The number Pi is transcendental
• The prime number theorem
• The brachistochrone is the curve of fastest descent
• The curve of the St. Louis Arch
• The intermediate value theorem
• The theorem of Lagrange
• It is impossible to trisect an angle
• It is impossible to square a circle
• It is impossible to duplicate a cube
• Fermat's last theorem
• The four color theorem
• There is no Eulerian path over the Königsburg bridges
• The isoperimetric problem (and solution)
• Taylor's theorem
• The strong law of large numbers
• Alternate theories of logic
• Babylonian mathematics
• The influence of Euclid
• The genius Archimedes
• The "cult" of Pythagoras
• The Hindu-Arabic numeral system
• Fibonacci and a tale of two rabbits
• The surprising solution of the cubic equation
• Descartes marries geometry to algebra
• Lobachevsky divorces geometry from reality
• Kepler finds geometry in the heavens
• Napier's "bones"
• The strange algebra of matrices
• Cantor's infinite arithmetic
• Great mathematicians and their mathematics
General Topics - Applications
• Astronomy
• Space Exploration
• Medicine
• Population biology
• Setting environmental standards
• Weather prediction
• Cryptology
• Music
• Art & Architecture
• Design of Buildings, Bridges, Airplanes etc.
• Marketing, Finance
• Scheduling
• Deciding authorship of disputed manuscripts
• Census, polls, etc.
• Setting odds on sporting events