Division of Complex Numbers

Complex numbers are numbers with two parts: real and imaginary. They are represented as a + bi, where a and b are the real and imaginary parts respectively. We use complex numbers to solve equations that cannot be solved only with real numbers. 
 

Complex numbers can be divided by using a method that involves the conjugate of the denominator. Two complex numbers, z₁ = a₁ + iy₁ and z₂ = a₂ + iy₂, can be expressed as 
z₁/z₂=a₁ + iy₁/a₂ + iy2

To find the formula for dividing two complex numbers, let's consider the complex numbers as z₁ = a + ib and z₂ = c + id. So, it can be calculated as 
z₁ / z₂=[ ac + bd/c² + d²] + i[(bc - ad/c² + d²)]  
 

Follow these steps to divide two complex numbers. 

Step 1: Make sure both numerator and denominator are complex numbers written correctly in standard form. 

Step 2: Determine the denominator’s conjugate. Example, if the denominator is c + id, then the conjugate is c - id. 

Step 3: The conjugate should be multiplied by both numerator and denominator.

Step 4: The denominator must be solved using the difference of squares formula.  

Step 5: Separate the result into its real and imaginary parts, expressed as a + bi.

Let us try dividing 3 + 4i/2 + i

Step 1: Checking for the standard form, we conclude that both numerator and denominator are in the standard form. 

Step 2: The denominator is 2 + i, so its conjugate is 2 - i.

Step 3: The conjugate must be multiplied with numerator and denominator.
So, (3 + 4i) x (2 - i)/(2 + i) x (2 - i)

Step 4: Expanding the numerator, we get: (3 + 4i) (2 - i) = 3(2) - 3(i) + 4i(2) - 4i(i) 
= 6 - 3i + 8i - 4i²
As i² = -1, 6 - 3i + 8i - 4i² becomes, 
= 6 - 3i + 8i + 4
= 10 + 5i
Applying the difference of squares formula to solve the denominator, 
(2 + i) (2 - i) = 2² - i²
= 4 - (-1) 
= 4 +1 = 5

Step 5: Divide the expanded numerator by the denominator
10 + 5i/5 = 10/5 + 5i/5 = 2 + i
 

Here, we will learn how to divide complex numbers in polar form. Let’s say we need to divide z₁ by z₂ where z₁ = r₁(cos Θ₁ + i sin Θ₁) and z₂ = r₂(cos Θ₂ + i sin Θ₂). 

So, z₁ / z₂ = r₁(cos Θ₁ + i sin Θ₁) / r₂ (cos Θ₂ + i sin Θ₂)
= r₁(cos Θ₁ + i sin Θ₁) / r₂ (cos Θ₂ + i sin Θ₂) × (cos Θ₂ - i sin Θ₂) / (cos Θ₂ - i sin Θ₂)
= r₁(cos Θ₁ + i sin Θ₁) (cos Θ₂ + i sin Θ₂) / r₂ (cos² Θ₂ - (i)₂ sin²Θ₂) 
= r₁/r₂[cos (Θ₁ - Θ₂) + i sin(Θ₁ - Θ₂)]
= r(cos Θ + i sin Θ)
Where, Θ = Θ₁ - Θ₂ and r = r₁/r₂
So, the formula for dividing complex number in polar form is:
z₁/z₂ = r(cos Θ + i sinΘ) 
 

Complex numbers are used daily in fields like engineering and quantum mechanics. Here are a few examples where the division of complex numbers is applied:

• In electrical engineering, complex numbers are divided to analyze the alternating current (AC) circuits. E.g., in electrical engineering, division of complex numbers is used in Ohm’s Law (V = IZ) to compute voltage or current when impedance Z is complex.

• In quantum mechanics, the division of complex numbers is used to understand wave function transformations and probability calculations.

• It is used to calculate ratios in the design of antennas, microwave circuits, and fiber optic communication systems. Here, engineers often work with signal reflection, impedance matching, and wave propagation, all of which involve complex numbers.