Chapter

The Real Number System1

Real numbers can be represented as points on a number line and can be used to count, measure, estimate, or approximate quantities. Most whole numbers have square roots that are irrational numbers.

BIG IDEA

Lessons

1.1 Representing Positive Square Roots and Irrational Numbers

1.2 Understanding the Real Number System

How far is it to the next exit?Have you ever been on a road trip? As you go along, you see

road signs telling you how far it is to the next town or exit.

But have you noticed the smaller kilometre marker signs along

some highways? They indicate the distance from the beginning

of the highway.

Emergency workers use the kilometre marker signs to locate

accidents, and drivers can use them to estimate how far away

an exit is. These signs are like the tick marks on a number line

and can be used to estimate or measure distances.

Do you think you could ever find all the numbers between two numbers?

1

RECALL PRIOR KNOWLEDGE

Representing rational numbers

Rational numbers are numbers that can be represented as fractions.

Examples

14, 35

8, 9

11

Rational numbers can also be represented as decimals using division:

14 = 1 ÷ 4

= 0.25

358 = 29

8 = 29 ÷ 8

= 3.625

911

= 9 ÷ 11

= 0.818 181 818 …

Some decimals repeat endlessly. You can use bar

notation to represent the repeating part.

In 0.818 181 818 …, the digits 81 repeat.

0.818 181 818 … = 0.81

In 0.154 545 454 …, only the digits 54 repeat.

0.154 545 454 … = 0.154

Quick Check

Represent each fraction as a decimal.

1 34

2 25

3 125

Represent each number using bar notation.

4 29

5 511

6 16

Write each number as a repeating decimal, rounded to six digits.

7 0.9 8 4.145 9 0.243

To write a mixed number as a decimal, you can write the mixed number as an improper fraction and then divide. Or, you can write the fraction in the mixed number as a decimal first, by dividing, and then add the decimal to the whole number.

358 = 3 + 58

= 3 + 0.625

= 3.625

2 Chapter 1 The Real Number System

Determining absolute values

The absolute value of a rational number n is denoted by | n |.

Examples

| 2 | = 2, | − 32 | = 3

2The absolute value of a number is a measure of its distance from 0.

The distance from − 32 to 0 is 32 units.

The distance from 2 to 0 is 2 units.

Quick Check

Use the following set of numbers for questions 10 to 14 .

1.25, − 35, 4, −4, −5.75

10 Determine the absolute value of each number.

11 Which number is closest to 0?

12 Which number is farthest from 0?

13 Name two numbers with the same absolute value.

14 Which number has the greatest absolute value?

−3 −2 −1 0 1 2 3

32

units2 units

Use a number line to determine the absolute value of each number.

15 | −15 | 16 65

17 | −2.1 |

Copy and complete by replacing each   ?   with >, =, or <.

18 | −7 |   ?   | −7.2 | 19 | 5 |   ?   | −5 | 20 | −26 |   ?   52

3Chapter 1 The Real Number System

Determining squares and square roots

Determine the square of 4. The square of a number is the product of the

number and itself.

42 = 4 × 4

  = 16

The square of 4 is 16.

Determine the square root of 4.

4 = 2

The square root of 4 is 2, because 22 = 4.

On the number line, √_

4 is closer to 0 than 42 is. To describe these numbers in relation

to one another, you can express them as follows:

4 < 42  

Quick Check

Determine the square of each number.

21 3 22 10

Determine the square root of each number.

23 64 24 49

Order the numbers from greatest to least. Use the > symbol.

25 √_

81 , 82, 52

0 2 16

42

10 20

4

Squares and square roots are inverse operations.

4 Chapter 1 The Real Number System

Ordering rational numbers on a number line

Order these rational numbers on a number line.

4, −5, 14, −312, 0.6, −0.1, 9

Quick Check

Order the numbers from least to greatest. Use the < symbol. Then graph numbers on a number line.

