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Understanding Complex Numbers and Operations with the Imaginary Unit

This section explains complex numbers, represented as a + bi, where a and b are real numbers. It introduces the imaginary unit i, defined as the square root of -1. The properties and operations of complex numbers are detailed, including addition, subtraction, and multiplication using the FOIL method. Furthermore, it discusses the conjugate of complex numbers and the product of conjugates. Lastly, it emphasizes the importance of simplifying expressions containing complex numbers, ensuring there are no imaginary units in the denominator.

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Understanding Complex Numbers and Operations with the Imaginary Unit

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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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