maths > complex-number

Properties of complex conjugate

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Overview

»  Complex conjugate

→  distributes into addition, multiplication, and power
→  $\overline{{z}_{1}+{z}_{2}}$$\overline{{z}_{1} + {z}_{2}}$$=\overline{{z}_{1}}+\overline{{z}_{2}}$$= \overline{{z}_{1}} + \overline{{z}_{2}}$
→  $\overline{{z}_{1}×{z}_{2}}$$\overline{{z}_{1} \times {z}_{2}}$$=\overline{{z}_{1}}×\overline{{z}_{2}}$$= \overline{{z}_{1}} \times \overline{{z}_{2}}$
→  $\overline{{z}^{n}}$$\overline{{z}^{n}}$$={\left(\overline{z}\right)}^{n}$$= {\left(\overline{z}\right)}^{n}$

→  modulus of the conjugate equals the modulus of the number
→  $|\overline{z}|$$| \overline{z} |$$=|z|$$= | z |$

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

conjugate of sum

Given ${z}_{1}={a}_{1}+i{b}_{1}$${z}_{1} = {a}_{1} + i {b}_{1}$ and ${z}_{2}={a}_{2}+i{b}_{2}$${z}_{2} = {a}_{2} + i {b}_{2}$, the conjugate of sum $\overline{{z}_{1}+{z}_{2}}$$\overline{{z}_{1} + {z}_{2}}$ $=\overline{{z}_{1}}+\overline{{z}_{2}}$$= \overline{{z}_{1}} + \overline{{z}_{2}}$

Conjugate of Sum or Difference: For complex numbers ${z}_{1},{z}_{2}\in ℂ$${z}_{1} , {z}_{2} \in \mathbb{C}$
$\overline{{z}_{1}±{z}_{2}}=\overline{{z}_{1}}±\overline{{z}_{2}}$$\overline{{z}_{1} \pm {z}_{2}} = \overline{{z}_{1}} \pm \overline{{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}$${z}_{1} = {a}_{1} + i {b}_{1}$ and ${z}_{2}={a}_{2}+i{b}_{2}$${z}_{2} = {a}_{2} + i {b}_{2}$ what is the conjugate $\overline{{z}_{1}×{z}_{2}}$$\overline{{z}_{1} \times {z}_{2}}$?

The answer is '$\overline{{z}_{1}}×\overline{{z}_{2}}$$\overline{{z}_{1}} \times \overline{{z}_{2}}$'

Conjugate of product or quotient: For complex numbers ${z}_{1},{z}_{2}\in ℂ$${z}_{1} , {z}_{2} \in \mathbb{C}$
$\overline{{z}_{1}×{z}_{2}}=\overline{{z}_{1}}×\overline{{z}_{2}}$$\overline{{z}_{1} \times {z}_{2}} = \overline{{z}_{1}} \times \overline{{z}_{2}}$
$\overline{{z}_{1}÷{z}_{2}}=\overline{{z}_{1}}÷\overline{{z}_{2}}$$\overline{{z}_{1} \div {z}_{2}} = \overline{{z}_{1}} \div \overline{{z}_{2}}$

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

conjugate of power

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

Conjugate of Power or Root: For a complex number $z\in ℂ$$z \in \mathbb{C}$
$\overline{{z}^{n}}={\left(\overline{z}\right)}^{n}$$\overline{{z}^{n}} = {\left(\overline{z}\right)}^{n}$
$\overline{{z}^{\frac{1}{n}}}={\left(\overline{{z}_{1}}\right)}^{\frac{1}{n}}$$\overline{{z}^{\frac{1}{n}}} = {\left(\overline{{z}_{1}}\right)}^{\frac{1}{n}}$

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

modulus of conjugate

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

Modulus of a Conjugate: For a complex number $z\in ℂ$$z \in \mathbb{C}$
$|\overline{z}|=|z|$$| \overline{z} | = | z |$

Modulus of a conjugate equals modulus of the complex number.

argument of conjugate

Given $z=a+ib$$z = a + i b$, the argument $\text{arg}\overline{z}$$\textrm{a r g} \overline{z}$$=-\text{arg}z$$= - \textrm{a r g} z$

Argument of a Conjugate: For a complex number $z\in ℂ$$z \in \mathbb{C}$
$\text{arg}\overline{z}=-\text{arg}z$$\textrm{a r g} \overline{z} = - \textrm{a r g} z$

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

conjugate of conjugate

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

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Conjugate or Conjugate: For a complex number $z\in ℂ$$z \in \mathbb{C}$
$\overline{\overline{z}}=z$$\overline{\overline{z}} = z$

Conjugate of a conjugate is the complex number itself.

product

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

Product of a number and its conjugate: For a complex number $z\in ℂ$$z \in \mathbb{C}$
$z\overline{z}={|z|}^{2}$$z \overline{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
→  $\overline{{z}_{1}+{z}_{2}}$$\overline{{z}_{1} + {z}_{2}}$$=\overline{{z}_{1}}+\overline{{z}_{2}}$$= \overline{{z}_{1}} + \overline{{z}_{2}}$
→  $\overline{{z}_{1}×{z}_{2}}$$\overline{{z}_{1} \times {z}_{2}}$$=\overline{{z}_{1}}×\overline{{z}_{2}}$$= \overline{{z}_{1}} \times \overline{{z}_{2}}$
→  $\overline{{z}^{n}}$$\overline{{z}^{n}}$$={\left(\overline{z}\right)}^{n}$$= {\left(\overline{z}\right)}^{n}$

→  modulus of the conjugate equals the modulus of the number
→  $|\overline{z}|$$| \overline{z} |$$=|z|$$= | z |$

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

Outline