1. Real Numbers and Algebraic Expressions
Recognize subsets of the real number.
Use inequality symbols.
Evaluate absolute value to express distance.
Algebraic expressions and their evaluation
Properties of algebraic operations
Understand and use of integer exponents
Properties of exponents
Simplify exponential expressions
Use of scientific notation.
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
HamestGevorgyan
Departmen of Mathematics and CS

2. A combination of variables and numbers using
Algebraic expression
A combination of variables and numbers using
the operations of addition, subtraction, division,
or multiplication, as well as powers or roots, is
called an algebraic expression.
Example: 7x, x+11, x-9, x4 -10,

3. Definition of a Natural Number Exponent
If b is a real number and n is a natural number,
bn is read “the nth power of b” or “ b to the nth power.” Thus, the nth power of b is defined as the product of n factors of b. Furthermore, b1 = b

4. The Order of Operations Agreement
Perform operations within the innermost parentheses and work outward. If the algebraic expression involves division, treat the numerator and the denominator as if they were each enclosed in parentheses.
Evaluate all exponential expressions.
Perform multiplication or division as they occur, working from left to right.
4. Perform addition or subtraction as they occur, working from left to right.

5. The set {1, 3, 5, 7, 9} has five elements.
The Basics About Sets
The set {1, 3, 5, 7, 9} has five elements.
A set is a collection of objects whose contents can be clearly determined.
The objects in a set are called the elements of the set.
We use braces to indicate a set and commas to separate the elements of that set.
For example,
The set of counting numbers can be represented by {1, 2, 3, … }.
The set of even counting numbers are {2, 4, 6, …}.
The set of even counting numbers is a subset of the set of counting numbers, since each element of the subset is also contained in the set.

6. Set operations
The union of two sets is the set of all elements formed by combining all the elements of set A and all the elements of set B into one set.
The symbolism used is The Venn Diagram representing the union of A and B is the entire region shaded yellow. A B

7. Union of Sets Combination of everything in both sets
A = {all tall children}
B = {all girls}
A union B = {all girls OR tall children} = {all girls and all tall boys}

8. Intersection of sets A and B
The intersection of sets A and B is the set of elements that is common to both sets A and B. It is symbolized as
{ x l x ∈A and x ∈B }
Represented by Venn Diagrams: B A
Intersection

9. Intersection of Sets
What they have in common
A = {all tall children}
B = {all girls}
A intersect B = {all tall girls}
All children that are girls AND are tall

10. Students in biology, chemistry, & physics. Students in chemistry.
A group of biology majors are taking Biology I & Chem. I. A group of chemistry majors are taking Calculus, Chem. I and Physics I. The Physics majors enrolled in Calculus, Physics I, and Chem I. What is the intersection of the 3 groups?
Students in biology, chemistry, & physics.
Students in chemistry.
Students in calculus.
Students in physics.

11. Important Subsets of the Real Numbers
-15, -7, -4, 0, 4, 7
{…, -2, -1, 0, 1, 2, 3, …}
Add the negative natural numbers to the whole numbers
Integers Z
0, 4, 7, 15
{0, 1, 2, 3, … }
Add 0 to the natural numbers
Whole Numbers W
4, 7, 15
{1, 2, 3, …}
These are the counting numbers
Natural Numbers N
Examples
Description
Name

12. Important Subsets of the Real Numbers
This is the set of numbers whose decimal representations are neither terminating nor repeating. Irrational numbers cannot be expressed as a quotient of integers.
Irrational Numbers I
These numbers can be expressed as an integer divided by a nonzero integer:
Rational numbers can be expressed as terminating or repeating decimals.
Rational
Numbers Q
Examples
Description
Name

13. Rational numbers Irrational numbers Integers Whole numbers
The Real Numbers
The set of real numbers is formed by combining the rational numbers and the irrational numbers.
Rational numbers
Irrational numbers
Integers
Whole numbers
Natural numbers

14. Units to the left of the origin are negative. Positive numbers
The Real Number Line
The real number line is a graph used to represent the set of real numbers. An arbitrary point, called the origin, is labeled 0;
Negative numbers
Units to the left of the origin are negative.
Positive numbers
Units to the right of the origin are positive.
the Origin

15. Graphing on the Number Line
Real numbers are graphed on the number line by placing a dot at the location for each number. –3, 0, and 4 are graphed below.

