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In the case of negative real numbers, the largest real part is shared by the two nonreal cube roots, and the principal cube root is the one with positive imaginary part. So, for negative real numbers, the real cube root is not the principal cube root. For positive real numbers, the principal cube root is the real cube root.
A square root of a number x is a number r which, when squared, becomes x: =. Every positive real number has two square roots, one positive and one negative. For example, the two square roots of 25 are 5 and −5. The positive square root is also known as the principal square root, and is denoted with a radical sign:
Roots are a special type of exponentiation using a fractional exponent. For example, the square root of a number is the same as raising the number to the power of and the cube root of a number is the same as raising the number to the power of .
In the case of two nested square roots, the following theorem completely solves the problem of denesting. [2]If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that + = if and only if is the square of a rational number d.
Here is an angle in the unit circle; taking 1 / 3 of that angle corresponds to taking a cube root of a complex number; adding −k 2 π / 3 for k = 1, 2 finds the other cube roots; and multiplying the cosines of these resulting angles by corrects for scale.
Including 0, the set has a semiring structure (0 being the additive identity), known as the probability semiring; taking logarithms (with a choice of base giving a logarithmic unit) gives an isomorphism with the log semiring (with 0 corresponding to ), and its units (the finite numbers, excluding ) correspond to the positive real numbers.
Geometric representation of the 2nd to 6th root of a general complex number in polar form. For the nth root of unity, set r = 1 and φ = 0. The principal root is in black. An n th root of unity, where n is a positive integer, is a number z satisfying the equation [1] [2] =
A non-example is in the ring of integers modulo ; while () and thus is a cube root of unity, + + meaning that it is not a principal cube root of unity. The significance of a root of unity being principal is that it is a necessary condition for the theory of the discrete Fourier transform to work out correctly.