1. Complex Numbers 2015 Imagine That!

2. Warm-Up Find all solutions to the polynomial. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

3. Warm-Up Find all solutions to the polynomial. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3

4. Quiz aftermath Factor given is a factor.

5. Quiz aftermath Sketch a graph of the polynomial. Consider end behavior, zeros and the number of turns.

6. 2.4 Complex Numbers Students will use the imaginary unit i to write complex numbers. Students will add, subtract, and multiply complex numbers. Students will use complex conjugates to write the quotient of two complex numbers in standard form.

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Definition: Complex Number The letter i represents the numbers whose square is –1. i = Imaginary unit If a is a positive real number, then the principal square root of negative a is the imaginary number i. = i Examples: = i= 2i = i= 6i The number a is the real part of a + bi, and b is the imaginary part. A complex number is a number of the form a + bi, where a and b are real numbers and i =. i 2 = –1

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Examples of Complex Numbers Examples of complex numbers: Real Part Imaginary Part abibi+ 27i7i + 203i3i– Real Numbers: a + 0i Pure Imaginary Numbers: 0 + bi a + bi form + i= 4 + 5i= + i= Simplify using the product property of radicals. Simplify: = i = 3i 1. = i= 8i 2. + 3.

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Add or Subtract Complex Numbers To add or subtract complex numbers: 1. Write each complex number in the form a + bi. 2. Add or subtract the real parts of the complex numbers. 3. Add or subtract the imaginary parts of the complex numbers.

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Adding Complex Numbers (10 + ) + (21 – ) = (10 + i ) + (21 – i ) i = = 31 Group real and imaginary terms. a + bi form = (10 + 21) + (i – i ) (11 + 5i) + (8 – 2i ) = 19 + 3i Group real and imaginary terms. a + bi form = (11 + 8) + (5i – 2i ) Adding Complex Numbers

11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Subtracting Complex Numbers (– 21 + 3i ) – (7 – 9i) = (– 21 – 7) + [(3 – (– 9)]i = (– 21 – 7) + (3i + 9i) = –28 + 12i (11 + ) – (6 + ) = (11 + i ) – (6 + i ) = (11 – 6) + [ – ]i = (11 – 6) + [ 4 – 3]i = 5 + i Group real and imaginary terms. a + bi form Subtracting Complex Numbers

12. a) b) c) d) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 You Try: Adding and Subtracting Complex Numbers

13. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Product of Complex Numbers To find the product of two complex numbers, distribute and combine like terms. 1. Use the FOIL method to find the product. 2. Replace i 2 by – 1. 3. Write the answer in standard form: a + bi. (a + bi)(c + di )

14. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Examples = 5i 2 = 5 (–1) = –5 2. 7i (11– 5i) = 77i – 35i 2 = 35 + 77i 3. (2 + 3i)(6 – 7i ) = 12 – 14i + 18i – 21i 2 = 12 + 4i – 21i 2 = 12 + 4i – 21(–1) = 12 + 4i + 21 = 33 + 4i Examples: 1. = i i = 5i i = 77i – 35 (– 1)

15. Examples: Multiplying Complex Numbers a) b) c) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15

16. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 Product of Conjugates The complex numbers a + bi and a - bi are called complex conjugates. Example: (5 + 2i)(5 – 2i) = (5 2 – 4i 2 ) = 25 – 4 (–1) = 29 The product of complex conjugates is the real number a 2 + b 2. (a + bi)(a – bi)= a 2 – b 2 i 2 = a 2 – b 2 (– 1) = a 2 + b 2

17. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17 Replace i 2 by –1 and simplify. Dividing Complex Numbers A rational expression, containing one or more complex numbers, is in simplest form when there are no imaginary numbers remaining in the denominator. Multiply the expression by. Write the answer in the form a + bi. Example: –1

18. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18 Example: (5 +3i)/(2+i) Replace i 2 by –1 and simplify. Multiply the numerator and denominator by the conjugate of 2 + i. Write the answer in the form a + bi. In 2 + i, a = 2 and b = 1. a 2 + b 2 = 2 2 + 1 2 Simplify: –1–1

19. Examples: Dividing Complex Numbers Write the quotientin standard form a + bi. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19

20. Example 5: Plotting Complex Numbers Plot each complex number in the complex plane. a)b) c)d) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20 y x –2 2

21. Complex Numbers Engineers use imaginary numbers to study stresses on beams and to study resonance. Complex numbers help us study the flow of fluid around objects, such as water around a pipe. They are used in electric circuits, and help in transmitting radio waves. So, if it weren’t for i, we might not be able to talk on cell phones, or listen to the radio! Imaginary numbers also help in studying infinite series. Lastly, every polynomial equation has a solution if complex numbers are used. Clearly, it is good that i was created. http://rossroessler.tripod.com/ Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 21

22. Applications of Complex Numbers Control theory Improper integrals Fluid dynamics Dynamic equations Electromagnetism and electrical engineering Signal analysis Quantum mechanics Relativity Geometry – Fractals – Triangles Algebraic number theory - Analytic number theory

23. Control Systems Input Output Feedback Process

24. HOMEWORK Section 2.4, pg 133: 15-61 odd 65-71 odd Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 24