2. 5.4 – Complex Numbers Our Imaginary Friends

4. Uses of Imaginary Numbers Contrary to its name, the “imaginary number” exists in a similar fashion to the “real number,” often describing physical characteristics that we can detect, observe, and measure. Common applications are circuit analysis in electrical engineering and vibration analysis in mechanical engineering. Want to learn more? http://www.picomonster.com/imaginary- numbers-lesson-1/

5. Fractals: What are they? A fractal is an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales. The object need not exhibit exactly the same structure at all scales, but the same "type" of structures must appear on all scales.self-similarity

6. Examples of Fractals Examples include clouds, river networks, fault lines, mountain ranges, craters, snow flakes, crystals, lightning, cauliflower or broccoli, and systems of blood vessels and pulmonary vessels, and ocean waves. DNA and heartbeats can be analyzed as fractals.river networks fault linesmountain rangescraters snow flakescrystalslightning cauliflowerbroccoli blood vesselspulmonary vesselsocean wavesDNAheartbeats

11. 5.4 – Complex Numbers Objectives: 1. Be able to find, identify, and name imaginary numbers 2. Be able to add and subtract imaginary numbers 3. Be able to multiply and divide imaginary numbers Vocabulary: imaginary unit - i, complex numbers, standard form, imaginary numbers, pure imaginary numbers, conjugate

12. Objective #1 – Be able to find, identify, and name imaginary numbers. 5.4 – Complex Numbers

13. complex number = a + bi a is the real part b is the imaginary part a + 0i are the real numbers a + bi are the imaginary numbers 0 + bi are the pure imaginary numbers Objective #1 – Be able to find, identify, and name imaginary numbers. 5.4 – Complex Numbers

14. Examples of Complex Numbers 4 + 0i = 4 (Real Number) 2 – 3i (Imaginary Number  Real Number and Imaginary Number Combo) 0 + 6i = 6i (Pure imaginary Number)

15. The Square Root of a Negative Number Properties

16. Examples

17. Examples

18. Examples

19. Examples

20. Objective #2 – Be able to add and subtract imaginary numbers. 5.4 – Complex Numbers Real partImaginary part

21. Examples

22. Examples

23. Examples

24. Solving Quadratic Equations with i

26. 1. Simplify the radical expression. 1. Solve the equation. 1. Write the expression as a complex number in standard form. Exit Slip Complete the following problems individually. You may use your notes. Remain quiet until all exit slips have been collected.

27. pg. 277 #18-28 even, 39-42 Homework Homework…

28. 5.4 – Complex Numbers Our Imaginary Friends

29. REMEMBER Objective #3 – Be able to multiply and divide imaginary numbers. 5.4 – Complex Numbers

30. Example

31. Example

32. Example

33. Example This answer is a REAL number!

34. Conjugates: a + bi and a - bi multiplying conjugates will always give you a real number! Objective #3 – Be able to multiply and divide imaginary numbers. 5.4 – Complex Numbers

35. Example

36. Example

37. Future electrical engineer? Check out pg. 279 #95, 96 Homework pg. 278 #47-62 (every third problem) That lesson was…