Definition

1. Noun:
   • A mathematical concept in complex number theory: A "complex conjugate" is one of a pair of complex numbers. The two numbers in this pair have identical real parts, but their imaginary parts are equal in magnitude and opposite in sign.
   • The specific number formed by negating the imaginary part: For a given complex number a + bi (where a and b are real numbers, and i is the imaginary unit), its "complex conjugate" is the number a - bi.

Usage Examples

• Noun:
  • The complex conjugate of 3 + 4i is 3 - 4i.
  • To find the magnitude of a complex number, you multiply it by its complex conjugate.
  • In the equation, the roots appeared as a complex conjugate pair.

Advanced Usage

• "Complex conjugate pair": A set of two complex numbers that are conjugates of each other.
  • The polynomial had non-real coefficients, resulting in complex conjugate pair solutions.
• "Complex conjugate transpose" (or "Hermitian transpose"): An operation in linear algebra involving both conjugation and transposition of a matrix.
  • A unitary matrix is one whose inverse is equal to its complex conjugate transpose.

Variants and Related Words

• Conjugate (verb/noun/adjective): As a verb, it means to form the conjugate of a complex number. As a noun or adjective, it can refer to the conjugate itself or things joined together, especially in pairs.
  • You must conjugate the complex number before proceeding with the division.
• Conjugation (noun): The process or result of forming a complex conjugate.
  • Complex conjugation is a fundamental operation.

Synonyms

• None: "Complex conjugate" is a precise, technical term with no direct single-word synonyms in mathematics. Related descriptive phrases include "conjugate pair" or simply "conjugate" when the context is clear.

Related Phrases

• Take the conjugate: An instructional phrase meaning to find the complex conjugate of a given number.
  • The next step is to take the conjugate of the denominator.
• Complex conjugate zeros: Refers to the property that non-real roots of polynomials with real coefficients always occur in conjugate pairs.
  • The theorem guarantees the polynomial will have complex conjugate zeros.

Related Idioms

• None: This is a technical mathematical term and is not used in idiomatic expressions.

Noun

1. either of two complex numbers whose real parts are identical and whose imaginary parts differ only in sign