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Complex Numbers

Complex Numbers. Understand complex numbers Simplify complex number expressions. Imaginary Numbers. What is the square root of 9?. What is the square root of -9?. Imaginary Numbers. There is no real number that when multiplied by itself gives a negative number.

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Complex Numbers

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  1. Complex Numbers • Understand complex numbers • Simplify complex number expressions

  2. Imaginary Numbers • What is the square root of 9? • What is the square root of -9?

  3. Imaginary Numbers • There is no real number that when multiplied by itself gives a negative number. • A new type of number was defined for this purpose. It is called an Imaginary Number. Imaginary numbers are NOT in the Real Set.

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

  5. Imaginary Numbers • The square root of -9 is an imaginary number... • When we simplify a radical with a negative coefficient inside the radical, we write it as an imaginary number.

  6. Imaginary Numbers • Simplify these radicals:

  7. Multiples of i • Consider multiplying two imaginary numbers: • So...

  8. Multiples of i • Powers of i:

  9. Multiples of i • This pattern repeats:

  10. 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? 84 + ? = 85 1 So the answer is:

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

  12. Solutions Involving i • Solve:

  13. Solutions Involving i • Solve:

  14. Solutions Involving i • Solutions:

  15. Complex Numbers • When we add a real number and an imaginary number we get a Complex Number . • Since the real and imaginary numbers are not like terms, we write complex numbers in the form a + bi • Examples: 3 - 7i, -2 + 8i, -4i, 5 + 2i

  16. Complex Numbers: A/S • To add or subtract two complex numbers, combine like terms (the real & imaginary parts). • Example: (3 + 4i) + (-5- 2i) = -2 + 2i

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

  18. Complex Numbers: M • To multiply two complex numbers, FOIL them: • Replace i2 with -1:

  19. Practice Multiply: • 5i(3 - 4i) • (1 - 3i)(2 - i) • (7 - 4i)(7 + 4i) = 20 + 15i = -1 - 7i = 65

  20. Complex Numbers: D • We leave complex quotients in fraction (rational) form: • But since i represents a square root, we cannot leave an i term in the denominator...

  21. Complex Numbers: D • We must rationalize any fraction with i in the denominator. Monomial Denominator: Binomial Denominator:

  22. Complex #: Rationalize • If the denominator is a monomial, multiply the top & bottom by i.

  23. Complex #: Rationalize • If the denominator is a binomial multiply the numerator and denominator by the conjugate of the denominator ...

  24. Complex #: Rationalize • When you multiply conjugate complex numbers, the imaginary part drops out:

  25. Practice • Simplify:

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