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7-7 Imaginary and Complex Numbers

7-7 Imaginary and Complex Numbers. Why Imaginary Numbers?. What is the square root of 9?. What is the square root of -9?. no real number. New type of number was defined for this purpose. It is called an Imaginary Number Imaginary numbers are NOT in the Real Set.

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7-7 Imaginary and Complex Numbers

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  1. 7-7 Imaginary and Complex Numbers

  2. Why Imaginary Numbers? • What is the square root of 9? • What is the square root of -9? no real number New type of number was defined for this purpose. It is called an Imaginary Number Imaginary numbers are NOT in the Real Set.

  3. The constant, i, is defined as the square root of negative 1: • Multiples of i are called Imaginary Numbers

  4. The square root of -9 is an imaginary number... • To simplify a square root with negative coefficient inside radical, write it as an imaginary number.

  5. Powers of i:

  6. This pattern repeats:

  7. 84 Multiples of i • We can find higher powers of i using this repeating pattern: i, -1, -i, 1 What is the highest number less than or equal to 85 that is divisible by 4? So the answer is:

  8. i28 i75 i113 i86 i1089 1 -i i -1 i Powers of i - Practice

  9. Negative Exponents Ex: Ex: Odd negative powers are opposite Even negative powers are the same!

  10. Simplify: Ex 1: Ex 2:

  11. Multiply • Ex 3 • Ex 4 • Ex 5

  12. Complex Numbers • Complex Number : a + bi , Where a and b are real #s and i is imaginary part • real and imaginary numbers are not like terms, • Examples: 3 - 7i, -2 + 8i, -4i, 5 + 2i

  13. Complex #s Imaginary #s Real #s Rational #s Irrational #s

  14. Add and Subtract • Combine Like Terms (the real & imaginary parts). • Example: (3 + 4i) + (-5- 2i) = -2 + 2i

  15. Practice Add these Complex Numbers: • (4 + 7i) - (2 - 3i) • (3 - i) + (7i) • (-3 + 2i) - (-3 + i) = 2 +10i = 3 + 6i = i

  16. Assignment • 7-7/323/1-41, 43-48, 56-64

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