complex conjugate

(noun)

Of a complex number x, the complex number <math>\overline x</math> formed by changing the sign of the imaginary part: The complex conjugate of a + bi is a - bi.

Related Terms

Examples of complex conjugate in the following topics:

  • Complex Conjugates and Division

    • The complex conjugate of x + yi is x - yi, and the division of two complex numbers can be defined using the complex conjugate.
    • The complex conjugate of the complex number z = x + yi is defined as x - yi.
    • Specifically, conjugating twice gives the original complex number: z** = z .
    • Moreover, a complex number is real if and only if it equals its conjugate.
    • Geometric representation of z and its conjugate in the complex plane.

  • Complex Conjugates

    • The complex conjugate of the number $a+bi$ is $a-bi$.
    • The complex conjugate (sometimes just called the conjugate) of a complex number $a+bi$ is the complex number $a-bi$.
    • Since the conjugate of a conjugate is the original complex number, we say that the two numbers are conjugates of each other.
    • The symbol for the complex conjugate of $z$ is $\overline{z}$.
    • One important fact about conjugates is that whenever a complex number is a root of polynomial, its complex conjugate is a root as well.

  • Division of Complex Numbers

    • Division of complex numbers is accomplished by multiplying by the multiplicative inverse of the denominator.
    • For complex numbers, the multiplicative inverse can be deduced using the complex conjugate.
    • We have already seen that multiplying a complex number $z=a+bi$ with its complex conjugate $\overline{z}=a-bi$ gives the real number $a^2+b^2$.
    • So the multiplicative inverse of $z$ must be the complex conjugate of $z$ divided by its modulus squared.
    • Suppose you wanted to divide the complex number $z=2+3i$ by the number $1+2i$.

  • Biosynthesis of Tetrapyrroles

    • Porphyrins are the conjugate acids of ligands that bind metals to form complexes.
    • Complexes of meso-tetraphenylporphyrin, e.g., the iron(III) chloride complex (TPPFeCl), catalyze a variety of reactions in organic synthesis.
    • For example, complexes of meso-tetraphenylporphyrin, e.g., the iron(III) chloride complex (TPPFeCl), catalyze a variety of reactions of potential interest in organic synthesis.
    • Thus, porphyrin macrocycles are highly conjugated systems.
    • Porphyrins are the conjugate acids of ligands that bind metals to form complexes.

  • Conjugation

  • The Four Conjugations

  • Conjugation of "Sum"

  • First (or A) Conjugation

  • Second (or E) Conjugation

  • Third (or Consonant) Conjugation