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Euler's Formula / Euler's Equation or Identity - HTML5 Animation

Animated illustration: Proof of Euler's formula using Taylor series expansions of the exponential function ez

Leonhard Euler (Swiss mathematician and physicist, 1707-1783) and his beautiful and extraordinary formula that links the 5 fundamental constants in Mathematics, namely, e, the base of the natural logarithms, i, the square root of -1, Pi, the ratio of the circumference of a circle to its diameter, 1 and 0, together!

Euler's Equation or Identity

Euler Formula

Euler's formula is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function.

Euler's formula states that, for any real number x:

(1) \(e^{ix}=\cos x+i\cdot \sin x\)  (Euler's formula)


  • e is the base of the natural logarithm

  • i is the imaginary unit

  • sin and cos are trigonometric functions.

Euler's equation or identity is a special case of the Euler' formula, where:

(2) \(x=\pi \)

By substitution in (1):

(3) \(e^{i\pi }=\cos \pi +i\cdot \sin \pi \)

(4) \(e^{i\pi }=-1+0 \)

(5) Therefore: \(e^{i\pi }=-1 \) (Euler's Equation or Identity)

Euler Formula e. i, Pi, 1, 0

Benjamin Peirce (1809-1880, American mathematician, professor at Harvard) gave a lecture proving "Euler's equation", and concluded:

"Gentlemen, that is surely true,
it is absolutely paradoxical;
we cannot understand it,
and we don't know what it means.
But we have proved it,
and therefore we know it must be the truth."

Reference: The Changing Shape of Geometry. Celebrating a Century of Geometry and Geometry Teaching. Edited on behalf of The Mathematical Association UK by Chris Pritchard. Cambridge University Press, (Cambridge 2003).

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