1. Complex Numbers in Polar Form; DeMoivre’s Theorem
The slides for this text are organized into chapters. This lecture covers Chapter 1.
Chapter 1: Introduction to Database Systems
Chapter 2: The Entity-Relationship Model
Chapter 3: The Relational Model
Chapter 4 (Part A): Relational Algebra
Chapter 4 (Part B): Relational Calculus
Chapter 5: SQL: Queries, Programming, Triggers
Chapter 6: Query-by-Example (QBE)
Chapter 7: Storing Data: Disks and Files
Chapter 8: File Organizations and Indexing
Chapter 9: Tree-Structured Indexing
Chapter 10: Hash-Based Indexing
Chapter 11: External Sorting
Chapter 12 (Part A): Evaluation of Relational Operators
Chapter 12 (Part B): Evaluation of Relational Operators: Other Techniques
Chapter 13: Introduction to Query Optimization
Chapter 14: A Typical Relational Optimizer
Chapter 15: Schema Refinement and Normal Forms
Chapter 16 (Part A): Physical Database Design
Chapter 16 (Part B): Database Tuning
Chapter 17: Security
Chapter 18: Transaction Management Overview
Chapter 19: Concurrency Control
Chapter 20: Crash Recovery
Chapter 21: Parallel and Distributed Databases
Chapter 22: Internet Databases
Chapter 23: Decision Support
Chapter 24: Data Mining
Chapter 25: Object-Database Systems
Chapter 26: Spatial Data Management
Chapter 27: Deductive Databases
Chapter 28: Additional Topics
Dr .Hayk Melikyan
Departmen of Mathematics and CS

2. The Complex Plane
We know that a real number can be represented as a point on a number line. By contrast, a complex number z = a + bi is represented as a point (a, b) in a coordinate plane, shown below. The horizontal axis of the coordinate plane is called the real axis. The vertical axis is called the imaginary axis. The coordinate system is called the complex plane. Every complex number corresponds to a point in the complex plane and every point in the complex plane corresponds to a complex number.
Imaginary axis
Real axis a b
z = a + bi

3. Text Example Plot in the complex plane:
a. z = 3 + 4i b. z = -1 – 2i c. z = -3 d. z = -4i
Solution
We plot the complex number z = 3 + 4i the same way we plot (3, 4) in the rectangular coordinate system. We move three units to the right on the real axis and four units up parallel to the imaginary axis. -5 -4 -3 -2 1 2 3 4 5
z = 3 + 4i
The complex number z = -1 – 2i corresponds to the point (-1, -2) in the rectangular coordinate system. Plot the complex number by moving one unit to the left on the real axis and two units down parallel to the imaginary axis.
z = -1 – 2i

4. Text Example cont. Plot in the complex plane:
a. z = 3 + 4i b. z = -1 – 2i c. z = -3 d. z = -4i
Solution
Because z = -3 = i, this complex number corresponds to the point (-3, 0). We plot –3 by moving three units to the left on the real axis. -5 -4 -3 -2 1 2 3 4 5
z = 3 + 4i
z = -3
Because z = -4i = 0 – 4i, this complex number corresponds to the point (0, -4). We plot the complex number by moving three units down on the imaginary axis.
z = -4i
z = -1 – 2i

5. The Absolute Value of a Complex Number
The absolute value of the complex number a + bi is
Determine the absolute value of z=2-4i
Solution:

6. Polar Form of a Complex Number
The complex number a + bi is written in polar form as
z = r (cos  + i sin  )
Where
a = r cos  ,
b = r sin  ,
tan =b/a
The value of r is called the modulus (plural: moduli) of the
complex number z, and the angle  is called the argument of the
complex number z, with 0 <  < 2

7. Text Example
Plot z = -2 – 2i in the complex plane. Then write z in polar form.
Solution
The complex number z = -2 – 2i, graphed below, is in rectangular form a + bi, with a = -2 and b = -2. By definition, the polar form of z is r(cos  + i sin  ). We need to determine the value for r and the value for  , included in the figure below.
Imaginary
axis 2 è
Real
axis -2 2 r -2
z = -2 – 2i

