Euler's formula: e^ix = cosx + isinx
if x = pi or 2pi, the formula becomes an elegant identity:
e^ipi = -1 or e^2ipi = 1
Usually e^ip=-1 is referred to as Euler's identity
You can call either one Euler's identity or Euler's formula.
e is about 2.732... the base of natural logs, e is used for growth or decay rates, such as compound interest calculations.
i = square root of -1, which is an imaginary number
pi is about 3.14159... It's the ratio of a circle's diameter to its circumference, an irrational number.
pi, sine and cosines are all cyclical. pi is related to the circle which is by name cyclical, circular.
the formula relates 5 major fundamental math concepts in a surprising way,
cosine, sine, i, e, and pi.
Bertrand Russell, major philosopher, mathematician & social activist, called Euler's formula the "most beautiful" equation in mathematics. Some people find numbers and equations "beautiful" and "elegant."
Euler's formula could also refer to an unrelated equation in topology, F+V=E+2. The number of faces plus vertices equals the number of edges plus two. For example, a cube has 6 faces, 8 vertices and 12 edges. 6+8=12+2.
There's a story, perhaps a mythical legend, that Russia's Catherine the Great invited Euler and a controversial French atheist philosopher Denis Diderot to her court, to debate. She amused herself having intellectuals spar like boxers, sort of. Euler presented, for the first time in public, his proof of his Euler's identity. As he finished the end of the proof he wrote, "Therefore God exists." the French atheist philosopher had no response and left the court in disgrace. there is perhaps a pun involved, as his identity "proves" an "imaginary" power has real existence. Bertrand Russell probably would not have been amused though.