Complex Numbers

Representation of Complex Numbers

Real Part: The real part of a complex number is the component that does not involve the imaginary unit j.
Imaginary Part: The imaginary part is the component that is multiplied by j, where j is the square root of -1.

Example:

z = 3 + 4j  # 3 is the real part, 4 is the imaginary part

Accessing Real and Imaginary Parts

You can access the real and imaginary parts of a complex number using the .real and .imag attributes.

z = 3 + 4j
real_part = z.real      # real_part is 3.0
imaginary_part = z.imag # imaginary_part is 4.0

Complex Number Operations

Addition:
- Adds the real parts and the imaginary parts separately.
```
z1 = 3 + 4j
z2 = 1 + 2j
result = z1 + z2  # result is 4 + 6j
```
Subtraction:
- Subtracts the real parts and the imaginary parts separately.
```
z1 = 3 + 4j
z2 = 1 + 2j
result = z1 - z2  # result is 2 + 2j
```
Multiplication:
- Multiplies complex numbers using distributive property.
```
z1 = 3 + 4j
z2 = 1 + 2j
result = z1 * z2  # result is -5 + 10j
```
Division:
- Divides complex numbers using the formula for complex division.
```
z1 = 3 + 4j
z2 = 1 + 2j
result = z1 / z2  # result is 2.2 - 0.4j
```
Conjugate:
- The conjugate of a complex number is obtained by changing the sign of the imaginary part.
```
z = 3 + 4j
conjugate_z = z.conjugate()  # conjugate_z is 3 - 4j
```

Magnitude and Phase

Magnitude: The magnitude (or absolute value) of a complex number is the distance from the origin to the point represented by the number in the complex plane. It can be calculated using the abs() function.
```
z = 3 + 4j
magnitude = abs(z)  # magnitude is 5.0
```
Phase: The phase (or argument) of a complex number is the angle between the positive real axis and the line representing the number in the complex plane. You can calculate it using the cmath.phase() function.
```
import cmath

z = 3 + 4j
phase = cmath.phase(z)  # phase is approximately 0.93 radians
```

Polar Coordinates

Conversion to Polar Coordinates: A complex number can be represented in polar form as (r \times e^{i\theta}), where (r) is the magnitude and (\theta) is the phase.
```
r, theta = cmath.polar(z)  # r is 5.0, theta is approximately 0.93 radians
```
Conversion from Polar to Rectangular Coordinates: You can convert polar coordinates back to a complex number using cmath.rect().
```
z = cmath.rect(5.0, 0.93)  # z is approximately 3 + 4j
```

Use Cases for Complex Numbers

Electrical Engineering: Used in the analysis of AC circuits, where impedance is represented as a complex number.
Quantum Physics: Complex numbers are used in quantum mechanics to describe wave functions.
Control Systems: Complex numbers are used in the analysis of control systems, particularly in the context of transfer functions.

Complex numbers are a powerful mathematical tool that extends the concept of one-dimensional numbers to two dimensions, incorporating both a real and an imaginary component. Understanding how to work with complex numbers in Python is crucial for fields that require advanced mathematical calculations.

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