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6/11/2014 10:01 PM15-4 - Complex Numbers1 WARM-UP 1.4. 2.5. 3.

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Presentation on theme: "6/11/2014 10:01 PM15-4 - Complex Numbers1 WARM-UP 1.4. 2.5. 3."— Presentation transcript:

1 6/11/2014 10:01 PM15-4 - Complex Numbers1 WARM-UP 1.4. 2.5. 3.

2 6/11/2014 10:01 PM15-4 - Complex Numbers2 C OMPLEX N UMBERS

3 6/11/2014 10:01 PM15-4 - Complex Numbers3 IMAGINARY NUMBERS Imaginary Numbers are numbers can be written as a real number times i.

4 6/11/2014 10:01 PM10.7 - Complex Numbers4 I MAGINARY N UMBERS Imaginary Numbers are numbers that can be written as a real number times i. Steps: 1.View the radical without the negative and simplify it 2.After simplified form, attach an i onto the real number 3.Simplify further, if needed

5 6/11/2014 10:01 PM10.7 - Complex Numbers5 E XAMPLE 1 Simplify After simplified form, attach an i onto the real number

6 6/11/2014 10:01 PM10.7 - Complex Numbers6 E XAMPLE 1 x = 0 Where does it cross the x-axis? +2i

7 Simplify 6/11/2014 10:01 PM10.7 - Complex Numbers7 E XAMPLE 2

8 Simplify 6/11/2014 10:01 PM10.7 - Complex Numbers8 E XAMPLE 3

9 Simplify 6/11/2014 10:01 PM10.7 - Complex Numbers9 E XAMPLE 4

10 Simplify 6/11/2014 10:01 PM10.7 - Complex Numbers10 Y OUR T URN

11 Rules with Imaginaries: i 1 = i i 2 = –1 Answers CAN have an i but CAN NOT have two is Two is make –1 6/11/2014 10:01 PM10.7 - Complex Numbers11 I MAGINARY N UMBERS

12 6/11/2014 10:01 PM10.7 - Complex Numbers12 W HY IS I 2 EQUAL TO –1? How do we cancel the radicals?

13 Multiply What is another way of writing this problem? 6/11/2014 10:01 PM10.7 - Complex Numbers13 E XAMPLE 5

14 Multiply 6/11/2014 10:01 PM10.7 - Complex Numbers14 E XAMPLE 6

15 Multiply 6/11/2014 10:01 PM10.7 - Complex Numbers15 Y OUR T URN

16 Multiply 6/11/2014 10:01 PM10.7 - Complex Numbers16 E XAMPLE 7

17 Multiply 6/11/2014 10:01 PM10.7 - Complex Numbers17 E XAMPLE 8

18 Multiply 6/11/2014 10:01 PM10.7 - Complex Numbers18 E XAMPLE 9

19 Multiply 6/11/2014 10:01 PM10.7 - Complex Numbers19 Y OUR T URN

20 Rationalize 6/11/2014 10:01 PM10.7 - Complex Numbers20 R EVIEW

21 Rationalize 6/11/2014 10:01 PM10.7 - Complex Numbers21 E XAMPLE 10 No i s AND radicals in the denominator

22 Rationalize 6/11/2014 10:01 PM10.7 - Complex Numbers22 E XAMPLE 11 No i s AND radicals in the denominator

23 Rationalize 6/11/2014 10:01 PM10.7 - Complex Numbers23 E XAMPLE 12

24 Rationalize 6/11/2014 10:01 PM10.7 - Complex Numbers24 Y OUR T URN

25 6/11/2014 10:01 PM10.7 - Complex Numbers25 C OMPLEX N UMBERS Complex Numbers a + b i Conjugate is the complex numbers opposite sign Example: 2 + 3 i s conjugate is 2 – 3 i Remember: NO IMAGINARY NUMBERS in the denominator SIMPLIFY RADICALS FIRST then OPERATE Real Number Imaginary Number

26 Add 6/11/2014 10:01 PM10.7 - Complex Numbers26 R EVIEW

27 Add 6/11/2014 10:01 PM10.7 - Complex Numbers27 E XAMPLE 13

28 Subtract 6/11/2014 10:01 PM10.7 - Complex Numbers28 E XAMPLE 14

29 Subtract 6/11/2014 10:01 PM10.7 - Complex Numbers29 Y OUR T URN

30 Multiply FOIL (First Outer Inner Last) 6/11/2014 10:01 PM10.7 - Complex Numbers30 R EVIEW

31 FOIL (First Outer Inner Last) Multiply 6/11/2014 10:01 PM10.7 - Complex Numbers31 E XAMPLE 15

32 Multiply 6/11/2014 10:01 PM10.7 - Complex Numbers32 E XAMPLE 16

33 Multiply 6/11/2014 10:01 PM10.7 - Complex Numbers33 Y OUR T URN

34 Rationalize What is the problem with the bottom? 6/11/2014 10:01 PM10.7 - Complex Numbers34 R EVIEW We try to get rid of the radicals on the bottom so the bottom will be even With the CONJUGATE

35 Rationalize What is the problem with the bottom? 6/11/2014 10:01 PM10.7 - Complex Numbers35 E XAMPLE 17 We try to get rid of the radicals on the bottom so the bottom will be even With the CONJUGATE

36 Rationalize 6/11/2014 10:01 PM10.7 - Complex Numbers36 E XAMPLE 18

37 Rationalize 6/11/2014 10:01 PM10.7 - Complex Numbers37 E XAMPLE 19

38 Rationalize 6/11/2014 10:01 PM10.7 - Complex Numbers38 Y OUR T URN

39 6/11/2014 10:01 PMCOMPLEX NUMBERS39 I MAGINARY P OWERS Steps: 1.Look at the exponent of the imaginary number and divide by 4 2.View only the remainder or decimal 3.Convert the problem with the number REMEMBER: No is in the denominator

40 6/11/2014 10:01 PMCOMPLEX NUMBERS40 IMAGINARY NUMBERS Imaginary Exponent RemainderDecimal Imaginary Number i1i1 R10.25i i2i2 R20.50–1–1 i3i3 R30.75–i i4i4 R00.001

41 6/11/2014 10:01 PMCOMPLEX NUMBERS41 IMAGINARY NUMBERS

42 Simplify 6/11/2014 10:01 PM10.7 - Complex Numbers42 E XAMPLE 20 Imaginary Exponent RemainderDecimal Imaginary Number i1i1 R10.25i i2i2 R20.50–1–1 i3i3 R30.75–i i4i4 R00.001

43 Simplify 6/11/2014 10:01 PM10.7 - Complex Numbers43 E XAMPLE 21

44 Simplify 6/11/2014 10:01 PM10.7 - Complex Numbers44 Y OUR T URN

45 Simplify i 25 = i i 36 = 1 6/11/2014 10:01 PM10.7 - Complex Numbers45 E XAMPLE 22

46 Simplify 6/11/2014 10:01 PM10.7 - Complex Numbers46 Y OUR T URN

47 Simplify 6/11/2014 10:01 PM10.7 - Complex Numbers47 E XAMPLE 24


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