Euler's Identity, called "the most remarkable formula in the world" by an entry in the notebook of then almost 15 year old Richard Feynman, is:
- eiπ + 1 = 0
where e is the base of the natural logarithm, i is the imaginary unit (an imaginary number with the property i2 = -1), and π is Archimedes' Constant Pi (the ratio of the circumference of a circle to its diameter). The equation appears in Leonhard Euler's Introductio, published in Lausanne in 1748.
Feynman found this formula remarkable because it links some very fundamental mathematical constants:
- The numbers 0 and 1 are elementary for counting and arithmetic.
- The number π is a constant related to our world being Euclidean, on small scales at least (otherwise, the ratio of the length of the circumference of circle to its diameter would not be a universal constant, i.e. the same for all circumferences).
- The number e is important in describing growth behaviors, as the simplest solution to the simplest growth equation dy / dx = y is y = ex.
- Finally, the imaginary unit i was introduced to ensure that all non-constant polynomial equations would have solutions (see Fundamental Theorem of Algebra).
The formula is a consequence of Euler's formula from complex analysis, which states that
- eix = cos x + i · sin x
for any real number x. If we set x = π, then
- eiπ = cos π + i · sin π,
and since cos(π) = -1 and sin(π) = 0, we get
- eiπ = - 1
and
- eiπ + 1 = 0.