8. Text Example cont. Solution
Since tan p4 = 1, we know that  lies in quadrant III. Thus,

9. Product of Two Complex Numbers in Polar Form
Let z1 = r1 (cos 1+ i sin  1) and
z2 = r2 (cos  2 + i sin  2)
be two complex numbers in polar form. Their product, z1z2, is
z1z2 = r1 r2 (cos ( 1 +  2) + i sin ( 1 +  2))
To multiply two complex numbers, multiply moduli and add arguments.

10. z1 = 4(cos 50º + i sin 50º) z2 = 7(cos 100º + i sin 100º)
Text Example
Find the product of the complex numbers. Leave the answer in polar form.
z1 = 4(cos 50º + i sin 50º) z2 = 7(cos 100º + i sin 100º)
Solution
z1z2
= [4(cos 50º + i sin 50º)][7(cos 100º + i sin 100º)]
Form the product of the given numbers.
Multiply moduli and add
arguments.
= (4 · 7)[cos (50º + 100º) + i sin (50º + 100º)]
= 28(cos 150º + i sin 150º)
Simplify.

11. Quotient of Two Complex Numbers in Polar Form
Let z1 = r1 (cos 1 + i sin 1) and
z2 = r2 (cos 2 + i sin 2)
be two complex numbers in polar form.
Their quotient, z1/z2, is
To divide two complex numbers, divide moduli and subtract arguments.

12. DeMoivre’s Theorem
Let z = r (cos  + i sin ) be a complex numbers in polar form. If n is a positive integer, z to the nth power, zn, is

13. Text Example
Find [2 (cos 10º + i sin 10º)]6. Write the answer in rectangular form a + bi.
Solution By DeMoivre’s Theorem,
[2 (cos 10º + i sin 10º)]6
= 26[cos (6 · 10º) + i sin (6 · 10º)]
Raise the modulus to the 6th power and multiply the argument by 6.
= 64(cos 60º + i sin 60º)
Simplify.
Write the answer in rectangular form.
Multiply and express the answer in a + bi form.

14. DeMoivre’s Theorem for Finding Complex Roots
Let =r(cos+isin) be a complex number in polar form. If 0,  has n distinct complex nth roots given by the formula

15. Example
Find all the complex fourth roots of
81(cos60º+isin60º)

16. Objectives (sec 1.4 from math 1100 staff)
Add and subtract complex numbers.
Multiply complex numbers.
Divide complex numbers.
Perform operations with square roots of negative
numbers.

17. Complex Numbers and Imaginary Numbers
The imaginary unit i is defined as
The set of all numbers in the form
a + bi
with real numbers a and b, and i, the imaginary unit,
is called the set of complex numbers.
The standard form of a complex number is
a + bi.

18. Operations on Complex Numbers
The form of a complex number a + bi is like the binomial
a + bx. To add, subtract, and multiply complex numbers,
we use the same methods that we use for binomials.

19. Example: Adding and Subtracting Complex Numbers
Perform the indicated operations, writing the result
in standard form:

20. Example: Multiplying Complex Numbers
Find the product:

21. Conjugate of a Complex Number
For the complex number a + bi, we define its complex
conjugate to be a – bi.
The product of a complex number and its conjugate
is a real number.

22. Complex Number Division
The goal of complex number division is to obtain a real
number in the denominator. We multiply the numerator
and denominator of a complex number quotient by the
conjugate of the denominator to obtain this real number.

23. Example: Using Complex Conjugates to Divide Complex Numbers
Divide and express the result in standard form:
In standard form, the result is

24. Principal Square Root of a Negative Number
For any positive real number b, the principal square root of
the negative number – b is defined by

25. Example: Operations Involving Square Roots of Negative Numbers
Perform the indicated operations and write the result in
standard form: