Complex-number Formula Sheet

Complex numbers extend the real numbers by including imaginary numbers and are expressed as a+bi,
where a is the real part, b is the imaginary part, and i is the imaginary unit satisfying i$^2$ = −1.

Neetesh Kumar | May 14, 2024 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. Definition
• 2. Conjugate Complex
• 3. Representation of a Complex Number
• 4. Important Properties of Complex Conjugate
• 5. Properties of Modulus
• 6. Properties of Amplitude
• 7. De-Moiver's Theorem
• 8. Cube Root of Unity
• 9. Square root of a Complex number
• 10. Rotation of a Complex number
• 11. Geometrical formulas in complex numbers
• 12. Equation of line through $z_1$ and $z_2$ points
• 13. Equation of Circle in the argand plane
• 14. Standard Loci in the argand plane

1. Definition:

Complex numbers are defined as expressions of the form a + ib where a, b $\in$ R & i = $\sqrt{-1}.$ It is denoted by z, i.e. z = a + ib where 'a' is called the Real Part of z (Re z), and 'b' is called the Imaginary Part of z (Im z).

• Complex Number$= \begin{cases} Purely \space Real &\text{if } b = 0 \\ Imaginary &\text{if } b \ne 0 \\ Purely \space Imaginary &\text{if } a = 0 \end{cases}$

$\bold{Note:}$
$\bold{(i)}$ The set R of real numbers is a subset of the Complex Numbers. Hence, the Complex Number system is N $\subset$ W $\subset$ I $\subset$ Q $\subset$ R $\subset$ C
$\bold{(ii)}$ Zero is purely real and purely imaginary but not imaginary.
$\bold{(iii)}$ $i = \sqrt{-1}$ is called the imaginary unit. Also $i^2 = -1; i^3 = -i; i^4 = 1$ etc.
$\bold{(iv)}$ $\sqrt{a}\sqrt{b} = \sqrt{ab}$ only if at least one of a or b is non-negative.

2. Conjugate Complex:

If z = a + ib, then its conjugate complex is obtained by changing the sign of its imaginary part & is denoted by $\bar{z}$, i.e., $\bar{z}$ = a - ib.
$\bold{(i)}$ $z$ + $\bar{z}$ = 2Re(z)
$\bold{(ii)}$ $z$ - $\bar{z}$ = 2$i$Im(z)
$\bold{(iii)}$ $z\bar{z}$ = $a^2 + b^2$
$\bold{(iv)}$ If z is purely real then $z$ - $\bar{z} = 0$
$\bold{(v)}$ If z is purely imaginary then $z$ + $\bar{z} = 0$
$\bold{(vi)}$ Inverse of a complex number can be obtained by multiplying and dividing by its complex conjugate.

3. Representation of a Complex Number:

$\bold{(a) \space Cartesian \space Form:}$ (Geometric Representation)
Every complex number z = x + iy can be represented by a point on the cartesian plane known as a complex plane (Argand diagram) by the ordered pair (x, y). Length OA is called the modulus of the complex number denoted by |z| & $\theta$ is called the principal argument or amplitude where $\theta \in (-\pi, \pi]$

We can also write |z| = $\sqrt{x^2 + y^2}$ & $\theta = tan^{-1}(\frac{y}{x})$ (angle made by OA with positive x -axis, x > 0 Geometrically, |z| represents the distance of point A from the origin. (|z| $\ge$ 1)

$\bold{(b)}$ Polar Form: (Trignometric Representation)
z = r(cos$\theta$ + isin$\theta$) where |z| = r; arg(z) = $\theta;$ $\bar{z}=$ r(cos$\theta$ - isin$\theta$)
we can also write cos $\theta$ + isin $\theta$ as CIS $\theta$
$\bold{Euler's \space Formula:}$
The formula $e^{i\theta} = cos\theta + isin\theta$ is called Euler's Formula.
We can also write Cos$\theta = \frac{e^{i\theta} + e^{-i\theta}}{2}$ & Sin$\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}$ are known as Euler's Identities.

$\bold{(c) \space Exponential \space Representation:}$
Let z be a complex number such that |z| = r and arg z = $\theta$, then z = r.$e^{i\theta}$

4. Important Properties of Complex Conjugate:

$\bold{(a)}$ $\overline{\bar{z}} = z$
$\bold{(b)}$ $\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}$
$\bold{(c)}$ $\overline{z_1 - z_2} = \overline{z_1} - \overline{z_2}$
$\bold{(d)}$ $\overline{z_1 z_2} = \overline{z_1}. \overline{z_2}$
$\bold{(e)}$ $\overline{(\frac{z_1}{z_2})} = \frac{\overline{z_1}}{\overline{z_2}};$ where $z_2 \ne 0$
$\bold{(f)}$ If f is a polynomial with a real coefficient such that f($\alpha + i\beta$) = x + iy, then f($\alpha - i\beta$) = x - iy

