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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

Dividing complex numbers involves multiplying the numerator and the denominator by the complex conjugate of the denominator. This process eliminates the imaginary part in the denominator, resulting in a real number

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]

 

 

 

 

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