Imagine listing all real numbersbetween 0 and 1 in any order.

You can always make an unlisted real number by changing every

digit on the diagonal,e.g., change .8731… to .9842…

Suppose were rational

is even is even

is even is even

is not reduced

is not rational

is divisible by 4

, reduced

[from Nagel and Newman, Gödel's Proof]

Fun arithmetic with thenumber nine.

Fun arithmetic with thenumber seven.

A magic square.All rows, columns, and

diagonals have the same sum.

The ratio of the circumferenceof a circle to its diameter is pi.

Pi is transcendental, i.e.,irrational and non-algebraic.

Area and volume formulas.Archimedes solved the sphere.

Pi, expressed asan infinite series andan infinite product.

Stirling's approximationof n factorial.

Euler's gamma function givesfactorials for integers but hassurprising values for fractions.

The sum of the numbersfrom 1 to n.

The product of the numbersfrom 1 to n is called n factorial.

A prime number is divisibleonly by one and itself.

The sieve of Eratosthenesfinds primes.

The prime number theoremof Gauss and Legendre

approximates the numberof primes less than x.

The zeta function of Euler andRiemann, expressed as an

infinite series and a curiousproduct over all primes.

The binomial theorem expandspowers of sums.

The binomial coefficient is thenumber of ways to choose

k objects from a set ofn objects, regardless of order.

Pascal's triangle shows thebinomial coefficients.

The quadratic equationdefines a parabola.

The trigonometric functions.Another form of

the Pythagorean theorem.Proof that the square root

of two is irrational.The Pythagorean theorem.A proof by rearrangement.

The golden rectangle,a classical aesthetic ideal.Cutting off a square leavesanother golden rectangle.

A logarithmic spiral isinscribed.

The golden ratio, phi.The ratio of a whole to its

larger part equals the ratio ofthe larger part to the smaller.phi is irrational and algebraic.

The pentagram contains manypairs of line segments that

have the golden ratio.The golden ratio, expressed

as a continued fraction.

Each Fibonacci number is thesum of the previous two.

The number of spirals in asunflower or a pinecone is a

Fibonacci number.

The ratio of successiveFibonacci numbers approaches

the golden ratio.An exact formula for thenth Fibonacci number.

Napier's constant, e,is the base of natural

logarithms and exponentials.e is transcendental.

Euler's formula relatingexponentials to sine waves.A special case relating the

numbers pi, e, and theimaginary square root of −1.

Calculus, developed byNewton and Leibniz, is based

on derivatives (slopes) andintegrals (areas) of curves.

The derivative of ex is ex.The integral of ex is ex.

The Gaussian ornormal probability distribution

is a bell-shaped curve.

Gibbs's vector cross product.Del operates on scalar and vector

fields in 3D, box in 4D.e, expressed as a limitand an infinite series.

The five regular polyhedra.Euler's formula for the number of

vertices, edges, and facesof any polyhedron.

Fractals of Mandelbrot,Koch, and Sierpinski have infinite

levels of detail.

Cantor's proof that the infinityof real numbers is greater than

the infinity of integers.

Gödel proved thatif arithmetic is consistent,

it must be incomplete,i.e., it has true propositionsthat can never be proved.

The Möbius strip hasonly one side.

The Klein bottle's insideis its outside.

The hypercube.Schläfli's formula for vertices,

edges, faces, and cells ofany 4-dimensional polytope.

Math Gems

© 2003-2016 Keith Enevoldsen thinkzone.wlonk.com Creative Commons Attribution-ShareAlike 4.0 International License

An assortment of mathematical marvels.

To find out more, look it up on the web or in the library.