5. Properties of Modulus:

$\bold{(a)}$ |z| $\ge$ 0
$\bold{(b)}$ |z| $\ge$ Re (z)
$\bold{(c)}$ |z| $\ge$ Im (z)
$\bold{(d)}$ |z| = | $\overline{z}$ | = | -z | = | $-\overline{z}$ |
$\bold{(e)}$ $z\overline{z}$ = $|z|^2$
$\bold{(f)}$ |$z_1z_2$| = $|z_1|.|z_2|$
$\bold{(g)}$ |$\frac{z_1}{z_2}$| = $\frac{|z_1|}{|z_2|},$ $z_2 \ne 0$
$\bold{(h)}$ |$z^n$| = $|z|^n$
$\bold{(i)}$ $|z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 + 2Re(z_1\overline{z_2})$ or $|z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 + 2|z_1||z_2|cos(\theta_1 - \theta_2)$
$\bold{(j)}$ $|z_1 + z_2|^2 + |z_1 - z_2|^2 = 2[|z_1|^2 + |z_2|^2]$
$\bold{(k)}$ $||z_1| - |z_2|| \le |z_1 + z_2| \le |z_1| + |z_2|$ [Triangular Inequality]
$\bold{(l)}$ $||z_1| - |z_2|| \le |z_1 - z_2| \le |z_1| + |z_2|$ [Triangular Inequality]
$\bold{(m)}$ If |z + $\frac{1}{z}$| = a (a>0), then max |z| = $\frac{a + \sqrt{a^2+4}}{2}$ & min |z| = $\frac{-a + \sqrt{a^2+4}}{2}$

6. Properties of Amplitude:

$\bold{(a)}$

• amp($z_1.z_2$) = amp $z_1 +$ amp $z_2 + 2k\pi; k \in I$
• amp($\frac{z_1}{z_2}$) = amp $z_1 -$ amp $z_2 + 2k\pi; k \in I$
• amp$(z^n)$ = n amp(z) + 2$k\pi$ where value of k is such that RHS lies in $(-\pi, \pi]$

$\bold{(b)}$ log(z) = log(r$e^{i\theta}$) = logr + $i\theta$ = log |z| + i amp(z)

7. De-Moiver's Theorem:

The value of $(cos\theta + isin\theta)^n$ is $(cosn\theta + isinn\theta)$ where n is an integer and it is one of the values of $(cos\theta + isin\theta)^n$ if n is a rational number of the form $\frac{p}{q}$, where p and q are Co-Prime numbers.

8. Cube Root of Unity:

$\bold{(a)}$ The cube roots of unity are 1, $\omega = \frac{-1+i\sqrt{3}}{2} = e^{i\frac{2\pi}{3}}$ and $\omega^2 = \frac{-1-i\sqrt{3}}{2} = e^{i\frac{4\pi}{3}}$
$\bold{(b)}$ 1 + $\omega + \omega^2 = 0, \omega^3 = 1$ (in general)
1 + $\omega^r + \omega^{2r}$ $= \begin{cases} 0 &\text{if } r \space is \space not \space integral \space multiple \space of \space 3 \\ 3 &\text{if } r \space is \space multiple \space of \space 3 \\ \end{cases}$
$\bold{(c)}$

• $a^2+b^2+c^2-ab-bc-ca = (a+b\omega+c\omega^2)(a+b\omega^2+c\omega)$
• $a^3+b^3 = (a+b)(a+\omega b)(a+\omega^2b)$
• $a^3-b^3 = (a-b)(a-\omega b)(a-\omega^2b)$
• $x^2 +x + 1 = (x-\omega)(x-\omega^2)$

9. Square root of a Complex number:

$\sqrt{a+ib}$ $= \begin{cases} \plusmn (\frac{\sqrt{|z|+a}}{2} + i\frac{\sqrt{|z|-a}}{2}) &\text{if } b > 0 \\ \plusmn (\frac{\sqrt{|z|+a}}{2} - i\frac{\sqrt{|z|-a}}{2}) &\text{if } b < 0 \\ \end{cases}$ where |z| = $\sqrt{a^2 + b^2}$

10. Rotation of a Complex number:

$\frac{z_2-z_0}{|z_2-z_0|} = \frac{z_1-z_0}{|z_1-z_0|}e^{i\theta}$ Take $\theta$ in anticlockwise direction.

11. Geometrical formulas in complex numbers:

$\bold{(a) \space Distance \space Formula:}$ $|z_1 - z_2|$ is represented as distance between the points $z_1$ and $z_2$ on the argand plane.
$\bold{(b) \space Section \space Formula:}$ If $z_1$ and $z_2$ are two complex numbers then the complex number z = $\frac{nz_1+mz_2}{m+n}$ divides the join of $z_1$ and $z_2$ in the ratio of m:n
$\bold{(c)}$ If the vertices A, B, and C of a triangle represent the complex numbers $z_1, z_2$ and $z_3$ respectively, then :

• Centroid of the $\Delta$ABC = $\frac{z_1+z_2+z_3}{3}$
  

• Orthocenter of the $\Delta$ABC = $\frac{(asecA)z_1 + (bsecB)z_2 + (csecC)z_3}{asecA + bsecB + csecC}$ or $\frac{z_1 tan A + z_2 tan B + z_3 tan C}{tanA + tanB + tanC}$
  

