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5.7.3 – Division of Complex Numbers

5.7.3 – Division of Complex Numbers. 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. Division.

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5.7.3 – Division of Complex Numbers

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  1. 5.7.3 – Division of Complex Numbers

  2. 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 • i2 = -1

  3. 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 can’t we have in the bottom of fractions? • Luckily, there is an easy to get rid of these using a specific property

  4. 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) = a2 – b2

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

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

  7. 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

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

  9. Example. Simplify (5 – 3i)/(1 + i)

  10. Example. Simplify (2 + i)/(1 – i)

  11. Assignment • Pg. 265 • 51 – 63 all

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