Complex Number Tutorial

Complex Number Tutorial

Definition:

Complex number have addition, subtraction, multiplication, division. A complex number is in the form of a + bi (a real number plus an imaginary number) where a and b are real numbers and i is the imaginary unit. When a single letter x = a + bi is used to denote a complex number it is sometimes called 'affix'.

Formula:

Multiplication = (a+bi) × (a+bi) Division = (a+bi) / (a+bi) Square root r = sqrt(a ² + b ²) r1 = x + yi r2 = -x - yi
where,
y = sqrt((r-a) / 2) x = b / 2y
Example 1:

Multiplying two complex numbers. Multiply (3 + 2i) and (4 + 5i)

Step 1:

The given problem is in the form of (a+bi) × (a+bi)

 (3 + 2i)(4 + 5i) = (3 × 4) + (3 × (5i)) + ((2i) × 4) + ((2i) × (5i)) = 12 + 15i + 8i + 10i² = 12 + 23i -10 (Remenber that 10i ² = 10(-1) = -10) = 2 + 23i

Therefore, (3 + 2i)(4 + 5i) = 2+23i

Example 2:

Dividing one complex number by another. Divide (2 + 6i) / (4 + i).

Step 1:

The given problem is in the form of (a+bi) / (a+bi) First write down the complex conjugate of 4+i ie., 4-i

Step 2:

Multiply both the top and bottom by that number

 Top = (2 + 6i)(4 - i) = 8 - 2i + 24i - 6i ² = 8 + 22i + 6 (Remember that -6i ² = -6(-1) = 6) = 14 + 22i Bottom = (4 + i)(4 - i) = 16 - 4i + 4i - i ² = 16 + 0 + 1 (Remenber that -i ² = 1) = 17
Step 3:

Carry out the division The ratio is now (14 + 22i) / 17 Therefore, (2 + 6i) / (4 + i) = 14/17 + 22i/17

Example 3:

Find the square root of 12 + 16i.

Step 1:

The given problem is in the form of (a+bi) r = sqrt(a² + b²)   = sqrt(12 ² + 16 ²)   = sqrt(144 + 256)   = sqrt(400) r = 20

Step 2:

For finding y we have to use the formula. y = sqrt((r - a) / 2)   = sqrt((20 - 12)/2)   = sqrt(8 / 2)   = sqrt(4) y = 2

Step 3:

Substitute the value of b and y in x. x = b / 2y   = 16 / 2 ×2   = 16 / 4 x = 4

Step 4:

To find the square root of 12 + 16i substitute x and y value in r1 and r2. r1 = x + yi = 4 + 2i r2 = -x - yi = -4 - 2i Therefore, square root of 12 + 16i is, r1 = 4 + 2i, r2 = -4 - 2i

Related Calculator:

This tutorial will help you to calculate the Complex Number Multiplication, Division, and Square root problems.