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'.
Multiplying two complex numbers. Multiply (3 + 2i) and (4 + 5i)
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
Dividing one complex number by another. Divide (2 + 6i) / (4 + i).
The given problem is in the form of (a+bi) / (a+bi) First write down the complex conjugate of 4+i ie., 4-i
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 |
Carry out the division The ratio is now (14 + 22i) / 17 Therefore, (2 + 6i) / (4 + i) = 14/17 + 22i/17
Find the square root of 12 + 16i.
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
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
Substitute the value of b and y in x. x = b / 2y = 16 / 2 ×2 = 16 / 4 x = 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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