Euler's identitymathematics, Euler's identity, a special case of Euler's formula, is the following:
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Perceptions of the identity
It was called "the most remarkable formula in mathematics" by Richard Feynman.
Feynman found this formula remarkable because it links some very fundamental mathematical constants:
- 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 is fundamental in trigonometry, 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 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 with initial condition is ).
- The imaginary unit (where i² = -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).
In addition, the result is remarkable to most students learning it for the first time because it is so highly counterintuitive. Consider that