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 melikyan@nccu.edu

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

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

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 = -3 + 0i, 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

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:

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

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

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

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.

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.

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.

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

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.

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

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

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.

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.

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.

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

Example: Multiplying Complex Numbers Find the product:

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.

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.

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

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

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