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Adding, Subtracting, Multiplying

Adding, Subtracting, Multiplying. Complex Numbers: z Î C. (4 + 3 i ) + (2 – 7 i ). Example: add. add the real part. 4 + 3 i + 2 – 7 i. add the imaginary part. 4 + 2 + 3 i – 7 i. – 4 i. = 6. Example: Subtract. (5 + 2 i ) – (7 – 3 i ). remove brackets. add the real part.

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Adding, Subtracting, Multiplying

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  1. Adding, Subtracting, Multiplying Complex Numbers: z ÎC (4 + 3i) + (2 – 7i) Example: add add the real part 4 + 3i + 2 – 7i add the imaginary part. 4 + 2+ 3i – 7i – 4i = 6

  2. Example: Subtract (5 + 2i) – (7 – 3i) remove brackets add the real part = 5 + 2i – 7 + 3i add the imaginary part. = 5 – 7+ 2i + 3i = – 2 + 5i

  3. Multiplying Complex Numbers Example: Multiply the following (2 + 5i)(4 – 3i) (use distributive property) = 8 – 6i + 20i – 15i2 (i2 = –1) = 8 + 14i – 15(–1) = 8 + 14i + 15 = 23 + 14i

  4. = –11 – 60i (5 – 6i)2 Simplify the following: (5 – 6i)(5 – 6i) = 25 – 30i – 30i + 36i2 = 25 – 60i + 36i2 (i2 = –1) = 25 – 60i + 36(–1) = 25 – 60i – 36

  5. Conjugates 5 – 2i 5 + 2i 3 + 4i 3 – 4i –5 + 2i –5 – 2i a + bi a – bi

  6. Simplify the following (conjugates): (5 – 6i)(5 + 6i) = 25 + 30i – 30i – 36i2 = 25 – 36i2 (i2 = –1) = 25 – 36(–1) = 25 + 36 = 61

  7. Multiplying conjugates: Example: (a – bi)(a + bi) (2 – 7i)(2 + 7i) = a2 + abi – abi – b2i2 a b (i2 = –1) = a2 – b2i2 = 22 + 72 = 4 + 49 = a2 – b2(–1) = 53 = a2 + b2 Product of conjugates is always real.

  8. Squaring Complex Numbers: Example: (3 + 5i)2 (a + bi)2 = (a + bi)(a + bi) a b = 32– 52 + 2(3)(5)i = a2 + abi + abi + b2i2 = 9 – 25 + 30i = a2 + b2i2 + 2abi = – 16 + 30i = a2 – b2 + 2abi (3 – 5i)2 (a + bi)2 = a2 – b2 + 2abi = – 16 – 30i (a – bi)2 = a2 – b2 – 2abi

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