16. Ordering the Real Numbers
On the real number line, the real numbers increase from left to right. The lesser of two real numbers is the one farther to the left on a number line. The greater of two real numbers is the one farther to the right on a number line.
Since 2 is to the left of 5 on the number line, 2 is less than < 5
Since 5 is to the right of 2 on the number line, 5 is greater than > 2

17. b is greater than or equal to a. b > a Because 7 =7 7 < 7
Inequality Symbols
Because -5 = -5
-5 > -5
Because 7 > 3
7 > 3
b is greater than or equal to a.
b > a
Because 7 =7
7 < 7
Because 3 < 7
3 < 7
a is less than or equal to b.
a < b
Explanation
Example
Meaning
Symbols

18. Absolute Value
Absolute value describes the distance from 0 on a real number line. If a represents a real number, the symbol |a| represents its absolute value, read “the absolute value of a.”
For example, the real number line below shows that
|-3| = 3 and |5| = 5.
The absolute value of –3 is 3 because –3 is 3 units from 0 on the number line.
|–3| = 3
The absolute value of 5 is 5 because 5 is 5 units from 0 on the number line.
|5| = 5

19. Definition of Absolute Value
The absolute value of x is given as follows:
|x| =
x if x > 0
-x if x < 0 {

20. Properties of Absolute Value
For all real number a and b,
1. |a| > 0
2. |-a| = |a|
3. a < |a| a b
|a|
|b|
4. |ab| = |a||b|
= , b not equal to 0
6. |a + b| < |a| + |b| (the triangle inequality)

21. Find the following: |-3| and |3|.
Example
Find the following: |-3| and |3|.
Solution:
Distance Between Two Points on the Real Number Line
If a and b are any two points on a real number line, then the distance between a and b is given by
|a – b| or |b – a|

22. Text Example
Find the distance between –5 and 3 on the real number line.
Solution Because the distance between a and b is given by |a – b|, the distance between –5 and 3 is |-5 – 3| = |-8| = 8. 8
We obtain the same distance if we reverse the order of subtraction:
|3 – (-5)| = |8| = 8.

23. Algebraic Expressions
A combination of variables and numbers using the operations of addition, subtraction, multiplication, or division, as well as powers or roots, is called an algebraic expression.
Here are some examples of algebraic expressions:
x + 6, x – 6, 6x, x/6, 3x + 5.

24. The Order of Operations Agreement
Perform operations within the innermost parentheses and work outward. If the algebraic expression involves division, treat the numerator and the denominator as if they were each enclosed in parentheses.
Evaluate all exponential expressions.
Perform multiplication or division as they occur, working from left to right.
4. Perform addition or subtraction as they occur, working from left to right.

25. Text Example
The algebraic expression 2.35x describes the population of the United States, in millions, x years after Evaluate the expression when x = 20. Describe what the answer means in practical terms.
Solution We begin by substituting 20 for x. Because x = 20, we will be finding the U.S. population 20 years after 1980, in the year 2000.
2.35x Replace x with 20.
= 2.35(20)
= Perform the multiplication.
= Perform the addition.
Thus, in 2000 the population of the United States was million.

26. Properties of the Real Numbers
3 + ( 8 + x)
= (3 + 8) + x
= 11 + x
If 3 real numbers are added, it makes no difference which 2 are added first.
(a + b) + c = a + (b + c)
Associative Property of Addition
x · 6 = 6x
Two real numbers can be multiplied in any order.
ab = ba
Commutative Property of Multiplication =
13x + 7 = x
Two real numbers can be added in any order.
a + b = b + a
Commutative Property of Addition
Examples
Meaning
Name

27. Properties of the Real Numbers
0 + 6x = 6x
Zero can be deleted from a sum.
a + 0 = a
0 + a = a
Identity Property of Addition
5 · (3x + 7)
= 5 · 3x + 5 · 7
= 15x + 35
Multiplication distributes over addition.
a · (b + c) = a · b + a · c
Distributive Property of Multiplication over Addition
-2(3x) = (-2·3)x = -6x
If 3 real numbers are multiplied, it makes no difference which 2 are multiplied first.
(a · b) · c = a · (b · c)
Associative Property of Multiplication
Examples
Meaning
Name

28. Properties of the Real Numbers
2 · 1/2 = 1
The product of a nonzero real number and its multiplicative inverse gives 1, the multiplicative identity.
a · 1/a = 1 and 1/a · a = 1
Inverse Property of Multiplication
(-6x) + 6x = 0
The sum of a real number and its additive inverse gives 0, the additive identity.
a + (-a) = 0 and (-a) + a = 0
Inverse Property of Addition
1 · 2x = 2x
One can be deleted from a product.
a · 1 = a and 1 · a = a
Identity Property of Multiplication
Examples
Meaning
Name

29. Definitions of Subtraction and Division
Let a and b represent real numbers.
Subtraction: a – b = a + (-b)
We call –b the additive inverse or opposite of b.
Division: a ÷ b = a · 1/b, where b = 0
We call 1/b the multiplicative inverse or reciprocal of b. The quotient of a and b, a ÷ b, can be written in the form a/b, where a is the numerator and b the denominator of the fraction.