26 1112, −13

5, − 0.3, 1.6, 1910, 4

−5 −4 −3 −2 −1 0 1 2 3 4 5

−312 −0.1 4−0.6

14−5 9

You can calculate the decimal value of a number to help you locate it on the number line.

5Chapter 1 The Real Number System

Lesson

You can use a calculator to determine greater square numbers.

Vocabularynon-perfect square number

approximate

irrational number

Goals• Estimate the positive square roots of whole numbers

that do not have whole-number square roots.

• Locate irrational numbers on the number line.

LEARN Identify perfect square numbers.

There are 12 perfect square numbers up to and including 144.

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144

You can visualize a perfect square number by picturing a square with whole-number sides.

The area of the square above is

11 × 11 = 112

= 121 cm2

Below are some of the perfect square numbers

beyond 144:

132 = 169 and 169 = 13

142 = 196 and 196 = 14

152 = 225 and 225 = 15

•

•

•

Representing Positive Square Roots and Irrational Numbers1.1

12 = 1 22 = 4 32 = 9 42 = 16 52 = 25 62 = 36 72 = 49 82 = 64 92 = 81 102 = 100112 = 121 122 = 144

11 cm

6 Chapter 1 The Real Number System

LEARN Identify non-perfect square numbers.

From the numbers in the previous Learn, you can see that there are whole numbers

that lie between two perfect square numbers:

2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, ...

Since the positive square roots of 1 and 4

are 1 and 2, the positive square roots of 2

and 3 must be between 1 and 2.

Hence, the whole numbers 2 and 3 do not have

whole-number square roots. The numbers 2

and 3 are examples of non-perfect square numbers.

The squares of whole numbers are always positive. You cannot determine the square root of a negative number.

Math Note

GUIDED LEARNINGAnswer the question.

1 Does a square with an area of 20 cm2 have a whole-number side length? Explain why.

Determine whether each number is a non-perfect square number.

2 16 3 48 4 2205

5 122 6 225 7 2025

HANDS-ON ACTIVITY

Work in pairs.

STEP 1 Draw a square that has a side length of 2 cm

on a piece of paper. Then cut out the square.

STEP 2 Determine the area of the square (square A).

Materials:

• paper

• ruler

• scissorsDETERMINE THE VALUE OF √

_ 2 USING A SQUARE

7Lesson 1.1 Representing Positive Square Roots and Irrational Numbers

STEP 3 Fold the four corners of square A toward the centre to form

square B, as shown below.

2 cm

square A square B

STEP 4 State how the areas of square A and square B are related. State

the area of square B. How can you represent the side length of

square B?

STEP 5 Using your answer in STEP 4, determine the side length of

square B with a calculator. Round your answer to two decimal

places.

Math Journal Measure a side of square B. Compare your measurement

with your answer from STEP 5. Did you get the same answer? Explain.

The value from the calculator is approximate because the number of digits the calculator shows is limited. The actual value has decimals that do not end or repeat.

Math Note

2 cm

210

Centimetres

LEARN Locate √_

2 on a number line.

The result from the calculator and your measurement give approximate values of √_

2 ,

the side length of square B. You can also think of the measure of √_

2 on the ruler as a

point on the right side of the number line.

8 Chapter 1 The Real Number System

The term rational number means that the number can be expressed as a ratio of two integers. So, the term irrational number means that the number cannot be expressed as a ratio of two integers.

Math Note

LEARN Understand irrational numbers on a number line.

Consider a small portion of a number line close to 0. If 14 is close to 0, then 18 is even

closer to 0 than 14. In the same way, 116 is closer to 0 than 18. If you continue in a similar

manner, you can see that there are infinitely many rational numbers clustered very

close to 0. In fact, this phenomenon appears at every other point on the number line.

116

18

14

0

However, there are spaces on this number line.

There are numbers on this number line that are

not rational numbers. One such example is 2.

