I have been re-reading Genius: The Life and Science of Richard Feynman by James Gleick (hence the recent Feynman-themed post), which reminded me of a special formula in mathematics; one that Feynman himself described as follows in his famous Feynman Lectures on Physics:
In our study of oscillating systems, we shall have occasion to use one of the most remarkable, almost astounding, formulas in all of mathematics. From the physicists’ point of view, we could bring forth this formula in two minutes or so and be done with it. But science is as much for intellectual enjoyment as for practical utility, so instead of just spending a few minutes, we shall surround the jewel by its proper setting in the grand design of that branch of mathematics called elementary algebra.
This remarkable formula? Euler’s Identity:
In analytical mathematics, Euler’s identity (named for the pioneering Swiss-German mathematician, Leonhard Euler), is an equality renowned for its mathematical beauty, linking five fundamental mathematical constants:
- The number 0, the additive identity.
- The number 1, the multiplicative identity.
- The number , which is ubiquitous in trigonometry, the geometry of Euclidean space, and analytical mathematics ( = 3.14159265…)
- The number , the base of natural logarithms, which occurs widely in mathematical and scientific analysis ( = 2.718281828…); both and are transcendental numbers.
- The number , the imaginary unit of the complex numbers, whose study leads to deeper insights into many areas of algebra and calculus.
The identity is a special case of Euler’s Formula from complex analysis, which states that:
for any real number . The derivation to the identity follows, as and .
Euler’s Identity reinforces the underlying beauty and interconnectedness of mathematics, pulling together three seemingly disparate areas into one simple formula, perhaps deservingly being known as “the most remarkable formula in all of mathematics”.