Algebra I Part 1
Chapter 2: Rational Numbers
2-1 Integers and the Number Line
After studying this lesson, you should be able to
 State the coordinate of a point on a number line,
 Graph integers on a number line, and
 Add integhers by using a number line.
A Number Line is drawn by choosing a starting poisition on a line, and marking off equal distances from
the point. The set of whole numbers is often represented on a number line. This set can be written {0, 1, 2,
3, . . . } where ". . ." means that the set continues indefitely.
If a portion of a number line is shown, an arrowhead is used at the end or ends of the line to indicate it
continues.
--------------------------------------------------------------------
10 11 12 13 14 15 16 17 18 19 20 21
To include negative numbers on a number lines, extend the line to the left of Zero and mark off equal
distances. To avoid confusion, name the point to the right of zero using the positive sign (+) and the
numbers to the left of zero using the negative sign (-). Zero is neither positive nor negative.

-6
-5
-4
-3
-2
-1
0
+1
+2
+3
+4
The set of numbers used to name the point on the number line are called the set of integers. This set can be
written { . . ., -3, -2, -1, 0, 1, 2, 3 . . . }
The set of numbers are often represented by figures called Venn diagrams. The size and shape of a Venn
diagram are unimiportant. However usually a rectangle is used to represent the universal set, and circles or
ovals are used to represent subsets of the universal set
Sets
Real Numbers
Integers
Whole Numbers
Natural Numbers
Examples
includes all positive and negative whole and fractional/decimal numbers
{ . . . -1.5, -1.4, -1.3, -1.2, -1.1, 0 , 1.1, 1.2, 1.3, 1.4, 1.5 . . . }
includes all positive and negative whole numbers
{ . . -2, -1, 0, 1, 2, . . . }
includes all positive whole numbers beginning with zero
{ 0, 1, 2, 3, 4, . . . }
includes all positive whole numbers beginning with one
{ 1, 2, 3, 4, . . . }
Notice in the Venn Diagram below, that real numbers (which include all numbers including all whole
numbers all fractional numbers, and all negative whole and fractional numbers) contains the subset of
integers. The set of whole numbers is a subset of Integers, and the set of Natural numbers is a subset of the
set of Whole numbers.
Real Numbers
Integers
Whole Numbers
Natural Numbers
Venn Diagrams can also show how various sets are related.
The Intersection of two or more sets is the set of elements common to all sets
The symbol for intersection is 
The Intersection of sets A and B = A  B. In the diagram the yellow areas represent A  B - the
intersection of set A and set B.
Set A = {2, 3, 5, 1}
Set B = {4, 7, 9, 1}
4
2
3
1
A
5
A  B = { 1, 2, 3, 4, 5, 7, 9}
7
9
B
The union of two or more sets is the areas that the sets have in common. The symbol for union is .
The union of two or more sets includes the set of numbers that are in both or all sets.
The Union of sets A and B = A B. The yellow area represents AB.
1
AB
3
7
4
6
A
9
0
8
B
2
5
Set A = { 4, 6, 5, 9, 3, 7}
Set B = { 1, 0, 3, 7, 8, 2}
The Union (or common elements) is expressed as: A  B = {3, 7}
The Intersection (or combined elements) is expressed as: A  B = { 0,1, 2, 3, 4, 5, 6, 7, 8, 9}
If two or more sets do not share are common elements, the Union of those sets is the null set {}
7
A
3
5
4
6
2
8
B
Set A = { 7, 3, 5, 6}
Set B = { 4, 8, 2}
A  B = { 7 ,3, 5, 6, 4, 2, 8}
A  B = {} or A  B = 
2-2 Adding and Subtracting Integers
After studying this lesson, you should be able to
 Find the absolute value of a number
 Add integers without using a number line, and
 Subtract Integers
Definition of Absolute Value:
The absolute value of a number is the number of units it is from 0 on a number line.
The symbol for an absolute value is two vertical lines: |-2| = 2
Adding Integers:
To add integers with the same sign, add their absolute values. Give the result the same sign as the
integers.
