The most remarkable formula in the world

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At least according to an entry in the notebook of then almost 15 year old Richard Feynman, "the most remarkable formula in the world" 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


eiπ + 1 = 0.