30. Text Example Simplify: 6(2x – 4y) + 10(4x + 3y). Solution
= 6 · 2x – 6 · 4y + 10 · 4x + 10 · 3y Use the distributive property.
= 12x – 24y + 40x + 30y Multiply.
= (12x + 40x) + (30y – 24y) Group like terms.
= 52x + 6y Combine like terms.

31. Properties of Negatives
Let a and b represent real numbers, variables, or
algebraic expressions.
(-1)a = -a
-(-a) = a
(-a)(b) = -ab
a(-b) = -ab
-(a + b) = -a - b
-(a - b) = -a + b = b - a

32. Definition of a Natural Number Exponent
If b is a real number and n is a natural number,
bn is read “the nth power of b” or “ b to the nth power.” Thus, the nth power of b is defined as the product of n factors of b. Furthermore, b1 = b

33. The Negative Exponent Rule
If b is any real number other than 0 and n is a natural number, then

34. The Zero Exponent Rule
If b is any real number other than 0, b0 = 1.

35. The Product Rule
b m · b n = b m+n
When multiplying exponential expressions with the same base, add the exponents. Use this sum as the exponent of the common base.

36. The Power Rule (Powers to Powers)
(bm)n = bm•n
When an exponential expression is raised to a power, multiply the exponents. Place the product of the exponents on the base and remove the parentheses.

37. The Quotient Rule
When dividing exponential expressions with the same nonzero base, subtract the exponent in the denominator from the exponent in the numerator. Use this difference as the exponent of the common base.

38. Example
Find the quotient of 43/42
Solution:

39. Products to Powers
(ab)n = anbn
When a product is raised to a power, raise each factor to the power.

40. Text Example
Simplify: (-2y)4.
Solution
(-2y)4 = (-2)4y4 = 16y4

41. Quotients to Powers
When a quotient is raised to a power, raise the numerator to that power and divide by the denominator to that power.

42. Simplify by raising the quotient (2/3)4 to the given power.
Example
Simplify by raising the quotient (2/3)4 to the given power.
Solution:

43. Properties of Exponents

44. If the sum of q-6, q-3, and q is 0, what is the value of q.
QUIZ#1
If the sum of q-6, q-3, and q is 0, what is the
value of q.

45. Scientific Notation
The number 5.5 x 1012 is written in a form called scientific notation. A number in scientific notation is expressed as a number greater than or equal to 1 and less than 10 multiplied by some power of 10. It is customary to use the multiplication symbol, x, rather than a dot in scientific notation.

46. where the absolute value of a is greater than or
Scientific Notation
A number is written in scientific notation when
it is expressed in the form
a x 10n
where the absolute value of a is greater than or
equal to 1 and less than 10, and n is an integer.

47. Write each number in decimal notation: 2.6 X 107 b. 1.016 X 10-8
Text Example
Write each number in decimal notation:
2.6 X 107
b X 10-8
Solution:
a x 107 can be expressed in decimal notation by moving the decimal point in 2.6 seven places to the right. We need to add six zeros.
2.6 x 107 = 26,000,000.
b x 10-8 can be expressed in decimal notation by moving the decimal point in eight places to the left. We need to add seven zeros to the right of the decimal point.
1.016 x 10-8 =

48. Scientific Notation
To convert from decimal notation to scientific notation, we reverse the procedure.
Move the decimal point in the given number to obtain a number greater than or equal to 1 and less than 10.
The number of places the decimal point moves gives the exponent on 10; the exponent is positive if the given number is greater than 10 and negative if the given number is between 0 and 1.

49. Text Example
Write each number in scientific notation. a. 4,600, b
Solution
a. 4,600,000 = 4.6 x 10?
4.6 x 106
Decimal point moves 6 places
b = 2.3 x 10?
2.3 x 10-4
Decimal point moves 4 places