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Understanding Complex Numbers and Their Real Solutions

This guide delves into complex numbers, highlighting the distinction between real and imaginary components. We'll explore the concept of the imaginary unit, which facilitates the definition of complex solutions. By learning how to express complex numbers and perform operations such as division using conjugates, you will gain a solid understanding of how real and imaginary numbers are combined. Through examples and explanations, we clarify the format of complex numbers, the roles of real and imaginary parts, and the principles that govern their interactions.

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Understanding Complex Numbers and Their Real Solutions

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  1. Math 71B 7.7 – Complex Numbers

  2. Does have any solutions that are real numbers? Nope! Let’s define a solution (called the imaginary unit): ______ (so that )

  3. Does have any solutions that are real numbers? Nope! Let’s define a solution (called the imaginary unit): ______ (so that )

  4. Does have any solutions that are real numbers? Nope! Let’s define a solution (called the imaginary unit): ______ (so that )

  5. Does have any solutions that are real numbers? Nope! Let’s define a solution (called the imaginary unit): ______ (so that )

  6. Does have any solutions that are real numbers? Nope! Let’s define a solution (called the imaginary unit): ______ (so that )

  7. Ex 1.Write as a multiple of :

  8. Ex 1.Write as a multiple of :

  9. Ex 1.Write as a multiple of :

  10. Ex 1.Write as a multiple of :

  11. Ex 1.Write as a multiple of :

  12. Ex 1.Write as a multiple of :

  13. Ex 1.Write as a multiple of :

  14. Ex 1.Write as a multiple of :

  15. Real and imaginary numbers together make the ________________________, which can all be written in the form: __________ ( and are real #’s; is called the ____________; is called the ________________________) ex:

  16. Real and imaginary numbers together make the ________________________, which can all be written in the form: __________ ( and are real #’s; is called the ____________; is called the ________________________) ex: complex numbers

  17. Real and imaginary numbers together make the ________________________, which can all be written in the form: __________ ( and are real #’s; is called the ____________; is called the ________________________) ex: complex numbers

  18. Real and imaginary numbers together make the ________________________, which can all be written in the form: __________ ( and are real #’s; is called the ____________; is called the ________________________) ex: complex numbers real part

  19. Real and imaginary numbers together make the ________________________, which can all be written in the form: __________ ( and are real #’s; is called the ____________; is called the ________________________) ex: complex numbers real part imaginary part

  20. Real and imaginary numbers together make the ________________________, which can all be written in the form: __________ ( and are real #’s; is called the ____________; is called the ________________________) ex: complex numbers real part imaginary part

  21. Real and imaginary numbers together make the ________________________, which can all be written in the form: __________ ( and are real #’s; is called the ____________; is called the ________________________) ex: complex numbers real part imaginary part

  22. Ex 2. Ex 3.

  23. Note: When multiplying square roots with negatives inside, pull out the ’s first! Ex 4.

  24. Note: When multiplying square roots with negatives inside, pull out the ’s first! Ex 4.

  25. Note: When multiplying square roots with negatives inside, pull out the ’s first! Ex 4.

  26. Note: When multiplying square roots with negatives inside, pull out the ’s first! Ex 4.

  27. Note: When multiplying square roots with negatives inside, pull out the ’s first! Ex 4.

  28. Conjugates and Division The conjugate of is __________. To divide complex #’s, we can use the conjugate to help.

  29. Conjugates and Division The conjugate of is __________. To divide complex #’s, we can use the conjugate to help.

  30. Conjugates and Division Ex 5.Divide and simplify to the form . Ex 6.Divide and simplify to the form .

  31. Powers of Ex 7.Simplify:

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