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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]
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
A rational number has an indefinitely repeating sequence of finite length l, if the reduced fraction's denominator contains a prime factor that is not a factor of the base. If q is the maximal factor of the reduced denominator which is coprime to the base, l is the smallest exponent such that q divides b ℓ − 1.
Equivalently, the exponent of a prime p in () equals the number of nonnegative integers j such that the fractional part of k/p j is greater than the fractional part of n/p j. It can be deduced from this that ( n k ) {\displaystyle {\tbinom {n}{k}}} is divisible by n / gcd ( n , k ).
Rational numbers have irrationality exponent 1, while (as a consequence of Dirichlet's approximation theorem) every irrational number has irrationality exponent at least 2. On the other hand, an application of Borel-Cantelli lemma shows that almost all numbers, including all algebraic irrational numbers , have an irrationality exponent exactly ...
However, there are generalizations of this formula valid for other exponents. These can be used to give explicit expressions for the n th roots of unity, that is, complex numbers z such that z n = 1. Using the standard extensions of the sine and cosine functions to complex numbers, the formula is valid even when x is an arbitrary complex number.
Because rational numbers have degree 1, we must have n ≤ 2 or φ(n) = 2 and therefore the only possibilities are n = 1,2,3,4,6. Next, he proved a corresponding result for the sine using the trigonometric identity sin( θ ) = cos( θ − π /2) . [ 4 ]
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).
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