Real Numbers and the Number Line
 The numbers in the pattern are integers and their
squares.
Table of Perfect Squares
Integer
Perfect
Square
Integer
1
2
3
11
12
13
4
5
6
14
15
16
7
17
8
9
18
19
10
20
Perfect
Square
Real Numbers and the Number Line
 A number a is a square root of a number b if a2 = b.
 Example:
7  49, so 7 is a square root of 49
2
or 49  7.
 Together, the radical symbol and the radicand form a
radical.

Problem 1
 What is the simplified form of each expression?
A.
81
C.
9

16
B.
25
D.
1
36

Problem 2
Estimating a Square Root
 The square of an integer is called a perfect square.
 For example, 49 is a perfect square since
7  49
2
 When a radicand is not a perfect square, you can estimate the
square root.
 Method 1: Estimate

86 by finding the two closest perfect squares.

Problem 2
Estimating a Square Root
 Method 2: Estimate
86 by using a calculator.
Estimate
375 to the nearest integer
using both methods?
Classifying Real Numbers
 You can classify numbers using sets.
 A set is a well-defined collection of objects. Each object is
called an element of the set. A subset of a set consists of
elements from the larger set. You can list elements of a set
within braces {}.
 The set of Real Numbers is the largest set of numbers that
you will work with. It is divided into two subsets: the
Rational Numbers and the Irrational Numbers.
The Real Numbers
Rational
Numbers

a
, where a and b are integers and b  0
b
Real
Numbers
Irrational
Numbers
cannot be written as the quotient of
two integers
Rational Numbers
 The Rational Numbers have a subset called the integers.
 The integers are the set
...,2,1,0,1,2,3,...
 The Integers have a subset called the whole numbers.
 The Whole Numbers are the set
0,1,2,3,...
Rational Numbers
 The Whole Numbers have a subset called the natural
numbers.
 The Natural Numbers are the set
1,2,3,4,...
 Some square roots are rational numbers and some are
irrational numbers. If a whole number is not a perfect
square, then its square root is irrational.
Classifying Real Numbers
Problem 3
Classifying Real Numbers
 To which subsets of the real numbers does each number
belong to?
A.
15
B.
3
4
C.
1.4583
D.
12
Inequality
 An inequality is a mathematical sentence that compares the
values of two expressions using an inequality symbol.
 The inequality symbols are:
Problem 4
Comparing Real Numbers
 What is an inequality that compares the following two
numbers?
A.
1
17 and 4
3
B.
129 and 11.52
Problem 5
Graphing and Ordering Real Numbers
 Order the following set of numbers from least to greatest.
 Note: To perform this task, you can write every number as a
decimal rounded to the nearest hundredth, and then graph
each of the numbers on a number line. Then, read the graph
from left to right.
7
3.5,  2.1, 9, , and
2
5