Complex Numbers

1. Complex Numbers Chapter 12

2. The Imaginary Number j Previously, when we encountered an equation like x2 + 4 = 0, we said that there was no solution since solving for x yielded There is no real number that can be squared to produce -4. Ah… but mathematicians were not satisfied with these so-called unsolvable equations. If the set of real numbers was not up to the task, they would define an expanded system of numbers that could handle the job! Hence, the development of the set of complex numbers.

3. Definition of a Complex Number The imaginary number j is defined as , where j2= . A complex number is a number in the form a + bj, where a and b are real numbers. (a is the real part and bj is the imaginary part)

4. Simplify:

5. The rectangular Form of a Complex Number Each complex number can be written in the rectangular form a + bj. Example Write the complex numbers in rectangular form.

6. Addition & Subtraction of Complex Numbers To add or subtract two complex numbers, add/subtract the real parts and the imaginary parts separately. Example #1

7. Addition & Subtraction of Complex Numbers Example #2 Simplify and write the result in rectangular form.

8. Multiplying Complex Numbers Multiply complex numbers as you would real numbers, using the distributive property or the FOIL method, as appropriate. Simplify your answer, keeping in mind that j2= -1. Always write your final answer in rectangular form, a + bj. Example #1

9. MultiplyingComplex Numbers (continued) Example #2 Simplify and write the result in rectangular form.

10. MultiplyingComplex Numbers (continued) Example #3 Simplify and write the result in rectangular form.

11. Be careful Multiplying with Complex Numbers (continued) Example #4 Simplify and write the result in rectangular form.

12. Multiplying with Complex Numbers (continued) Example #5 Simplify each expression.

13. Multiplying with Complex Numbers (continued) Example #6 Simplify and write the result in rectangular form.

14. Powers of j Complete the following: “What pattern do you observe?”

15. Powers of j Examples Simplify and write the result in rectangular form. Note: If a complex expression is in simplest form, then the only power of jthat should appearin the expression is j1.

16. Dividing Complex Numbers • For a quotient of complex numbers to be in rectangular form, it cannot have jin the denominator. • Scenario 1: The denominator of an expression is in the form bj • Multiply numerator and denominator by j • Then use the fact that j2 = -1 to simplify the expression and write in rectangular form.

17. Dividing Complex Numbers (continued) Example #1

18. Dividing Complex Numbers (continued) Example #2 Write the quotient in rectangular form.

19. Complex Conjugates Pairs of complex numbers in the form a + bj and a – bj are called complex conjugates. These are important because when you multiply the conjugates together (FOIL), the imaginary terms drop out, leaving only a2 + b2. We will use this idea to simplify a quotient of complex numbers in rectangular form.

20. Complex Conjugates (continued) • Scenario 2: The denominator of an expression is in the form a+bj • Multiply numerator and denominator by the conjugate of the denominator • Then use the fact that j2 = -1 to simplify the expression and write in rectangular form.

21. Complex Conjugates (continued) Example #1

22. Complex Conjugates (continued) Example #2 Write the quotient in rectangular form.

23. Graphical Representation of Complex Numbers A complex number can be represented graphically as a point in the rectangular coordinate system. For a complex number in the form a + bj, the real part, a, is the x-value and the imaginary part, b, is the y-value. In the complex plane, the horizontal axis is the real axis and the vertical axis is the imaginary axis.

24. Graphical Representation of Complex Numbers Graph the points in the complex plane: A: -3 + 4j B: -j C: 6 D: 2 – 7j

25. Earlier, we saw that a point in the plane could be located by polar coordinates, as well as by rectangular coordinates, and we learned to convert between polar and rectangular. Polar Coordinates

26. imaginary r  real Now, we will use a similar technique with complex numbers, converting between rectangular and polar form*. *The polar form is sometimes called the trigonometric form. We’ll start by plotting the complex number x + yj, drawing a vector from the origin to the point. Polar Form of a Complex Number To convert to polar form, we need to know:

27. The polar form is found by substituting the values of x and y into the rectangular form. or A commonly used shortcut notation for the polar form is

28. For example,

29. Example Represent the complex numbers graphically and give the polar form of each. 1) 2 + 3j 2) 4

30. Example Represent the complex numbers graphically and give the polar form of each. 3) 4)

31. Example The current in a certain microprocessor circuit is given by Write this in rectangular form.

32. The exponential form of a complex number is written as This form is used commonly in electronics and physics applications, and is convenient for multiplying complex numbers (you simply use the laws of exponents). Remember, from the chapter on exponential and logarithmic equations, that e is an irrational number that is approximately equal to 2.71828. (It is called the natural base.) The Exponential Form of a Complex Number

33. The Exponential Form of a Complex Number  in radians known as Euler’s Formula

34. Example Write the complex number in exponential form.

35. Example Write the complex number in exponential form.

36. Example Write the complex number in exponential form.

37. Example Express the complex number in rectangular and polar forms.

38. Rectangular: Polar: Exponential: We have have now used three forms of a complex number: So we have,

39. End of Section