Because 2 is a non-terminating, non-repeating

decimal, it cannot be written in mn form. 2 is an

irrational number. As you will see later, 2 fits in

the space between 1.4 and 1.5 on the number line.

In other words, the positive side of the number line

has many spaces between the rational numbers.

There are also similar spaces on the negative side

of the number line.

The spaces between the rational numbers on the

number line are filled with irrational numbers,

such as 2.

− 2 is a negative irrational number.

All non-zero multiples of

2 , such as 2 2 , 3 2 ,

4 2 , and so on, are also irrational numbers.

−  2 is different from −2.

Caution

9Lesson 1.1 Representing Positive Square Roots and Irrational Numbers

GUIDED LEARNINGDetermine whether each number is an irrational number.

8 2 7 9 81 10 1.4810 11 π = 3.141 592 …

LEARN Use the areas of squares to estimate the location of an irrational number on a number line.

Perfect squares can help you estimate the location of an irrational number on a

number line without using a calculator.

Consider 2. When you square 2, you get 2. You can think of the area of

square B as 2 square units.

2 units2

square B

√__

2 units

2 is more than 1 but less than 4. Both 1 and 4 are perfect squares. So, 12 < 2 < 22.

You read 12 < 2 < 22 as 2 is between 12 and 22.

Not all square roots are irrational numbers! The square roots of perfect square numbers are rational.

4 = 2

Caution

A number is irrational if it cannot be expressed as a ratio of two numbers.

10 Chapter 1 The Real Number System

This means that the area of square B is more than the area of square C but less

than the area of square A.

22 units2

square A

2 units

12 units2

square C

1 unit2 units2

square B

√__

2 units

Compare the side lengths of squares A, B, and C. You get 1 < 2 > 2.

To approximate the location of 2, consider square D with a

side length of 1.5 units. The area of square D is 1.52, which is

2.25 square units. You can identify 12 < 2 < 1.52. Compare the

side lengths of squares B, C, and D. You get 1 < 2 < 1.5.

1 21.5

√__

2

This method gives only a rough estimate of the location of 2 on the number line. A calculator gives a better estimate.

GUIDED LEARNINGSolve.

12 The area of a square is 5 m2. Estimate the side length of this square, and locate the side

length on a number line.

Estimate the following irrational numbers, and locate them on a number line.

13 √_

3

14 √_

7

15 √_

12

11Lesson 1.1 Representing Positive Square Roots and Irrational Numbers

LEARN Use rational numbers to estimate the value of an irrational number.

Graph 15 on a number line using rational approximations.

Because 15 is a little less than 16, √_

15 should be a little less than √_

16 , or 4.

STEP 1 Determine an approximate value of √_

15 using a calculator.

√_

15 = 3.872 983 346 …

√_

15 lies between 3.8 and 3.9. So, 3.8 < √_

15 < 3.9.

STEP 2 Graph the interval from 3.8 to 3.9 on a number line.

3.8 3.9

STEP 3 Use the approximate value of √_

15 with two decimal places.

3.87 is closer to 3.9 than to 3.8. So, 15 is closer to 3.9.

STEP 4 Use 3.87 to locate √_

15 approximately on the number line.

3.8 3.9 √___

15

GUIDED LEARNINGComplete.

16 Graph 254 on a number line using rational approximations.

a) Which two whole numbers is 254 between? Justify your

reasoning. Using a calculator, 254 =    ?    .

b) The value of 254 with two decimal places is    ?   .    ?    is closer to    ?    than to    ?   .

So, 254 is located closer to    ?   .

c) By using an approximate value of 254, locate

254 on the number line.

   ?       ?   

   ?       ?       ?   

12 Chapter 1 The Real Number System

LEARN Use rational numbers to estimate the value of a negative irrational number.

Graph − 3 on a number line using rational approximations.

Because 12 = 1 and 22 = 4, 3 is between 1 and 2, and − 3 is between −1 and −2.

STEP 1 Determine an approximate value of − 3

using a calculator.

