Consider the quadratic equation x 2 + 1 = 0. Solving for x , gives x 2 = – 1

1. Complex Numbers Consider the quadratic equation x2 + 1 = 0. Solving for x , gives x2 = – 1 We make the following definition:

2. Complex Numbers Note that squaring both sides yields: therefore and so and And so on…

3. Real numbers and imaginary numbers are subsets of the set of complex numbers. Complex Numbers Imaginary Numbers Real Numbers

4. Definition of a Complex Number If a and b are real numbers, the number a + bi is a complex number, and it is said to be written in standard form. If b = 0, the number a + bi = a is a real number. If a = 0, the number a + biis called an imaginary number. Write the complex number in standard form

5. Addition and Subtraction of Complex Numbers If a + bi and c +diare two complex numbers written in standard form, their sum and difference are defined as follows. Sum: Difference:

6. Perform the subtraction and write the answer in standard form. ( 3 + 2i ) – ( 6 + 13i ) 3 + 2i – 6 – 13i –3 – 11i 4

7. Multiplying Complex Numbers Multiplying complex numbers is similar to multiplying polynomials and combining like terms. Perform the operation and write the result in standard form. ( 6 – 2i )( 2 – 3i ) F O I L 12 – 18i – 4i + 6i2 12 – 22i + 6 ( -1 ) 6 – 22i

8. Consider ( 3 + 2i )( 3 – 2i ) 9 – 6i + 6i – 4i2 9 – 4( -1 ) 9 + 4 13 This is a real number. The product of two complex numbers can be a real number. This concept can be used to divide complex numbers.

9. Complex Conjugates and Division Complex conjugates-a pair of complex numbers of the form a + bi and a – bi where a and b are real numbers. ( a + bi )( a – bi ) a 2 – abi + abi – b 2 i2 a 2 – b 2( -1 ) a 2 + b 2 The product of a complex conjugate pair is a positive real number.

10. To find the quotient of two complex numbers multiply the numerator and denominator by the conjugate of the denominator.

11. Perform the operation and write the result in standard form.

12. Perform the operation and write the result in standard form.

13. Expressing Complex Numbers in Polar Form Now, any Complex Number can be expressed as: X + Y i That number can be plotted as on ordered pair in rectangular form like so…

14. Expressing Complex Numbers in Polar Form Remember these relationships between polar and rectangular form: So any complex number, X + Yi, can be written in polar form: Here is the shorthand way of writing polar form:

15. Expressing Complex Numbers in Polar Form Rewrite the following complex number in polar form: 4 - 2i Rewrite the following complex number in rectangular form:

16. Expressing Complex Numbers in Polar Form Express the following complex number in rectangular form:

17. Expressing Complex Numbers in Polar Form Express the following complex number in polar form: 5i

18. Products and Quotients of Complex Numbers in Polar Form The product of two complex numbers, and Can be obtained by using the following formula:

19. Products and Quotients of Complex Numbers in Polar Form The quotient of two complex numbers, and Can be obtained by using the following formula:

20. Products and Quotients of Complex Numbers in Polar Form Find the product of 5cis30 and –2cis120 Next, write that product in rectangular form

21. Products and Quotients of Complex Numbers in Polar Form Find the quotient of 36cis300 divided by 4cis120 Next, write that quotient in rectangular form

22. Products and Quotients of Complex Numbers in Polar Form Find the result of Leave your answer in polar form. Based on how you answered this problem, what generalization can we make about raising a complex number in polar form to a given power?

23. De Moivre’s Theorem De Moivre's Theorem is the theorem which shows us how to take complex numbers to any power easily. De Moivre's Theorem – Let r(cos F+isin F) be a complex number and n be any real number. Then [r(cos F+isin F]n = rn(cos nF+isin nF) What is this saying? The resulting r value will be r to the nth power and the resulting angle will be n times the original angle.

24. De Moivre’s Theorem Try a sample problem: What is [3(cos 45°+isin45)]5? To do this take 3 to the 5th power, then multiply 45 times 5 and plug back into trigonometric form. 35 = 243 and 45 * 5 =225 so the result is 243(cos 225°+isin 225°) Remember to save space you can write it in compact form. 243(cos 225°+isin 225°)=243cis 225°

25. De Moivre’s Theorem Find the result of: Because of the power involved, it would easier to change this complex number into polar form and then use De Moivre’s Theorem.

26. De Moivre’s Theorem De Moivre's Theorem also works not only for integer values of powers, but also rational values (so we can determine roots of complex numbers).

27. De Moivre’s Theorem Simplify the following:

28. De Moivre’s Theorem Every complex number has ‘p’ distinct ‘pth’ complex roots (2 square roots, 3 cube roots, etc.) To find the p distinct pth roots of a complex number, we use the following form of De Moivre’s Theorem …where ‘n’ is all integer values between 0 and p-1. Why the 360? Well, if we were to graph the complex roots on a polar graph, we would see that the p roots would be evenly spaced about 360 degrees (360/p would tell us how far apart the roots would be).

29. De Moivre’s Theorem Find the 4 distinct 4th roots of -3 - 3i

30. De Moivre’s Theorem Solve the following equation for all complex number solutions (roots):