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(a² + b²)
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 10i² = 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 - 6i²
= 8 + 22i + 6 (Remember that -6i² = -6(-1) = 6)
= 14 + 22i
Bottom = (4 + i)(4 - i)
= 16 - 4i + 4i - i²
= 16 + 0 + 1 (Remenber that -i² = 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(12² + 16²)
= 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
This tutorial will help you to calculate the Complex Number Multiplication, Division, and Square root problems.
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