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Section 2.4

Section 2.4. Complex Numbers. The letter i represents the number whose square root is –1. Imaginary unit. i 2 = –1. If a is a positive real number, then the principal square root of negative a is the imaginary number . Examples :. = 2 i. = 6 i. Complex Number.

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Section 2.4

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  1. Section 2.4 Complex Numbers

  2. The letter i represents the number whose square root is –1. Imaginary unit i2 = –1 If a is a positive real number, then the principal square root of negative a is the imaginary number

  3. Examples: = 2i = 6i

  4. Complex Number A complex number is a number of the form a+ bi, where aand bare real numbers and The number a is the real part of a + bi, and biis the imaginary part.

  5. a + bi 2 7i + 20 – 3i Examples of complex numbers: Imaginary Part Real Part Real Numbers: a + 0i Imaginary Numbers: 0 + bi

  6. Simplify: 1. 2. = 8i 3. a + bi form

  7. To add or subtract complex numbers: 1. Write each complex number in the form a + bi. 2. Add or subtract the real parts of the complex numbers. 3. Add or subtract the imaginary parts of the complex numbers. (a + bi ) + (c + di ) = (a + c) + (b + d )i (a + bi ) – (c + di ) = (a – c) + (b – d )i

  8. Examples: Add (11 + 5i) + (8 – 2i ) = (11 + 8) + (5i – 2i ) Group real and imaginary terms. = 19 + 3i a + bi form Group real and imaginary terms. = 31 a + bi form

  9. Examples: Subtract: (– 21 + 3i ) – (7 – 9i) = (– 21 – 7) + [(3 – (– 9)]i = (– 21 – 7) + (3i + 9i) = –28 + 12i = 5+i

  10. The product of two complex numbers is defined as: 1. Use the FOIL method to find the product. 2. Replace i2 by –1. 3. Write the answer in the form a + bi. (a + bi)(c + di ) = (ac – bd) + (ad + bc)i

  11. Examples: 1.

  12. 2. 7i(11– 5i) = 77i– 35i2 = 77i– 35 (– 1) = 35 + 77i 3. (2 + 3i)(6 – 7i) = 12 –14i+18i–21i2 = 12 + 4i–21i2 = 12 + 4i–21(–1) = 12 + 4i + 21 = 33 + 4i

  13. The complex numbers a + bi and a− bi are called conjugates. The product of conjugatesis the real number a2+ b2. Example: (5 + 2i)(5 – 2i) = (52 – 4i2) = 25 – 4 (–1) = 29

  14. Dividing Complex Numbers

  15. A rational expression, containing one or more complex numbers, is in simplest form when there are no imaginary numbers remaining in the denominator.

  16. Example:

  17. Simplify: HW: pp. 167-168 (8-68 multiples of 4, 76)

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