Euler's Identity

$e^\{i\; \backslash pi\}\; +\; 1\; =\; 0\; \backslash ,\backslash !$

$e^\{ix\}\; =\; \backslash cos\; x\; +\; i\; \backslash sin\; x\; \backslash ,\backslash !$

$e^\{i\; \backslash pi\}\; =\; \backslash cos\; \backslash pi\; +\; i\; \backslash sin\; \backslash pi\; \backslash ,\backslash !$

$e^\{i\; \backslash pi\}\; =\; -1\; \backslash ,\backslash !$

$e^\{i\; \backslash pi\}\; +\; 1\; =\; 0\; \backslash ,\backslash !$

Perceptions of the identity

• The number 0, the identity element for addition (for all a, a+0=0+a=a). See Group (mathematics).
• The number 1, the identity element for multiplication (for all a, a×1=1×a=a).
• The number $\backslash pi$ is fundamental in trigonometry, $\backslash pi$ is a constant in a world which is 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 a fundamental in connections to the study of logarithms and in calculus (such as in describing growth behaviors, as the solution to the simplest growth equation $dy\; /\; dx\; =\; y$ with initial condition $y(0)\; =\; 1$ is $y\; =\; e^x$).
• The imaginary unit $i$ (where i ² = −1) is a unit in the complex numbers. Introducing this unit yields all non-constant polynomial equations soluble in the field of complex numbers (see fundamental theorem of algebra).

$e^\{\backslash pi\}\; \backslash approx\; 23.14069\; \backslash ,\backslash !$ while $e^\{i\; \backslash pi\}\; =\; -1\; \backslash ,\backslash !$

References

• Feynman, Richard P. The Feynman Lectures on Physics, vol. I - part 1. Inter European Editions, Amsterdam (1975)

External link

• Proof of Euler's Identity by Julius O. Smith III
• Proof of Euler's Identity for a Layman by Ian Henderson

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