complex Fundamentals¶

The complex type represents complex numbers, which include both a real part and an imaginary part.

A complex number has the form:

where:

• \(a\) is the real part
• \(b\) is the imaginary part
• j is Python’s symbol for the imaginary unit

Examples:

1 + 2j
3 - 4j
0 + 1j
````

```mermaid
flowchart LR
    A[complex number]
    A --> B[real part]
    A --> C[imaginary part]

1. Creating Complex Numbers¶

Complex numbers can be written directly.

z = 2 + 3j
print(z)

Output:

(2+3j)

They can also be created with complex().

z = complex(2, 3)
print(z)

Output:

(2+3j)

2. Real and Imaginary Parts¶

Python complex numbers expose .real and .imag attributes.

z = 4 + 5j

print(z.real)
print(z.imag)

Output:

4.0
5.0

3. Complex Arithmetic¶

Complex numbers support arithmetic operations.

a = 1 + 2j
b = 3 + 4j

print(a + b)
print(a - b)
print(a * b)

Output:

(4+6j)
(-2-2j)
(-5+10j)

4. Conjugates¶

The conjugate of a complex number changes the sign of the imaginary part.

z = 2 + 3j
print(z.conjugate())

Output:

(2-3j)

Conjugates are useful in algebra and engineering.

5. Complex Numbers and Magnitude¶

The magnitude of a complex number behaves like its distance from the origin in the complex plane.

z = 3 + 4j
print(abs(z))

Output:

5.0

This follows the Pythagorean relationship:

[ |a + bj| = \sqrt{a^2 + b^2} ]

flowchart TD
    A[origin]
    A --> B[real axis]
    A --> C[imaginary axis]
    B --> D[point a + bj]
    C --> D

6. Equality and Comparison¶

Complex numbers support equality comparison:

print((1 + 2j) == (1 + 2j))

But they do not support ordering comparisons such as < or >.

# (1+2j) < (2+3j)   # TypeError

This is because complex numbers are not ordered like real numbers.

7. Worked Examples¶

Example 1: addition¶

z1 = 1 + 1j
z2 = 2 + 3j

print(z1 + z2)

Output:

(3+4j)

Example 2: magnitude¶

z = 6 + 8j
print(abs(z))

Output:

10.0

Example 3: conjugate¶

z = 5 - 2j
print(z.conjugate())

Output:

(5+2j)

8. Common Uses¶

Complex numbers appear in:

• signal processing
• electrical engineering
• physics
• mathematics

They are less common in beginner Python programming, but Python includes them as a built-in numeric type.

9. Summary¶

Key ideas:

• complex represents numbers with real and imaginary parts
• complex numbers use j for the imaginary unit
• Python supports complex arithmetic directly
• .real, .imag, and .conjugate() expose useful properties
• abs() gives the magnitude

The complex type extends Python’s numeric model beyond ordinary real-number arithmetic.