Unit 7.3 Multiplying and Dividing Complex Numbers in Rectangular Format
Multiplying Complex Numbers
When multiplying a polynomial, we use the FOIL method (see unit 5.3 for a refresher) . This is also true when multiplying two complex numbers together
[latex](a + bi)(c + di) = ac + adi + bci + bdi^2 = ac + adi + bci - bd = (ac - bd) + (ad + bc)i[/latex]
Multiplying Polynomials
Multiply (2 + 3i) and (4 + 5i)
[latex](2 + 3i)(4 + 5i) = 2 \cdot 4 + 2 \cdot 5i + 3i \cdot 4 + 3i \cdot 5i[/latex]
[latex]= 8 + 10i + 12i + 15i^2[/latex]
[latex]= 8 + 22i + 15(-1)[/latex]
[latex]= 8 + 22i - 15[/latex]
[latex]= -7 + 22i[/latex]
Dividing Complex Numbers
a + bi complex conjugate would be a – bi
Dividing a + bi by x + yi
[latex]\frac{a + bi}{x + yi} = \frac{(ax + by) + (bx - ay)i}{x^2 + y^2}[/latex]
Division of Complex Numbers
Divide (3 + 4i) by (1 – 2i)
[latex]\frac{3 + 4i}{1 - 2i}[/latex]
Multiply by the complex conjugate of the denominator
[latex]\frac{3 + 4i}{1 - 2i} \cdot \frac{1 + 2i}{1 + 2i} = \frac{(3 + 4i)(1 + 2i)}{(1 - 2i)(1 + 2i)}[/latex]
Simplify the denominator
[latex](1 - 2i)(1 + 2i) = 1^2 - (2i)^2 = 1 - 4(-1) = 1 + 4 = 5[/latex]
Expand the Numerator
[latex](3 + 4i)(1 + 2i) = 3 \cdot 1 + 3 \cdot 2i + 4i \cdot 1 + 4i \cdot 2i[/latex]
[latex]= 3 + 6i + 4i + 8i^2 = 3 + 10i + 8(-1)[/latex]
[latex]= 3 + 10i - 8[/latex]
[latex]= -5 + 10i[/latex]
Finish the division
[latex]\frac{3 + 4i}{1 - 2i} = \frac{-5 + 10i}{5} = -1 + 2i[/latex]