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5.7.3 – Division of Complex Numbers

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We now know about adding, subtracting, and multiplying complex numbers Combining like terms Reals with reals Imaginary with imaginary When we multiply, treat as two binomials FOIL Sqaure i 2 = -1

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Division The division of two complex numbers is a little more challenging Rule: You cannot have a complex number as the denominator (bottom) of a fraction What else cant we have in the bottom of fractions? Luckily, there is an easy to get rid of these using a specific property

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Rationalize To rationalize the division of complex numbers, we will take use of the following property; If you have the number a +bi, the complex conjugate is a – bi Example. If you have 3 + 5i, the conjugate is 3 – 5i Property: (a + bi)(a – bi) = a 2 – b 2

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Example. Identify the complex conjugates of the following complex numbers A) 4 + 7i B) -3 – 9i C) 10i D) 4i + 3

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Example. Simplify the following expressions. A) (1 + 2i)(1 – 2i) B) (3 – 2i)(3 + 2i) C) (6 + 2i)(6 – 2i)

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Division of Complex Numbers Now, from this, we can now look into how we divide complex numbers If given the complex number 6/(5 + i), we must get rid of the complex number in the bottom To do this, we will multiply top and bottom by the complex conjugate

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Example. Simplify the expression 6/(5 + i) Short cut for multiplying the complex conjugates?

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Example. Simplify (5 – 3i)/(1 + i)

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Example. Simplify (2 + i)/(1 – i)

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Assignment Pg. 265 51 – 63 all

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Multiply Simplify Write the expression as a complex number.

Multiply Simplify Write the expression as a complex number.

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