Division of a Complex Number by a Complex Number

Introduction to Division in Complex Numbers

Division in Complex Numbers involves dividing a complex number by another complex number, just as in real numbers.
The main difference is that, in complex numbers, the denominator has to be a real number. So, the complex number in the denominator is converted into a real number before performing the division.

The principle of division involves two complex numbers, represented as a + bi and c + di.
To divide one complex number by another, we multiply both the numerator and the denominator by the conjugate of the denominator, c - di.

First, write the complex numbers in the form (a + bi) / (c + di).
Multiply both the numerator (the top number) and the denominator (the bottom number) by the conjugate, (c - di), of the denominator.
This will give you a new complex number as (a + bi)(c - di) / (c + di)(c - di).
Apply the FOIL (First, Outer, Inner, Last) method to multiply out.
The denominator becomes a real number as the product of a complex number and its conjugate always results in a real number, i.e., c² + d².
Simplify the numerator and denominator separately and then simplify the whole expression.
Finally, write your answer in the form of a + bi.

Consider (3 + 4i) / (1 - 2i); the conjugate of the denominator is (1 + 2i).
So, multiply numerator and denominator by (1 + 2i): (3 + 4i)(1 + 2i) / (1 - 2i)(1 + 2i).
Simplifying gives (3 + 2i + 4i - 8i²) / (1 - 4i²) = (3 + 6i + 8) / (1 + 4) = (11 + 6i) / 5 = 2.2 + 1.2i.

Division of complex numbers is fundamental in solving problems in physics, engineering, and computer science.
Understanding this concept is crucial as it allows for the simplification of problems involving complex equations.