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Complex Number Tutorial

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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 + 10i ²
                             = 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(a ² + b ²)
                  = 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



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