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For even values of n, positive numbers also have a negative nth root, while negative numbers do not have a real nth root. For odd values of n, every negative number x has a real negative nth root. For example, −2 has a real 5th root, = … but −2 does not have any real 6th roots. Every non-zero number x, real or complex, has n different ...
A simple arithmetic calculator was first included with Windows 1.0. [5]In Windows 3.0, a scientific mode was added, which included exponents and roots, logarithms, factorial-based functions, trigonometry (supports radian, degree and gradians angles), base conversions (2, 8, 10, 16), logic operations, statistical functions such as single variable statistics and linear regression.
From top to bottom: x 1/8, x 1/4, x 1/2, x 1, x 2, x 4, x 8. If x is a nonnegative real number, and n is a positive integer, / or denotes the unique nonnegative real n th root of x, that is, the unique nonnegative real number y such that =.
For each integer n > 2, the function n x is defined and increasing for x ≥ 1, and n 1 = 1, so that the n th super-root of x, , exists for x ≥ 1. However, if the linear approximation above is used, then = + if −1 < y ≤ 0, so + cannot exist.
[17] [18] For example, the fraction 1/(x 2 + 1) is not a polynomial, and it cannot be written as a finite sum of powers of the variable x. For polynomials in one variable, there is a notion of Euclidean division of polynomials , generalizing the Euclidean division of integers.
More generally, we find that + + + + is the positive real root of the equation x 3 − x − n = 0 for all n > 0. For n = 1, this root is the plastic ratio ρ, approximately equal to 1.3247. The same procedure also works to get as the real root of the equation x 3 + x − n = 0 for all n > 1.
In particular, this sequence has the combinatorial interpretation as being the number of ways to insert parentheses into the product x 0 · x 1 ·⋯· x n so that the order of multiplication is completely specified. For example, C 2 = 2 which corresponds to the two expressions x 0 · (x 1 · x 2) and (x 0 · x 1) · x 2.
The plus–minus sign, ±, is used as a shorthand notation for two expressions written as one, representing one expression with a plus sign, the other with a minus sign. For example, y = x ± 1 represents the two equations y = x + 1 and y = x − 1. Sometimes, it is used for denoting a positive-or-negative term such as ±x.