complex conjugate

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 $1+2i$.

• The Fundamental Theorem of Algebra

  • The fundamental theorem states that every non-constant, single-variable polynomial with complex coefficients has at least one complex root.
  • As it turns out, every polynomial with a complex coefficient has a complex zero.
  • admits one complex root of multiplicity $4$, namely $x_0 = 0$, one complex root of multiplicity $3$, namely $x_1 = i$, and one complex root of multiplicity $1$, namely $x_2 = - \pi$.
  • The complex conjugate root theorem says that if a complex number $a+bi$ is a zero of a polynomial with real coefficients, then its complex conjugate $a-bi$ is also a zero of this polynomial.
  • By dividing with the real polynomial$(x-(a+bi))(x-(a-bi))=(x-a)^2 +b^2$, we obtain another real polynomial, for which the complex conjugate root theorem again applies.

• Finding Zeros of Factored Polynomials

  • All polynomial functions of positive, odd order have at least one zero (this follows from the fundamental theorem of algebra), while polynomial functions of positive, even order may not have a zero (for example $x^4+1$ has no real zero, although it does have complex ones).
  • It follows from the fundamental theorem of algebra and a fact called the complex conjugate root theorem, that every polynomial with real coefficients can be factorized into linear polynomials and quadratic polynomials without real roots.

• Roots of Complex Numbers

• Exponentials With Complex Arguments: Euler's Formula

• Trigonometry and Complex Numbers: De Moivre's Theorem

• Complex Logarithms

• Standard Equations of Hyperbolas

  • A conjugate axis of length 2b, corresponding to the minor axis of an ellipse, is sometimes drawn on the non-transverse principal axis; its endpoints ±b lie on the minor axis at the height of the asymptotes over/under the hyperbola's vertices.
  • The perpendicular thin black line through the center is the conjugate axis.
  • The two thick black lines parallel to the conjugate axis (thus, perpendicular to the transverse axis) are the two directrices, D1 and D2.