Nonreal

Nonreal is a term used in mathematics to describe complex numbers whose imaginary part is not zero. A complex number is written as a + bi, where a and b are real numbers and i satisfies i^2 = -1. If b ≠ 0, the number is nonreal; if b = 0, the number is real. Pure imaginary numbers (where a = 0 and b ≠ 0) are included in the nonreal category.

The set of nonreal numbers constitutes the complement of the real axis in the complex plane. It is not a subfield, and it is not closed under addition or multiplication. For example, 1 + i and 1 − i are nonreal, but their sum is 2, which is real. The distance of a nonreal number a + bi from the origin is given by its modulus, |a + bi| = sqrt(a^2 + b^2).

Each nonreal number has a complex conjugate a − bi, which is also nonreal. In polynomials with real coefficients, nonreal roots occur in conjugate pairs: if a + bi is a root with b ≠ 0, then a − bi is also a root.

Terminology notes: some authors distinguish nonreal numbers from imaginary numbers by emphasizing that imaginary numbers have zero real part, while nonreal numbers include all complex numbers with nonzero imaginary part. Thus pure imaginary numbers are a subset of nonreal numbers.

Usage and context: the term nonreal is commonly employed in algebra and analysis to contrast with real numbers, particularly when discussing roots, values, or coefficients that involve the imaginary component.