What is a Complex Number?
A complex number is a number that includes a real part and an imaginary part, expressed as:
$$ z = a + bi $$
Where:
- a is the real part
- b is the imaginary part
- i represents √-1, also known as the imaginary unit.
Complex numbers are used in engineering, physics, signal processing, and many advanced mathematical fields.
Supported Operations
You can use this tool to perform the following operations on complex numbers:
1. Addition
To add two complex numbers:
$$ (a + bi) + (c + di) = (a + c) + (b + d)i $$
2. Subtraction
To subtract one complex number from another:
$$ (a + bi) – (c + di) = (a – c) + (b – d)i $$
3. Multiplication
Multiply two complex numbers:
$$ (a + bi) \times (c + di) = (ac – bd) + (ad + bc)i $$
4. Division
To divide one complex number by another:
$$ \frac{a + bi}{c + di} = \frac{(ac + bd) + (bc – ad)i}{c^2 + d^2} $$
5. Modulus (Magnitude)
The modulus of a complex number tells us how far it is from zero in the complex plane.
$$ |z| = \sqrt{a^2 + b^2} $$
6. Argument (Angle \( \theta \))
The argument of a complex number is the angle it makes with the positive x-axis.
$$ \arg(z) = \tan^{-1} \left(\frac{b}{a}\right) $$
7. Conjugate
The conjugate of a complex number flips the sign of the imaginary part.
$$ \overline{z} = a – bi $$