1. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
§7.7 Complex Numbers
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer

2. 7.6 Review § Any QUESTIONS About Any QUESTIONS About HomeWork
MTH 55
Review §
Any QUESTIONS About
§7.6 → Radical Equations
Any QUESTIONS About HomeWork
§7.6 → HW-37

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
Thus for any positive number p we can now define the square root of a negative number as follows .

5. Imaginary Numbers
An imaginary number is a number that can be written in the form bi, where b is a real numbers 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. Complex numbers that are real numbers: a + bi, b = 0
Rational numbers:
Complex numbers that are real numbers:
a + bi, b = 0
Irrational numbers:
The complex
numbers:
a = bi
Complex numbers (Imaginary)
Complex numbers that are not real numbers:
a + bi, b ≠ 0
Complex numbers

13. Add/Subtract Complex No.s
The 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

14. 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

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

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

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

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

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

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

21. 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

22. 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)

23. 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

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

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

26. Example  Complex Division
SOLN

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

28. 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

29. 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.

30. 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)

31. Electrical Engrs Use j instead of i
All Done for Today
Electrical Engrs Use j instead of i

32. Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu
Chabot Mathematics
Appendix
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer –

33. Graph y = |x|
Make T-table