Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

1. Chabot Mathematics §7.7 ComplexNumbers Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. MTH 55 7.6 Review § • Any QUESTIONS About • §7.6 → Radical Equations • Any QUESTIONS About HomeWork • §7.6 → HW-29

3. Imaginary & Complex Numbers • Negative numbers do not have square roots in the real-number system. • A larger number system that contains the real-number system is designed so that negative numbers do have square roots. That system is called the complex-number system. • The complex-number system makes use of i, a number that with the property (i)2 = −1

4. The “Number” i • i is the unique number for which i2 = −1 and so • Thus for any positive number p we can now define the square root of a negative number using the product-rule as follows .

5. Imaginary Numbers • An imaginary number is a number that can be written in the form bi, where b is a real number that is not equal to zero • Some Examples • i is called the “imaginary unit”

6. Example  Imaginary Numbers • Write each imaginary number as a product of a real number and i a) b) c) • SOLUTION • a) b) c)

7. ReWriting Imaginary Numbers • To write an imaginary number in terms of the imaginary unit i: Separate the radical into two factors Replace with i Simplify

8. Example  Imaginary Numbers • Express in terms of i: a) b) • SOLUTION • a) • b)

9. Complex Numbers • The union of the set of all imaginary numbers and the set of all real numbers is the set of all complex numbers • A complex number is any number that can be written in the form a + bi, where a and b are real numbers. • Note that a and b both can be 0

10. Complex Number Examples • The following are examples of Complex numbers Here a = 7, b =2.

11. Rational numbers: Complex numbers thatare real numbers: a + bi, b = 0 Irrational numbers: The complex numbers: a = bi Complex numbers (Imaginary) Complex numbers thatare not real numbers: a + bi, b ≠ 0 Complex numbers

12. Add/Subtract Complex No.s • Complex numbers obey the commutative, associative, and distributive laws. • Thus we can add and subtract them as we do binomials; i.e., • Add Reals-to-Reals • Add Imaginaries-to-Imaginaries

13. Example  Complex Add & Sub • Add or subtract and simplify a+bi (−3 + 4i) − (4 − 12i) • SOLUTION: We subtract complex numbers just like we subtract polynomials. That is, add/sub LIKE Terms → Add Reals & Imag’s Separately • (−3 + 4i) − (4 − 12i) = (−3 + 4i) + (−4 + 12i) • = −7 + 16i

14. Combining real and imaginary parts Example  Complex Add & Sub • Add or subtract and simplify to a+bi a) b) • SOLUTION • a) • b)

15. Complex Multiplication • To multiply square roots of negative real numbers, we first express them in terms of i. For example,

16. Caveat Complex-Multiplication • CAUTION • With complex numbers, simply multiplying radicands is incorrect when both radicands are negative: • The Correct Multiplicative Operation

17. Example  Complex Multiply • Multiply & Simplify to a+bi form a) b) c) • SOLUTION • a)

18. Example  Complex Multiply • Multiply & Simplify to a+bi form a) b) c) • SOLUTION: Perform Distribution • b)

19. Example  Complex Multiply • Multiply & Simplify to a+bi form a) b) c) • SOLUTION : Use F.O.I.L. • c)

20. Complex Number CONJUGATE • The CONJUGATE of a complex number a+bi is a–bi, and the conjugate of a–bi is a+bi • Some Examples

21. Example  Complex Conjugate • Find the conjugate of each number a) 4 + 3i b) −6 − 9i c) i • SOLUTION: • a) The conjugate is 4 − 3i • b) The conjugate is −6 + 9i • c) The conjugate is −i (think: 0 + i)

22. Conjugates and Division • Conjugates are used when dividing complex numbers. The procedure is much like that used to rationalize denominators. • Note the Standard Form for Complex Numbers does NOT permit i to appear in the DENOMINATOR • To put a complex division into Std Form, Multiply the Numerator and Denominator by the Conjugate of the DENOMINATOR

23. Example  Complex Division • Divide & Simplify to a+bi form • SOLUTION: Eliminate i from DeNom by multiplying the Numer & DeNom by the Conjugate of i

24. Example  Complex Division • Divide & Simplify to a+bi form • SOLUTION: Eliminate i from DeNom by multiplying the Numer & DeNom by the Conjugate of 2−3i •  NEXT SLIDE for Reduction

25. Example  Complex Division • SOLN

26. Example  Complex Division • Divide & Simplify to a+bi form • SOLUTION: Rationalize DeNom by Conjugate of 5−i

27. Powers of i → in • Simplifying powers of i can be done by using the fact that i2 = −1 and expressing the given power of i in terms of i2. • The First 12 Powers of i • Note that (i4)n = +1 for any integer n

28. Example  Powers of i • Simplify using Powers of i a) b) • SOLUTION : Use (i4)n = 1 • a) • b) = 1 Write i40 as (i4)10. Write i32 as (i4)8. Replace i4 with 1.

29. WhiteBoard Work • Problems From §7.7 Exercise Set • 32, 50, 62, 78, 100, 116 • Ohm’s Law of Electrical Resistance in the Frequency Domain uses Complex Numbers (See ENGR43)

30. All Done for Today ElectricalEngrs Use j insteadof i

31. Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu –

33. Graph y = |x| • Make T-table