conjugate

Examples of 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 $a-bi$.
  • Thus, for example, the conjugate of $2+3i$ is $2-3i$ and the conjugate of $1-5i$ is $1+5i$.
  • Since the conjugate of a conjugate is the original complex number, we say that the two numbers are conjugates of each other.
  • The product of two conjugates is always a real number.

• Division of Complex Numbers

  • 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.

• 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.

• The Fundamental Theorem of Algebra

  • 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

  • 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.

• Ellipses

  • The distance between antipodal points on the ellipse, or pairs of points whose midpoint is at the center of the ellipse, is maximum along the major axis or transverse diameter, and a minimum along the perpendicular minor axis or conjugate diameter.

• What Are Conic Sections?

  • The transverse axis is also called the major axis, and the conjugate axis is also called the minor axis.