# Math Gems

An assortment of mathematical marvels. A magic square. All rows, columns, and diagonals have the same sum. The ratio of the circumference of a circle to its diameter is pi. Pi is transcendental, i.e., irrational and non-algebraic. The product of the numbers from 1 to n is called n factorial. Stirling's approximation of n factorial. Euler's gamma function gives factorials for integers but has surprising values for fractions. A prime number is divisible only by one and itself. The sieve of Eratosthenes finds primes. The prime number theorem of Gauss and Legendre approximates the number of primes less than x. The zeta function of Euler and Riemann, expressed as an infinite series and a curious product over all primes. The binomial theorem expands powers of sums. The binomial coefficient is the number of ways to choose k objects from a set of n objects, regardless of order. The golden ratio, phi. The ratio of a whole to its larger part equals the ratio of the larger part to the smaller. phi is irrational and algebraic. The golden rectangle, a classical aesthetic ideal. Cutting off a square leaves another golden rectangle. A logarithmic spiral is inscribed. The pentagram contains many pairs of line segments that have the golden ratio. Each Fibonacci number is the sum of the previous two. The number of spirals in a sunflower or a pinecone is a Fibonacci number. The ratio of successive Fibonacci numbers approaches the golden ratio. An exact formula for the nth Fibonacci number. Napier's constant, e, is the base of natural logarithms and exponentials. e is transcendental. Calculus, developed by Newton and Leibniz, is based on derivatives (slopes) and integrals (areas) of curves. The derivative of ex is ex. The integral of ex is ex.