Complex numbers- College Algebra
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The document explains the progression of number systems starting from natural numbers to integers, rationals, irrationals, and finally complex numbers. It introduces the concept of complex numbers as encompassing real and imaginary parts, illustrated with examples and operations such as addition and multiplication. The document also touches on the conjugate of complex numbers and techniques for simplifying them.
Complex numbers- College Algebra
- 2. In the beginning there were counting numbers (Natural Numbers) And then we needed Integers 1 2
- 3. In the beginning there were counting numbers And then we needed integers 1 -1 2 -3
- 4. In the beginning there were counting numbers And then we needed integers And rationals 1 -1 2 -3 0.41
- 5. In the beginning there were counting numbers And then we needed integers And rationals And irrationals 1 -1 2 -3 0.41 2
- 6. In the beginning there were counting numbers And then we needed integers And rationals And irrationals And reals 1 -1 2 -3 0.41 2 0
- 7. 1 2
- 8. 8
- 9. 9
- 10. By definition Consider powers if i 1 i i2 1 10 2 i 1 i 3 i 2 i i i 4 i 2 i 2 i 5 i 4 i i i 1 1 1 1 ... It's any number you can imagine
- 11. i i 2 3 4 5 6 7 i i i i i i
- 12. i i 3 4 5 6 7 2 2 1 1 i i i i i i
- 13. i i 1 1 i i 1 i i 3 4 5 6 7 2 2 1 1 1 i i i i
- 14. i i i i i i i i i i i i i i 1 1 1 5 9 13 2 6 10 14 3 7 11 15 4 8 12 16 i i i i
- 15. 15
- 16. Now we can handle quantities that occasionally show up in mathematical solutions What about 16 a 1 a i a 49 18
- 17. Combine real numbers with imaginary numbers ◦ a + bi Examples 17 Real part Imaginary part 3 4i 3 6 i 2 4.5 i 2 6
- 18. We have always used them. 6 is not just 6 it is 6 + 0i. Complex numbers incorporate all numbers. 1 2i 3 + 4i -1 2 -3 0.41 2 0
- 19. Note: A number such as 3i is a purely imaginary number A number such as 6 is a purely real number a + ib is the general form of a complex number 6 + 3i is a complex number -2 + 7i is also a complex number
- 20. If x + iy = 6 – 4i then x = 6 and y = -4 The ‘real part’ of 6 – 4i is 6 The ‘imaginary part’ of 6 – 4i is -4 20
- 21. 3i = 0 + 3i -7= -7 + 0i 0= 0 + 0i
- 22. Write these complex numbers in standard form a + bi 1. 2. 3. 4. 22 9 75 16 7 5 144 100
- 23. Complex numbers can be combined with ◦ Addition ◦ Subtraction ◦ Multiplication ◦ Division 3i8 2i 912i715i 23 24i43i i i 3 5 2
- 24. Example 1: 24 24i43i
- 25. Example 2: 25 3i8 2i
- 26. Example 3: (Use FOIL method) 26 912i715i
- 27. We need to know Conjugate of a+bi= a-bi (Conjugate of a complex number is obtained by changing the sign of the imaginary part) For example: 27
- 28. 28
- 29. To solve Multiply and divide by its conjugate 29
- 30. Qs. Simplify 2 3i 7 The trick is to make the denominator real:
- 31. 6. Simplify 2 37i The trick is to make the denominator real: i i 2 3 7 2(3 7) 3 7 3 7 58 i i i (3 7) 29 i 7 3 29
- 32. Example 4: i i Division technique ◦ Multiply numerator and denominator by the conjugate of the denominator 32 3 52 i i i i i i 3 5 2 5 2 5 2 15 6 2 25 4 i 2 6 15 i 6 15 29 29 29 i 3 5 2 i i
- 33. 1. Simplify 4
- 37. 6. Simplify 2 37i
- 38. 7. 6 13 0 2 6 36 52 2 6 16 2 6 16 1 2 x x x i complexsolutions Conjugates 3 2 ( ) x Solve x x
- 39. Use the correct principles to simplify the following: 39 3 121 4 81 4 81 2 3 144
- 40. Lesson P.6 Page - 65 Exercises 1 – 32, 37, 38, 41, 42, 55-62 40