Introduction to Complex Numbers

1. Introduction to Complex Numbers AC Circuits I

2. Outline • Complex numbers basics • Rectangular form • Addition • Subtraction • Multiplication • Division • Rectangular to Polar form conversion • Polar form • Multplication • Division • Euler’s formula • Polar form to Rectangular form conversion AC Circuits I

3. Solutions to Quadratic Equations • What are the roots of the following quadratic equation ? • Roots are complex!!!! What does this mean? AC Circuits I

4. The meaning of i • i represents • Mathematicians use i to represent • Since i already represents current, from now on we shall use j instead of i i.e. AC Circuits I

5. The complex plane Imaginary axis Real axis AC Circuits I

6. Complex, real and imaginary • A complex number is typically comprised of a real and imaginary portions i.e. • X1 is a complex number • -1 is the real number • 1.414 is the imaginary number AC Circuits I

7. Examples • Simplify the following: AC Circuits I

8. Rectangular Representation of Complex Numbers • Complex numbers can be represented in both rectangular and polar formats • Lets start with the Rectangular formats • Addition - • E.g. AC Circuits I

9. Rectangular Representation of Complex Numbers • Subtraction – • E.G. AC Circuits I

10. Rectangular Representation of Complex Numbers • Multiplication – • E.g. AC Circuits I

11. Rectangular Representation of Complex Numbers • Division – • In dividing a+jb by c+jd, we rationalized the denominator using the fact that: • The complex numbers c+jd and c−jd are called complex conjugates • If z = c+jd then z* =c-jd AC Circuits I

12. Rectangular Representation of Complex Numbers • Division Example AC Circuits I

13. Polar Representation of Complex Numbers • Consider the point represented by 4+j6 in the complex plane • Can be represented as a coordinate i.e. (4+j6) or via r (radius) and angle θ • i.e. Or r y =r sin θ θ x = r cosθ AC Circuits I

14. Converting from Rectangular to Polar • For a complex number x+jy: • Note that θ is defined - to  r y =r sin θ θ x = r cosθ AC Circuits I

15. Converting from Rectangular to Polar • Example – Convert 4+j6 to polar notation • Plot the polar points 6+j8, 4-j3, -2-j6 on the complex plane and then convert these complex values to their polar form AC Circuits I

16. Polar Representation of Complex Numbers • In polar number representation, we can only multiply and divide. • Multiplication – • E.g – easy!! • Try: AC Circuits I

17. Polar Representation of Complex Numbers • Division – Easy! • E.g. AC Circuits I

18. Euler’s Formula • Euler’s Formula can be written as: • A complex number in terms of this formula can be written as: • This allows us to convert a complex number in polar form back to rectangular form!!!! AC Circuits I

19. Converting from Polar to Rectangular • Convert the following complex number from polar form to rectangular form: • Try converting these 7@40o, 15@65o, 25@1.01rad AC Circuits I

20. Calculators • Many scientific and graphic calculators have built in complex number functionality. • Use it to aid you in your calculations. • But make sure you know how to manipulate complex numbers without a calculator as well!!! AC Circuits I