Rational and Irrational Numbers

Honors Math – Grade 8

Get ready for the Lesson

During a 4-month period, an environmentalist recorded the following changes to a lake’s water level:

.8

3.,41.0,

4

12.,4 inandininin

These numbers recorded above are called rational numbers.

A rational number is the quotient of two

integers, a and b, written as

where .b

a

0bRemember:Natural or Counting Numbers{1, 2, 3, 4, …}Whole Numbers{0, 1, 2, 3, …}

We have already studied some types of rational numbers.

Now let’s examine and discuss some other subsets of rational numbers.

The following types of numbers are rational numbers…

Terminating Decimals have a finite number of digits.

0.41, 3.0, -5.7

Repeating Decimals have a sequence of one or more digits that repeat indefinitely.

Decimals – can also be positive or negative. They are either terminating or repeating.

Fractions and mixed numbers – can be positive or negative

{…, -3, -2, -1, 0, 1, 2, 3, …}Integers – are whole numbers and their opposites.

Example SetType of Number

8

3

7

11

4

12

3.2 ...333.2

Multiplying a number by itself, or raising it to the second power, is finding the square of that number.

8

3ofsquare

2

8

3

8

3

8

3

64

9

5.0ofsquare 25.0 5.05.0 25.0

7ofsquare 27 77 49

4ofsquaretheofopposite

24 44 16

648497366255164934211

2

2

2

2

2

2

2

2

Perfect squares, or square numbers, are the square of natural numbers.

The set of perfect squares is {1, 4, 9, 16, 25, 36, 49, 64, 81, …}

A square root of a number is one of two equal factors of that number.

The positive square root of a number is called the principal square root. It is indicated with the a symbol called a radical sign.

The expression under a radical sign is called the radicand.

The negative square root of a number is indicated by writing a negative sign if front of the radical.

25Read as “the principal square root of 25.”

555

25 5Read as “the negative square root of 25.”

Find each square root.

225Another way to find the principal square root of a perfect square is to use the prime factorization.

259225 5533225

)53)(53(

Arrange the factors into groups.

)53( 15

9

4 When the radicand is a fraction, find the principal square root of the numerator and the denominator.

9

4

9

4

3

2

A nonperfect square is a number that is not the square of a natural number.

To approximate the square of a nonperfect square, find two consecutive integers that the square root is between.

Find two nearby perfect squares that 19 is between.

List the perfect squares in order.

,...49,36,25,16,9,4,1

1919 is between 16 and 25

THINK

525416

is between 4 and 5.19

19Between what two consecutive integers is ?

Square roots of nonperfect square are examples of numbers that are not rational. Numbers that are not rational are called irrational.

KEY CONCEPT Irrational Numbers

Irrational numbers are numbers that cannot be expressed in

the form where a and b are integers and .b

a0b

The following are examples of irrational numbers.

1. Positive or negative decimals that are nonterminating and nonrepeating or have a pattern in their digits but do not repeat exactly.

4.13216582…

-0.505005005…

2. Square roots of nonperfect squares.

3. Pi, symbolized by a Greek letter.

19 42

250

Please note, 3.14 and 22/7 are rational approximate values.

For each number, list all the terms that apply: whole number, integer, rational number, and irrational number.

-321.11

Rational number

45

Rational number

Whole number

Integer

0.010203…

Irrational number

358.1

Rational number

7

22

Rational number

144

Rational number

Whole number

Integer

36

Rational number

Integer

102

Irrational number

Find each square root. If the radicand is a nonperfect square, between with two consecutive integers would the square root fall?

4001040400

2558400 255222400

)225)(225( )225(

20

205415205

This is a nonperfect square.

,225,196,,169,144

205Between 14 and 15.

Find each square root. If the radicand is a nonperfect square, between with two consecutive integers would the square root fall?

81

49

81

499

7

77

11777 This is a nonperfect square.

49,64,81,100

-77

Between -9 and -8.