DEFINITION:

A complex number is a number that can be expressed in the form a + ib, where a, b are real numbers and i is the imaginary unit that satisfies the equation i = $\sqrt{-1}$

• Complex number a + ib is denoted by Z. Where a = Real part and b = Imaginary Part
• Real part is denoted by Re(Z) = a and Imaginary part is denoted by Im(Z) = b

NOTE :

• If b = 0, Complex number is real.
• If a = 0, Complex number is purely imaginary.
• Two complex numbers a + ib = c + id are equal if only a = c and b = d.

Conjugate Complex Number

Complex numbers a + ib and a - ib are said to be conjugate to each other and it is denoted by Z = a + ib and Z̅ = a - ib

Properties of Conjugate Complex Number

If Z₁ and Z₂ be any two complex numbers, then

• $\overline{\text{Z}_1 + \text{Z}_2} = \overline{\text{Z}_1} + \overline{\text{Z}_2}$
• $\overline{\text{Z}_1 - \text{Z}_2} = \overline{\text{Z}_1} - \overline{\text{Z}_2}$
• $\overline{\text{Z}_1 \cdot \text{Z}_2} = \overline{\text{Z}_1} \cdot \overline{\text{Z}_2}$
• $\overline{\left(\cfrac{\text{Z}_1}{\text{Z}_2}\right)} = \cfrac{\overline{\text{Z}_1}}{\overline{\text{Z}_2}}$ where Z₂ ≠ 0
• ${\text{Z}_1 - \text{Z}_2}$ is purely imaginary
• $|{\text{Z}_1 \cdot \text{Z}_2}|=$ $ |{\text{Z}_1}| \cdot |{\text{Z}_2}|$
• ${\left|\cfrac{\text{Z}_1}{\text{Z}_2}\right |}=$ $\cfrac{|\text{Z}_1 |}{|\text{Z}_2|}$ where Z₂ ≠ 0
• $ \text{arg}(Z_1 \cdot \text{Z}_2)=$ $\text{arg}(\text{Z}_1)+\text{arg}(Z_2)$
• $ \text{arg} \left(\cfrac{\text{Z}_1}{\text{Z}_2}\right)=$ $\text{arg}(\text{Z}_1)-\text{arg}(\text{Z}_2)$

Modulus and Argument

• If Z = a + ib, then modulus of Z, $|\text{Z}|=\sqrt{\text{a}^2+\text{b}^2}$
• $\text{Z} =$ $\text{r}(\text{Cos}{\theta}+\text{i}\text{Sin}{\theta})$. This form is known as Polar Form of a complex number.
• $\text{Z}=\text{a}+\text{ib}=$ $\text{r}(\text{Cos}{\theta}+\text{i}\text{Sin}{\theta})$. Where $\theta$ is called argument and it can be in degrees and radians.
• $|\text{Z}|=$ $\sqrt{\text{a}^2+\text{b}^2}=$ $\text{r}$. Where modulus is a non-negative real number and r ≥ 0.
• $\text{arg}(\text{Z})={\theta} \\ \Rightarrow \text{tan}{\theta}=\cfrac{\text{b}}{\text{a}} \\ \Rightarrow {\theta}= \text{tan}^{-1}{\left(\cfrac{\text{b}}{\text{a}}\right)}$

θ denotes the angle measured counterclockwise from the positive real axis.

NOTE :

• $|1|=1,$ $\text{arg}(1)=0$
• $|-1|=1,$ $\text{arg}(-1)=\pi$
• $|\text{i}|=1,$ $\text{arg}(\text{i})=\cfrac{\pi}{2}$
• $|-\text{i}|=1,$ $\text{arg}(-\text{i})=$ $-\cfrac{\pi}{2}$