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More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation (+) = has no real solution, because the square of a real number cannot be negative, but has the two nonreal complex solutions + and .
The complex numbers are a generalization of the real numbers, which are a generalization of the rational numbers, which are a generalization of the integers, which are a generalization of the natural numbers. A polygon is a generalization of a 3-sided triangle, a 4-sided quadrilateral, and so on to n sides.
The imaginary unit or unit imaginary number (i) is a mathematical constant that is a solution to the quadratic equation x 2 + 1 = 0. Although there is no real number with this property, i can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of i in a complex ...
A negative real number −x has no real-valued square roots, but when x is treated as a complex number it has two imaginary square roots, + and , where i is the imaginary unit. In general, any non-zero complex number has n distinct complex-valued n th roots, equally distributed around a complex circle of constant absolute value .
The set of all algebraic integers A is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers. The ring of integers of a number field K, denoted by O K, is the intersection of K and A: it can also be characterised as the maximal order of the field K. Each algebraic integer belongs to ...
A complex number is equal to its complex conjugate if its imaginary part is zero, that is, if the number is real. In other words, real numbers are the only fixed points of conjugation. Conjugation does not change the modulus of a complex number: | ¯ | = | |. Conjugation is an involution, that is, the conjugate of the conjugate of a complex ...
When every term of a series is a non-negative real number, for instance when the terms are the absolute values of another series of real numbers or complex numbers, the sequence of partial sums is non-decreasing. Therefore a series with non-negative terms converges if and only if the sequence of partial sums is bounded, and so finding a bound ...
Conversely, given a complex vector space W with a complex conjugation χ, W is isomorphic as a complex vector space to the complexification V C of the real subspace = {: =}. In other words, all complex vector spaces with complex conjugation are the complexification of a real vector space.