real numbers and the number line the numbers in the pattern are integers and their squares

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

€

or 49 = 7.

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

Problem 1What is the simplified form of each

expression?

€

A. 81

€

B. 25

€

C. 9

16

€

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

€

72 = 49When 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 2Estimating a Square Root

€

86 by using a calculator.

Method 2: Estimate

€

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

€

a

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

€

cannot be written as the quotient of

two integers

Rational NumbersThe 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

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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?

€

A. 15

€

B. 3

4

€

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?

€

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.

€

3.5, − 2.1, 9,−7

2, and 5