Complex Numbers

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1. What Is a Complex Number?

Definition (Mathematics)

A complex number is a number of the form:

$$z=a+bi$$

Where:

• a is the real part,

• b is the imaginary part,

• i is the imaginary unit where:

$$i^{2}=-1$$

Example:

$$z=3+4i$$

• Real part: 3

• Imaginary part: 4

Python Representation

z = complex(3, 4)
print("Complex Number:", z)
print("Real Part:", z.real)
print("Imaginary Part:", z.imag)

Output:

Complex Number: (3+4j)
Real Part: 3.0
Imaginary Part: 4.0

2. Basic Operations

Addition

$$(3+4i)+(1+2i)=(3+1)+(4+2)i=4+6i$$

Subtraction

$$(3+4i)−(1+2i)=(3−1)+(4−2)i=2+2i$$

Multiplication

$$(3+4i)(1+2i)=3+6i+4i+8i^{2}=3+10i+8\left( -1 \right)=-5+10i$$

Division

$$\frac{3+4i}{1+2i}=\frac{3+4i}{1+2i}\times \frac{1-2i}{1-2i}=\frac{11-2i}{5}$$

Python Operations

z1 = complex(3, 4)
z2 = complex(1, 2)

print("Addition:", z1 + z2)
print("Subtraction:", z1 - z2)
print("Multiplication:", z1 * z2)
print("Division:", z1 / z2)

Output:

Addition: (4+6j)
Subtraction: (2+2j)
Multiplication: (-5+10j)
Division: (2.2-0.4j)

3. Modulus of a Complex Number

Formula:

$$\left| z \right|=\sqrt{a^{2}+b^{2}}$$

Example:

$$\left| 3+4i \right|=\sqrt{3^{2}+4^{2}}=5$$

Python:

import math

z = complex(3, 4)
modulus = abs(z)
print("Modulus of", z, "is", modulus)

Output:

Modulus of (3+4j) is 5.0

4. Complex Conjugate

Definition:

The conjugate of

$$z=a+bi$$

is

$$\bar{z}=a-bi$$

Python:

z = complex(3, 4)
conjugate = z.conjugate()
print("Conjugate of", z, "is", conjugate)

Output:

Conjugate of (3+4j) is (3-4j)

5. Polar Form of a Complex Number

Formula:

Convert

$$z=a+bi$$

into:

$$z=r(cos\theta+isin\theta)=rcis\theta$$

Where:

$$\begin{matrix} r=\left| z \right|=\sqrt{a^{2}+b^{2}} \\ \\ \theta=arg\left( z \right)=tan^{-1}\left( \frac{b}{a} \right) \end{matrix}$$

Python:

import cmath

z = complex(3, 4)
r, theta = abs(z), cmath.phase(z)

print("Modulus (r):", r)
print("Angle (θ in radians):", theta)

Output:

Modulus (r): 5.0
Angle (θ in radians): 0.9272952180016122

6. Euler's Formula

$$e^{i\theta}=cosθ+isinθ$$

So:

$$z=re^{i\theta}$$

7. Convert Between Cartesian and Polar Forms

Python:

Convert to Polar:

z = complex(1, 1)
r, theta = cmath.polar(z)
print("Polar Form: (r =", r, ", theta =", theta, ")")

Output:

Polar Form: (r = 1.4142135623730951 , theta = 0.7853981633974483 )

Convert to Rectangular:

z = cmath.rect(r, theta)
print("Rectangular Form:", z)

Output:

Rectangular Form: (1.0000000000000002+1.0000000000000002j)

8. Powers and Roots

Power:

$$z^{n}=\left[ re^{i\theta} \right]^{n}=\left[ r\left( cos\theta+isin\theta \right) \right]^{n}=r^{n}cis\left( n\theta \right)$$

n-th Roots:

$$\sqrt[n]{z}=\sqrt[n]{rcis(\theta)}=\sqrt[n]{r}cis\left( \frac{\theta+2k\pi}{n} \right);k=0,1,2,...,\left( n-1 \right)$$

Python (Using cmath):

# z^n
z = complex(2, 3)
power = z**3
print("z^3 =", power)

# Square root
root = cmath.sqrt(z)
print("√z =", root)

Output:

z^3 = (-46+9j)
√z = (1.6741492280355401+0.8959774761298381j)

9. Plotting Complex Numbers (Optional Visualization)

import matplotlib.pyplot as plt

z = complex(3, 4)
plt.plot([0, z.real], [0, z.imag], marker='o')
plt.text(z.real, z.imag, f'{z}', fontsize=12)
plt.axhline(0, color='gray', linestyle='--')
plt.axvline(0, color='gray', linestyle='--')
plt.xlabel('Real')
plt.ylabel('Imaginary')
plt.grid(True)
plt.title('Complex Number on the Complex Plane')
plt.axis('equal')
plt.show()

Output:

Summary Cheat Sheet

Concept	Python Function
Complex number	complex(a, b)
Real part	z.real
Imaginary part	z.imag
Conjugate	z.conjugate()
Modulus	abs(z)
Phase/angle	cmath.phase(z)
Polar form	cmath.polar(z)
Rectangular form	cmath.rect(r, θ)
Square root	cmath.sqrt(z)
Plotting	matplotlib

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Keywords

complex numbers, imaginary numbers, real part, imaginary part, complex arithmetic, addition, subtraction, multiplication, division, modulus, complex conjugate, polar form, rectangular form, Euler’s formula, Python complex type, cmath module, plotting complex numbers, argument of complex number, powers of complex numbers, roots of complex numbers, nerd cafe , نرد کافه

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