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:

*e*^{iπ}+ 1 = 0

where *e* is the base of the natural logarithm, *i* is the imaginary unit (an imaginary number with the property *i*^{2} = -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*=*e*^{x}. - 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

*e*^{ix}= cos*x*+*i*· sin*x*

for any real number *x*. If we set *x* = π, then

*e*^{iπ}= cos π +*i*· sin π,

and since cos(π) = -1 and sin(π) = 0, we get

*e*^{iπ}= - 1

and

*e*^{iπ}+ 1 = 0.