− 3 = −1.732 050 808 …

− 3 lies between −1.8 and −1.7. So, −1.8 , − 3 , −1.7.

STEP 2 Graph the interval from −1.8 to −1.7 on a

number line.

STEP 3 Use the approximate value of − 3 with two decimal places.

−1.73 is closer to −1.7 than to −1.8. So, − 3 is located closer to −1.73.

STEP 4 Use −1.73 to locate − 3 approximately on the

number line.

GUIDED LEARNINGCopy and complete.

17 Graph − 2 on a number line using rational approximations.

Which two integers is − 2 between? Justify your reasoning.

Using a calculator, − 2 = ? .

Graph an interval where − 2 is located.

The value of − 2 with two decimal places is ? .

? is closer to ? than to ? .

So, − 2 is located closer to ? .

Using an approximate value of − 2 , locate − 2 on the number line.

Solve.

18 Graph − 7 on a number line using rational approximations.

−1.8 −1.7

− 3−1.8 −1.7

Just as −1 means “the opposite of

1,” − √_

3 means the opposite of √_

3 .

Math Note

? ?

???

13Lesson 1.1 Representing Positive Square Roots and Irrational Numbers

height

LET’S PRACTISELocate each positive square root on a number line using rational approximations. First tell which two whole numbers the square root is between.

1 3

2 7

3 11

4 26

5 34

6 48

Locate each negative square root on a number line using rational approximations. First tell which two integers the square root is between.

7 − 5

8 − 6

9 − 17

10 − 26

11 − 53

12 − 80

Use a calculator. Locate each irrational number, rounded to three decimal places, on a number line using rational approximations.

13 47 14 − 15 15 94

Locate each irrational number on a number line using rational approximations.

16 101 17 − 132 18 2255

Solve.

19 Locate the value of the constant π on a number line using rational numbers.

20 A triangle is cut from a square as shown in the diagram. The area of the square is 59 cm2. Approximate the height of the triangle to three decimal places.

21 Math Journal When do you need to approximate an irrational number with a rational

value? Explain and illustrate with an example.

14 Chapter 1 The Real Number System

Lesson

Understanding the Real Number System1.2

Goals• Show that irrational numbers are characterized by a

non-terminating and non-repeating decimal representation.

• Understand the real number system and the real number line.

Vocabularyreal number

real number line

LEARN Understand the real number system and a real number line.

A number line is made up of both rational and irrational numbers. Together, these

numbers can be used to label every point on the number line.

−3 −2 −1 0 1 2 3 4 5 6

− 74 5.69π2

Rational and irrational numbers are collectively known as real numbers. A number

line that contains all real numbers is called a real number line. The diagram below

summarizes the relationship among the types of numbers you have learned. You can

see that a real number is either rational or irrational.

Real Numbers

Rational Numbers

Integers

Whole Numbers

Non-integers(Terminating or

Repeated Decimals)

Negative Integers

Positive Integers(Natural Numbers)

Zero

Irrational Numbers

15Lesson 1.2 Understanding the Real Number System

LEARN Order real numbers on a real number line.

Irrational numbers can be located on a number line. So, you can use their absolute

values to indicate their distances from 0.

| − 15 | = | 15 |

This equation is true because 15 and − 15 are equidistant from 0.

− 4 − 3 − 2 − 1 0 1 2 3 4

| −  15 |

−  15 15

| 15 |

The absolute value of a rational number can be measured exactly. The absolute

value of an irrational number cannot because the value of the irrational number can

only be approximated.

You have seen that rational numbers can be written as terminating or repeating

decimals. Mathematicians have shown algebraically that every repeating decimal

is also a rational number. All non-terminating and non-repeating decimals are

irrational numbers.

Some decimal approximations of a few irrational numbers are shown below. For

each irrational number, you can see that the digits after the decimal point do not

terminate or have a repeating pattern in the first 32 digits. Mathematicians have

also shown that no matter how many digits you calculate, they still will not

terminate or repeat.

