A Beautiful Identity
e ip +1 = 0
This is probably the most beautiful identity in mathematics. It is often attributed to the most prolific mathematician in history, Leonhard Euler (1707-1783). However, Roger Cotes (1682-1716) had published the equivalence of the above identity in 1714, namely, iq = ln (cos q + i sin q ).
This identity involves three of the most basic operations, addition, multiplication and exponentiation. It contains 5 of the most commonly used constants, representing the four major branches of mathematics. 0, the additive identity and 1, the multiplicative identity (arithmetic). The Euler constant, e, which is the base of the natural logarithm (analysis). The exponent contains the imaginary unit, i , which is Ö(-1) (algebra), and one of the most fascinating number in the history of mathematics, p (geometry).
And this is the very same equality that was used by Lindemann to prove the transcendence of p, (A transcendental number is one that is not the root of any polynomial equation with rational coefficients.) thus, settling the two thousand year-old problem, that a circle cannot be squared with a pair of compass and a straight edge.
Using this identity, there are a few surprises. Try to calculate i i and iÖ i. These two numbers are actually real numbers! They are e-p/2 and ep/2 respectively. And don't think 1x always equals 1. Try to use the above identity to compute 1p , it is not just 1!!