Real Numbers and the Number Line

The numbers in the pattern are integers and their squares.

Table of Perfect SquaresInteger Perfect

Square

1

2

3

4

5

6

7

8

9

10

Integer Perfect Square

11

12

13

14

15

16

17

18

19

20

Real Numbers and the Number Line

A number a is a square root of a number b if a2 = b.

Example:

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72 = 49, so 7 is a square root of 49

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or 49 = 7.

Together, the radical symbol and the radicand form a radical.

Problem 1What is the simplified form of each

expression?

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A. 81

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B. 25

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C. 9

16

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D. 1

36

Problem 2Estimating a Square RootThe square of an integer is called a perfect

square.For example, 49 is a perfect square since

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72 = 49When a radicand is not a perfect square,

you can estimate the square root.Method 1: Estimate

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86 by finding the two closest perfect squares.

Problem 2Estimating a Square Root

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86 by using a calculator.

Method 2: Estimate

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Estimate 375 to the nearest integer

using both methods?

Classifying Real NumbersYou 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

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a

b, where a and b are integers and b ≠ 0

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cannot be written as the quotient of

two integers

Rational NumbersThe Rational Numbers have a subset called

the integers.The integers are the set

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...,−2,−1,0,1,2,3,...{ }

The Integers have a subset called the whole numbers.

The Whole Numbers are the set

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0,1,2,3,...{ }

Rational NumbersThe Whole Numbers have a subset called

the natural numbers.The Natural Numbers are the set

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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 3Classifying Real NumbersTo which subsets of the real numbers does

each number belong to?

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A. 15

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B. 3

4

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C. −1.4583

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D. 12

InequalityAn inequality is a mathematical sentence

that compares the values of two expressions using an inequality symbol.

The inequality symbols are:

Problem 4Comparing Real NumbersWhat is an inequality that compares the

following two numbers?

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A. 17 and 41

3

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B. 129 and 11.52

Problem 5Graphing and Ordering Real NumbersOrder 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.

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3.5, − 2.1, 9,−7

2, and 5