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Imaginary & Complex Numbers

Imaginary & Complex Numbers. Once upon a time…. -In the set of real numbers, negative numbers do not have square roots. -Imaginary numbers were invented so that negative numbers would have square roots and certain equations would have solutions.

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Imaginary & Complex Numbers

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  1. Imaginary & Complex Numbers

  2. Once upon a time…

  3. -In the set of real numbers, negative numbers do not have square roots. -Imaginary numbers were invented so that negative numbers would have square roots and certain equations would have solutions. -These numbers were devised using an imaginary unit named i.

  4. -The imaginary numbers consist of all numbers bi, where b is a real number and i is the imaginary unit, with the property that i² = -1. -The first four powers of i establish an important pattern and should be memorized. Powers of i

  5. Powers of i Divide the exponent by 4 No remainder: answer is 1. remainder of 1: answer is i. remainder of 2: answer is –1. remainder of 3:answer is –i.

  6. Powers of i 1.) Find i23 2.) Find i2006 3.) Find i37 4.) Find i828

  7. Complex Number System Reals Imaginary i, 2i, -3-7i, etc. Rationals (fractions, decimals) Integers (…, -1, -2, 0, 1, 2, …) Irrationals (no fractions) pi, e Whole (0, 1, 2, …) Natural (1, 2, …)

  8. 4.) 5.) -Express these numbers in terms of i. Simplify. 3.)

  9. You try… 6. 7. 8.

  10. To multiply imaginary numbers or an imaginary number by a real number, it is important first to express the imaginary numbers in terms of i.

  11. Multiplying 9. 10. 11.

  12. Complex Numbers a + bi imaginary real The complex numbers consist of all sums a + bi, where a and b are real numbers and i is the imaginary unit. The real part is a, and the imaginary part is bi.

  13. Add or Subtract 12. 13. 14.

  14. Multiplying & Dividing Complex Numbers Part of 7.9 in your book

  15. REMEMBER: i² = -1 Multiply 1) 2)

  16. You try… 3) 4)

  17. Multiply 5)

  18. You try… 6)

  19. You try… 7)

  20. Conjugate -The conjugate of a + bi is a – bi -The conjugate of a – bi is a + bi

  21. Find the conjugate of each number… 8) 9) 10) 11)

  22. Divide… 12)

  23. You try… 13)

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