Complex Number Tutorial

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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(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

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