Complex Numbers : Properties of complex conjugate

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Properties of complex conjugate

what you'll learn...

Overview

» Complex conjugate

    → distributes into addition, multiplication, and power
    → $\bar{z_{1} + z_{2}}$ $bar(z_1+z_2)$ $= \bar{z_{1}} + \bar{z_{2}}$ $= bar(z_1) + bar(z_2)$
    → $\bar{z_{1} \times z_{2}}$ $bar(z_1xxz_2)$ $= \bar{z_{1}} \times \bar{z_{2}}$ $= bar(z_1) xx bar(z_2)$
    → $\bar{z^{n}}$ $bar(z^n)$ $= {(\bar{z})}^{n}$ $= (bar(z))^n$

    → modulus of the conjugate equals the modulus of the number
    → $| \bar{z} |$ $| bar(z)|$ $= | z |$ $= |z|$

    → argument of the conjugate is negative of the argument of the number
    → $a r g (\bar{z})$ $arg (bar(z))$ $= - a r g (z)$ $= - arg(z)$

conjugate of sum

Given $z_{1} = a_{1} + i b_{1}$ $z_1 = a_1+ib_1$ and $z_{2} = a_{2} + i b_{2}$ $z_2 = a_2+ib_2$ , the conjugate of sum $\bar{z_{1} + z_{2}}$ $bar(z_1+z_2)$ $= \bar{z_{1}} + \bar{z_{2}}$ $= bar(z_1)+bar(z_2)$

Conjugate of Sum or Difference: For complex numbers $z_{1}, z_{2} \in ℂ$ $z_1, z_2 in CC$
$\bar{z_{1} \pm z_{2}} = \bar{z_{1}} \pm \bar{z_{2}}$ $bar(z_1 +- z_2) = bar(z_1) +- bar(z_2)$

Conjugate of sum is sum of conjugates.
Conjugate of difference is difference of conjugates.

conjugate of product

Given $z_{1} = a_{1} + i b_{1}$ $z_1 = a_1+ib_1$ and $z_{2} = a_{2} + i b_{2}$ $z_2 = a_2+ib_2$ what is the conjugate $\bar{z_{1} \times z_{2}}$ $bar(z_1 xx z_2)$ ?

The answer is ' $\bar{z_{1}} \times \bar{z_{2}}$ $bar(z_1) xx bar(z_2)$ '

Conjugate of product or quotient: For complex numbers $z_{1}, z_{2} \in ℂ$ $z_1, z_2 in CC$
$\bar{z_{1} \times z_{2}} = \bar{z_{1}} \times \bar{z_{2}}$ $bar(z_1 xx z_2) = bar(z_1) xx bar(z_2)$
$\bar{z_{1} \div z_{2}} = \bar{z_{1}} \div \bar{z_{2}}$ $bar(z_1 -: z_2) = bar(z_1) -: bar(z_2)$

Conjugate of product is product of conjugates.
Conjugate of quotient is quotient of conjugates.

conjugate of power

Given $z = a + i b$ $z = a+ib$ , the conjugate $\bar{z^{n}}$ $bar(z^n)$ $= {(\bar{z})}^{n}$ $=(bar(z))^n$

Conjugate of Power or Root: For a complex number $z \in ℂ$ $z in CC$
$\bar{z^{n}} = {(\bar{z})}^{n}$ $bar(z^n) = (bar(z))^n$
$\bar{z^{\frac{1}{n}}} = {(\bar{z_{1}})}^{\frac{1}{n}}$ $bar(z^(1/n)) = (bar(z_1))^(1/n)$

Conjugate of a power is power of conjugate.
Conjugate of a root is root of conjugate.

modulus of conjugate

Given $z = a + i b$ $z = a+ib$ , the modulus $| \bar{z} |$ $|bar(z)|$ $= | z |$ $=|z|$

Modulus of a Conjugate: For a complex number $z \in ℂ$ $z in CC$
$| \bar{z} | = | z |$ $|bar(z)| = |z|$

Modulus of a conjugate equals modulus of the complex number.

argument of conjugate

Given $z = a + i b$ $z = a+ib$ , the argument $arg \bar{z}$ $text(arg) bar(z)$ $= - arg z$ $=-text(arg)z$

Argument of a Conjugate: For a complex number $z \in ℂ$ $z in CC$
$arg \bar{z} = - arg z$ $text(arg) bar(z) = - text(arg)z$

Argument of a conjugate equals negative of the argument of the complex number

conjugate of conjugate

Given $z = a + i b$ $z = a+ib$ , the conjugate of conjugate $\bar{\bar{z}}$ $bar bar(z)$ $= z$ $=z$

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Conjugate or Conjugate: For a complex number $z \in ℂ$ $z in CC$
$\bar{\bar{z}} = z$ $bar (bar(z)) = z$

Conjugate of a conjugate is the complex number itself.

product

Given $z = a + i b$ $z = a+ib$ , what is the product $z \bar{z}$ $z bar(z)$ $= {| z |}^{2}$ $=|z|^2$

Product of a number and its conjugate: For a complex number $z \in ℂ$ $z in CC$
$z \bar{z} = {| z |}^{2}$ $z bar(z) = |z|^2$

Product of a number and its conjugate is the square of the modulus.

summary

Complex conjugate

    → distributes into addition, multiplication, and power
    → $\bar{z_{1} + z_{2}}$ $bar(z_1+z_2)$ $= \bar{z_{1}} + \bar{z_{2}}$ $= bar(z_1) + bar(z_2)$
    → $\bar{z_{1} \times z_{2}}$ $bar(z_1xxz_2)$ $= \bar{z_{1}} \times \bar{z_{2}}$ $= bar(z_1) xx bar(z_2)$
    → $\bar{z^{n}}$ $bar(z^n)$ $= {(\bar{z})}^{n}$ $= (bar(z))^n$

    → modulus of the conjugate equals the modulus of the number
    → $| \bar{z} |$ $| bar(z)|$ $= | z |$ $= |z|$

    → argument of the conjugate is negative of the argument of the number
    → $a r g (\bar{z})$ $arg (bar(z))$ $= - a r g (z)$ $= - arg(z)$

Outline

The outline of material to learn "complex numbers" is as follows.

Note : Click here for detailed overview of Complex-Numbers

→ Complex Numbers in Number System

→ Representation of Complex Number (incomplete)

→ Euler's Formula

→ Generic Form of Complex Numbers

→ Argand Plane & Polar form

→ Complex Number Arithmetic Applications

→ Understanding Complex Artithmetics

→ Addition & Subtraction

→ Multiplication, Conjugate, & Division

→ Exponents & Roots

→ Properties of Addition

→ Properties of Multiplication

→ Properties of Conjugate

→ Algebraic Identities