Complex Number Calculator

What is a Complex Number?

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, satisfying the equation i² = −1. The real part of the complex number is a, and the imaginary part is b.

Complex Number Operations

Here are some common operations on complex numbers:

1. Addition and Subtraction

\[ (a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i \]

2. Multiplication

\[ (a + bi)(c + di) = (ac - bd) + (ad + bc)i \]

3. Division

\[ \frac{a + bi}{c + di} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2} \]

4. Absolute Value (Magnitude)

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

5. Argument (Phase)

\[ \arg(a + bi) = \tan^{-1}\left(\frac{b}{a}\right) \]

6. Conjugate

\[ \overline{a + bi} = a - bi \]

Example Calculation

Let's multiply two complex numbers: (3 + 2i) and (1 - i)

1. Use the multiplication formula: (ac - bd) + (ad + bc)i
2. a = 3, b = 2, c = 1, d = -1
3. (3 × 1 - 2 × (-1)) + (3 × (-1) + 2 × 1)i
4. (3 + 2) + (-3 + 2)i
5. 5 - i

Visual Representation

Complex numbers can be visualized on a complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. Each complex number is represented as a point or vector on this plane.