complex conjugate

Thân thiện

The student writes the complex conjugate of a number on the whiteboard.

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.

Học thuật

Noun

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