The Complex Number Division Calculator is a mathematical tool that simplifies the process of dividing two complex numbers. Complex numbers include a real part and an imaginary part, written in the form z = a + bi, where i is the square root of -1. This calculator performs division of two complex numbers quickly and accurately by using the conjugate method. It eliminates the need for manual calculations, ensuring precision while saving time. This tool belongs to the Mathematics Calculators category and is ideal for students, engineers, and professionals who work with complex number computations.
Formula of Complex Number Division Calculator
If the two complex numbers are:
- Numerator: z₁ = a + bi
- Denominator: z₂ = c + di
The division is performed using the formula:
z₁ / z₂ = [(a + bi) × (c – di)] / (c² + d²)
Where:
- a = Real part of the numerator
- b = Imaginary part of the numerator
- c = Real part of the denominator
- d = Imaginary part of the denominator
Steps to Calculate:
- Multiply the numerator and denominator by the conjugate of the denominator:
- The conjugate of (c + di) is (c – di).
So, z₁ × conjugate(z₂) = (a + bi) × (c – di).
- The conjugate of (c + di) is (c – di).
- Use the distributive property: (a + bi)(c – di) = ac – adi + bci – bdi².
- Simplify using i² = -1: Replace i² with -1, leading to: (ac + bd) + (bc – ad)i.
- Divide the result by the modulus squared of the denominator: The modulus squared of z₂ is |z₂|² = c² + d².
- Combine and simplify: z₁ / z₂ = [(ac + bd) + (bc – ad)i] / (c² + d²).
Precalculated Results Table
Below is a table showing general results for dividing some common complex numbers without manual calculations.
| Numerator (z₁) | Denominator (z₂) | Result (z₁ / z₂) |
|---|---|---|
| 1 + i | 1 + i | 1 |
| 2 + 3i | 1 + i | 2.5 + 0.5i |
| 1 + 2i | 2 – i | 0 + 1i |
| 3 + 4i | 1 + 2i | 2.2 – 0.4i |
| 2 + i | 2 + 2i | 0.75 – 0.25i |
This table allows users to cross-check their division results quickly.
Example of Complex Number Division Calculator
Let’s divide two complex numbers: z₁ = 3 + 2i and z₂ = 1 – i.
- Write the conjugate of the denominator: Conjugate of 1 – i is 1 + i.
- Multiply the numerator and denominator by the conjugate:
(3 + 2i)(1 + i) = 3 + 3i + 2i + 2i². - Simplify using i² = -1:
3 + 3i + 2i + 2(-1) = 3 + 3i + 2i – 2 = 1 + 5i. - Find the modulus squared of the denominator: (1 – i)(1 + i) = 1 – i² = 1 – (-1) = 2.
- Divide the result by the modulus squared:
z₁ / z₂ = (1 + 5i) / 2 = 0.5 + 2.5i.
The final result is 0.5 + 2.5i.
Most Common FAQs
The conjugate of a complex number z = a + bi is z̅ = a – bi. It has the same real part but the opposite imaginary part.
Multiplying by the conjugate eliminates the imaginary part in the denominator, making the division simpler and resulting in a real denominator.
The modulus squared of z = c + di is |z|² = c² + d². It represents the sum of the squares of the real and imaginary parts.