To add integers with different signs, subtract the lesser absolute value from the greater absolute
value. Give the result the sign of the greater absolute value.
Additive Inverse Property:
For every number a, a + (-a) = 0
Subtracting Integers:
To subtract a number, add its additive inverse.
For any numbers a and b, a - b = a + (-b)
2-3 Inequalities and the Number Line
After studying this lesson you should be able to
 Compare Numbers
 Write Inequalities for graphs on number lines, and
 Graph inequalities on number lines.
Comparison Property
For any two numbers a and b, exactly one of the following sentences is true:
A<B A=B A >B
Transitive Property of Order
For all numbers a, b, and c.
If a < b and b < c, then a < c and
If a > b and b > c, then a > c
Comparing Numbers on the Number Line
If a and b represent any numbers, and the graph of a is to the left of the graph of b, then a < b.
If the graph of a is to the right of the graph of b, then a > b.
(**MOVE TO NEXT CHAPTER ON REVISION**)
Definition of a Rational Number
A Rational Number is a number that can be expressed in the form of a/b, where a and b are
integers and b is not equal to zero.
2-4 Comparing and Ordering Rational Numbers
After studying this lesson, you should be able to
 Compare Rational Numbers
 Write rational numbers in increasing or decreasing order, and
 Find a number between two rational numbers.
Comparison Property for Rational Numbers
For any rational numbers a/ b and c/d with b > 0 and d > 0;
If a/b < c/d then ad < bc
If ad < bc, the a/b < c/d
Density Property for Rational Numbers
Between every pair of distinct rational numbers, there are infinitely man rational numbers.
2-5 Adding and Subtracting Rational Numbers
After studying this lesson, you should be able to:
 Add two or more rational numbers
 Subtract rational numbers, and
 Simplify expressions that contain rational numbers.
Adding Numbers with the Same Sign
The sum of numbers with the Same Sign is the sum of those numbers with the same sign as
the numbers
The sum of negative numbers is negative; the sum of positive numbers is positive.
Adding Numbers with different signs
The sum of number with different signs is the difference between the positive and the
negative values, using the sign of the greater value (positive greater than negative is positive; negative
numbers greater than positive number is negative.)
Subtracting Numbers with the same sign
The difference between positive numbers is the difference between the numbers using the
sign of the greater number.
The difference between negative numbers is teh difference between the numbers using the
sign of the greater number.
Subtracting with different signs.
To find the difference between numbers of different signs, reverse the sign of the number or
numbers after the subtraction sign, then find the difference between the positive and negative values,
using the sign of the greater number.
2-6 Multiplying Rational Numbers
After studying this lesson, you should be able to
 Multiply rational numbers.
Multiplying Two Numbers with Different Signs:
The product of two numbers having different signs is negative.
Multiplying Two Numbers with the Same Sign
The product of two numbers having different signs is positive.
Multiplicative Property of -1:
The product of any number and -1 is it's additive inverse -1(a) = -a and a(-1) = -a
2-7 Dividing Rational Numbers
After studying this lesson, you should be able to:
 Divide rational numbers
Dividing Rational Numbers
The Quotient of two numbers is positive if the numbers have the same sign. The quotient of two
numbers is negative if the numbers have different signs
Multiplicative Inverse Property
For every nonzero number a, there is exactly one number 1/a such that a(1/a) = 1/a(a) = 1
Division Rule
For any numbers a and b, with b  0, a  b = a/b = a(1/b) = (1/b)a
2-8 Problem Solving Strategy: Write an Equation
After studying this lesson, you should be able to:
 Define variables and write equations for verbal problems, and
 Write verbal problems for equations.
Problem Solving Plan
Explore the problem
Plan the solution
 Read the problem again. Decide how the unspecified numbers relate to each other and to the
other given information.
 Write an equation to represent the relationship.
Make up your own verbal problem if your are given an equation or two. First you need to know
what the variable in the equation represents. Then you can use the equation to establish the condition of the
problem.