Euler's identity

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:

e^iπ + 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 = 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.

There has been substantial debate in the philosophy of mathematics on the "real meaning" or "deep meaning" or even sacred geometry reflected by the Identity's relationship of key constants and operations (multiplication, exponentiation, addition, equality). Some assert it is simply a fundamental fact of the physical universe, and that algebra itself is a natural consequence of its structure. If so, the formula would be more than simply remarkable - it might be 'divine'.