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Exponentiation with negative exponents is defined by the following identity, which holds for any integer n and nonzero b: =. [1] Raising 0 to a negative exponent is undefined but, in some circumstances, it may be interpreted as infinity (). [22]
Exponential functions with bases 2 and 1/2. In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative equal to its value. . The exponential of a variable is denoted or , with the two notations used interchangeab
Solving for , = = = = = Thus, the power rule applies for rational exponents of the form /, where is a nonzero natural number. This can be generalized to rational exponents of the form p / q {\displaystyle p/q} by applying the power rule for integer exponents using the chain rule, as shown in the next step.
Negative numbers: Real numbers that are less than zero. Because zero itself has no sign, neither the positive numbers nor the negative numbers include zero. When zero is a possibility, the following terms are often used: Non-negative numbers: Real numbers that are greater than or equal to zero. Thus a non-negative number is either zero or positive.
If K is a field (such as the complex numbers), a Puiseux series with coefficients in K is an expression of the form = = + / where is a positive integer and is an integer. In other words, Puiseux series differ from Laurent series in that they allow for fractional exponents of the indeterminate, as long as these fractional exponents have bounded denominator (here n).
This can be expressed more concisely by using summation notation: = That is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero terms. Each term consists of the product of a number – called the coefficient of the term [ a ] – and a finite number of indeterminates, raised to non-negative integer powers.
A number is positive if it is greater than zero. A number is negative if it is less than zero. A number is non-negative if it is greater than or equal to zero. A number is non-positive if it is less than or equal to zero. When 0 is said to be both positive and negative, [citation needed] modified phrases are used to refer to the sign of a number:
0 (zero) is a number representing an empty quantity.Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and complex numbers, as well as other algebraic structures.
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