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For positive real numbers, exponentiation to real powers can be defined in two equivalent ways, either by extending the rational powers to reals by continuity (§ Limits of rational exponents, below), or in terms of the logarithm of the base and the exponential function (§ Powers via logarithms, below).
The multiplication of two odd numbers is always odd, but the multiplication of an even number with any number is always even. An odd number raised to a power is always odd and an even number raised to power is always even, so for example x n has the same parity as x. Consider any primitive solution (x, y, z) to the equation x n + y n = z n.
The original proof is based on the Taylor series expansions of the exponential function e z (where z is a complex number) and of sin x and cos x for real numbers x . In fact, the same proof shows that Euler's formula is even valid for all complex numbers x .
Since the right-most expression is defined if n is any real number, this allows defining for every positive real number b and every real number x: = (). In particular, if b is the Euler's number e = exp ( 1 ) , {\displaystyle e=\exp(1),} one has ln e = 1 {\displaystyle \ln e=1} (inverse function) and thus e x = exp ...
The law of non-contradiction (alternately the 'law of contradiction' [20]): 'Nothing can both be and not be.' [19] The law of excluded middle: 'Everything must either be or not be.' [19] In accordance with the law of excluded middle or excluded third, for every proposition, either its positive or negative form is true: A∨¬A.
Euler's identity is a special case of Euler's formula, which states that for any real number x, e i x = cos x + i sin x {\displaystyle e^{ix}=\cos x+i\sin x} where the inputs of the trigonometric functions sine and cosine are given in radians .
For example, the order does not matter in the multiplication of real numbers, that is, a × b = b × a, so we say that the multiplication of real numbers is a commutative operation. However, operations such as function composition and matrix multiplication are associative, but not (generally) commutative.
For all other values of , the expression is not well-defined for <, as was covered above, or is not a real number, so the limit does not exist as a real-valued derivative. For the two cases that do exist, the values agree with the value of the existing power rule at 0, so no exception need be made.