• Circumcenter of the $\Delta$ABC = $\frac{z_1 sin 2A + z_2 sin 2B + z_3 sin 2C}{sin2A + sin2B + sin2C}$

$\bold{(d) \space Equilateral \space Triangle:}$

• $\frac{z_1 - z_2}{l} = \frac{z_3 - z_2}{l}e^{\frac{i\pi}{3}}.....$ (i)
• $\frac{z_2 - z_3}{l} = \frac{z_1 - z_3}{l}e^{\frac{i\pi}{3}}.....$ (ii)
  
• from (i) and (ii)
  
• $\Rightarrow$ $z_1^2+z_2^2+z_3^2 = z_!z_2+z_2z_3+z_3z_1$ or $\frac{1}{z_1-z_2} + \frac{1}{z_2-z_3} + \frac{1}{z_3-z_1} = 0$

$\bold{(e) \space Isocelles \space Triangle:}$
then 4cos$^2\alpha(z_1-z_2)(z_3-z_1) = (z_3-z_2)^2$

$\bold{(f)}$ Area of $\Delta$ABC = $\frac{1}{4} \begin{vmatrix} z_1 & \overline{z_1} & 1 \\ z_2 & \overline{z_2} & 1 \\ z_3 & \overline{z_3} & 1 \end{vmatrix}$

12. Equation of line through $z_1$ and $z_2$ points:

$\begin{vmatrix} z & \overline{z} & 1 \\ z_1 & \overline{z_1} & 1 \\ z_2 & \overline{z_2} & 1 \end{vmatrix} = 0 \Rightarrow z(\overline{z_1} - \overline{z_2}) + \overline{z}(z_2 - z_1) + z_1\overline{z_2} - \overline{z_1}z_2 = 0 \Rightarrow z(\overline{z_1} - \overline{z_2})i + \overline{z}(z_2 - z_1)i + z_1\overline{z_2} - \overline{z_1}z_2 = 0$

Let ($z_2 - z_1$)i = a, then equation of line is: $\overline{a}z + a\overline{z} + b = 0$ where a $\in$ C and b $\in$ R.

• Complex Slope of the line joining points $z_1$ and $z_2$ is $\ \frac {(z_2 - z_1)}{(\overline{z_2-z_1})}$, and slope of a line in the cartesian plane is different from the argand plane.
• Complex slope of a line $\overline{a}z + a\overline{z} + b = 0$ is $\frac{a}{\overline{a}}, b \in R$
• Two lines with complex slope $\mu_1$ & $\mu_2$ are parallel if $\mu_1$ = $\mu_2$ and perpendicular if $\mu_1$ + $\mu_2$ = 0
• Length of perpendicular from point A($\alpha$) to line $\overline{a}z + a\overline{z} + b = 0$ is $\frac{\overline{a}\alpha + a\overline{\alpha} + b}{2|a|}$

13. Equation of Circle in the argand plane:

$\bold{(a)}$ Circle whose centre is $z_0$ radius = r = |$z - z_0$|
$\bold{(b)}$ General equation of circle is $z\overline{z} + a\overline{z} + b = 0$ where center is -a and radius = $\sqrt{|a|^2 - b}$
$\bold{(c)}$ Diameter form $(z - z_1)(\overline{z} - \overline{z_2}) + (z - z_2)(\overline{z} - \overline{z_1}) = 0$ or $arg(\frac{z-z_1}{z-z_2}) = \plusmn \frac{\pi}{2}$
$\bold{(d)}$ Equation $|\frac{z - z_1}{z - z_2}| = k$ represent a circle if k $\ne$ 1 and a straight line if k = 1.
$\bold{(e)}$ Equation $|z-z_1|^2 + |z-z_2|^2 = k$ represent circle if k $\ge \frac{1}{2}|z_1 - z_2|^2$
$\bold{(f)}$ arg$(\frac{z - z_1}{z - z_2}) = \alpha$ where $0 < \alpha < \pi, \alpha \ne \frac{\pi}{2}$ represent a segment of circle passing through A($z_1$) and B($z_2$)

14. Standard Loci in the argand plane:

$\bold{(a)}$ $|z - z_1| + |z - z_2| = 2k$ (constant) represents $= \begin{cases} An \space Ellipse &\text{if } 2k > |z_1 - z_2| \\ A \space line \space segment &\text{if } 2k = |z_1 - z_2| \\ No \space Solution &\text{if } 2k < |z_1 - z_2| \end{cases}$

$\bold{(b)}$ $|z - z_1| - |z - z_2| = 2k$ (constant) represents $= \begin{cases} A \space Hyperbola &\text{if } 2k < |z_1 - z_2| \\ A \space line \space ray &\text{if } 2k = |z_1 - z_2| \\ No \space Solution &\text{if } 2k > |z_1 - z_2| \end{cases}$

$\color{red} \bold{Related \space Pages:}$
Operation on Complex Numbers
Operation on Matrices
Matrices Formula Sheet