Examples

2 = 1.414 213 562 373 095 048 801 688 724 209 6 ...

π = 3.141 592 653 589 793 238 462 643 383 279 50 ...

Real numbers appear in many forms, such as fractions, negative decimals, and irrational numbers. When placing real numbers on a number line, you may find it helpful to compare and place them using their decimal forms.

16 Chapter 1 The Real Number System

Order these numbers on a number line.

5 1113

, −  30 , − 8425

, − 8.283, π2

a) Represent each real number in decimal form, rounded to three

decimal places.

From a calculator,

5 1113 < 5.846, −  30 < − 5.477, −

8425 = −3.360,

−8.283 < −8.284, π2 < 9.870

b) Order the real numbers from least to greatest using the symbol , .

−8.283 , − 30 , − 8425 , 5

1113 , π2

c) Locate each real number approximately on a real number line.

−4−5−6−7−8−9−10 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10

π25 1113−8.283 − 30 − 84

25

The symbol < means “approximately equal to.”

Math Note

GUIDED LEARNINGRepresent each real number as a decimal rounded to two decimal places.

1 208 1219 2 − 457

37

3 4π

Represent each real number below as a decimal rounded to four decimal places when necessary. Locate each number on a real number line.

4 19923 , −12.054, −π3, π

2, 200 , − 289

17Lesson 1.2 Understanding the Real Number System

LET’S PRACTISEUse a calculator. Compare each pair of real numbers using either < or >.

1 18 and 19

3 6.1640 and 38

2 −2.23 and − 5

4 −87.098 12 and −87.098 126 …

Use the irrational numbers below for questions 5 to 7 .

26 , 46 , π, π

5 Determine the absolute value of each irrational number with three decimal places.

6 Graph each irrational number on a real number line.

7 Order the irrational numbers from greatest to least using the symbol ..

Use the real numbers below for questions 8 and 9 .

10, 100, 1000 , 10 000 , – π2, 25 , 64 , − 8, π3

8 Copy and complete the table using the real numbers above.

Rational numbers Irrational numbers

? ?

9 Order the real numbers from least to greatest using the symbol ,.

Solve.

10 Using a formula from physics, a sky diver knows that she can free fall for 875 s before opening her parachute.

a) About how many seconds (to the nearest 0.01 s) can she free fall?

b) For her next jump, she can free fall for 29.55 s. Does she have more time on her first or second jump? Explain using a number line.

18 Chapter 1 The Real Number System

PUT ON YOUR THINKING CAP!

PROBLEM SOLVING

Use the decimal representation of √_

3 to answer the following questions.

You can use a calculator to help you.

a) Give a decimal number, a, that is at a distance of 0.01 from √_

3 .

b) Give another decimal number, b, that is even closer to √_

3 than a.

c) Graph the positions of √_

3 , a, and b on a number line.

d) What can you conclude about the number of decimals on the number

line?

What tools and strategies can you use to find more numbers

between a and 3? How many more can you find?

19Chapter 1 The Real Number System

CHAPTER WRAP UP

Concept Map

Key Concepts

▶ Irrational numbers are found in every segment on a number line.

▶ Irrational numbers cannot be expressed in the form mn

.

▶ The decimal form of an irrational number is non-terminating and non-repeating.

▶ An irrational number can be located on a segment of a number line between

two rational numbers.

Example: 1.41 < 2 < 1.42

can be

mapped

to

include

are

represented by

can be expressed as

measure the

distances of

numbers from

0 on

The Real Number System

Number

line

absolute

valuesintegers fractions

Rational

numbers

terminating or

repeating decimals

Irrational

numbers

consists of

are approximately located on a real number line by

m _ n , where m and n are

integers with n ≠ 0

non-terminating,

non-repeating

decimals

are represented bycannot be expressed as

What do you think now? Can you find all the numbers between two numbers?

20 Chapter 1 The Real Number System