Describe and order rational and irrational numbers

Ontario Grade 8 Mathematics

Number

Ontario Grade 8 Mathematics — B1.2

Imagine placing numbers along a path from left to right. Numbers farther left are smaller, and numbers farther right are larger. Some numbers can be written as fractions of integers. Others cannot. In this lesson, you will describe both kinds of numbers and decide where they belong when ordering a set.

What you will learn

Describe rational and irrational numbers.
Recognize rational numbers written as fractions or decimals.
Use a number line and decimal approximations to order rational and irrational numbers.

1. Describe the numbers

An integer is a whole number, its opposite, or zero. Examples include

-3

,

0

, and

5

. A rational number can be written as a fraction of two integers, with a denominator other than zero. For example,

\frac{3}{4}

is rational. The integers

5

and

-2

are rational too, because they can be written as

\frac{5}{1}

and

\frac{-2}{1}

.

A decimal that ends is called a terminating decimal. It is rational. For example,

0.6=\frac{6}{10}=\frac{3}{5}

. A decimal that repeats a digit or group of digits forever is called a repeating decimal. It is rational too. For example,

0.333\ldots

has the repeating digit

3

. The dots show that the pattern continues.

An irrational number cannot be written as a fraction of two integers. Its decimal continues without ending and without repeating a fixed pattern. For example,

\sqrt{2}

is irrational, and its decimal begins

1.4142\ldots

. Here, the root notation is simply the exact name of the number being compared; no root calculation is needed.

A decimal approximation is a value close to a number, but not necessarily its exact value. For example,

1.41

is an approximation of

\sqrt{2}

, not its exact value. The symbol

\approx

means “is approximately equal to.”

Rational numbers can be written as fractions of integers.
Terminating and repeating decimals are rational.
Irrational numbers cannot be written as fractions of integers; their decimals do not end or repeat.

2. Use a number line

A number line shows numbers in order. As you move to the right, the numbers get larger. As you move to the left, they get smaller. So a number to the left of another number is less than it.

To place an irrational number, use a decimal approximation that is accurate enough for the comparison. For example,

\sqrt{2}\approx1.41

and

\sqrt{5}\approx2.24

. These approximations help show where the exact numbers belong. Keep the exact forms when naming the irrational numbers.

The same left-to-right rule works with negative numbers. For example,

-1.5

is less than

-1.2

because

-1.5

is farther left. When ordering a set, read from left to right to list its values from least to greatest.

a<b

Left means less; right means greater.
Use decimal approximations when they make the positions clear.
Keep exact forms, such as $\sqrt{2}$ , when naming irrational numbers.

3. A method for ordering a mixed set

First, describe each value as rational or irrational. Then write helpful decimal forms for rational numbers and approximations for irrational numbers. Compare the values and list them from least to greatest.

Use enough decimal places to make the order clear. If two values are close, a rough estimate may not be enough. Also take care with negative values: the one farther left is smaller, even if its digits after the negative sign are larger.

A number line can make your comparison visible. Place the values in their estimated positions, then read the labels from left to right. The order should agree with the decimal comparisons.

Describe, compare, and then order.
Use enough decimal places to distinguish nearby values.
Check that the ordered list moves from smaller to larger.

4. Try the ideas

For each set, describe which values are rational and which are irrational. Then order the values from least to greatest. Explain how the number line or decimal estimates support your order.

Set A contains

\frac{3}{4}

,

0.8

, and

\sqrt{2}

. Set B contains

-1.5

,

-\sqrt{2}

, and

-1.3

. For Set B, the estimate

-\sqrt{2}\approx-1.41

helps compare the negative values.

Check your work in two ways. Confirm that the descriptions match the forms of the numbers. Then read your ordered list from left to right and make sure each value is greater than the one before it.

For Set A, compare $0.75$ , $0.8$ , and about $1.41$ .
For Set B, compare $-1.5$ , about $-1.41$ , and $-1.3$ .

Worked example

Order two rational numbers and an irrational number

Describe each number as rational or irrational, then order

\frac{3}{4}

,

0.8

, and

\sqrt{2}

from least to greatest.

Comparing three numbers

From left to right: $\frac{3}{4}=0.75$, $0.8$, and $\sqrt{2}\approx1.41$.

Describe each number
$\frac{3}{4}$ is rational because it is a fraction of integers. The terminating decimal $0.8$ is rational. The number $\sqrt{2}$ is irrational.
Compare their positions
Write the fraction as a decimal. The approximation for $\sqrt{2}$ places it to the right of both rational numbers.
$\frac{3}{4}=0.75<0.8<\sqrt{2}\approx1.41$
State the order
Read the values from left to right to give the order from least to greatest.
$\frac{3}{4},\ 0.8,\ \sqrt{2}$

Answer: The rational numbers are

\frac{3}{4}

and

0.8

. The irrational number is

\sqrt{2}

. From least to greatest:

\frac{3}{4}<0.8<\sqrt{2}

.

Check: The decimal values are

0.75

,

0.8

, and about

1.41

, in increasing order.

Worked example

Order negative and positive numbers

Describe and order

-\sqrt{2}

,

-1.3

, and

0

from least to greatest.

Ordering negative values and zero

From left to right: $-\sqrt{2}\approx-1.41$, $-1.3$, and $0$.

Describe each number
$-1.3$ is rational because it is a terminating decimal. Zero is rational because it can be written as a fraction of integers. The number $-\sqrt{2}$ is irrational.
Compare the negative values
The estimate $-\sqrt{2}\approx-1.41$ places it to the left of $-1.3$ . Both negative values are to the left of zero.
$-\sqrt{2}\approx-1.41<-1.3<0$
State the order
Reading from left to right gives the order from least to greatest.
$-\sqrt{2},\ -1.3,\ 0$

Answer: The rational numbers are

-1.3

and

0

. The irrational number is

-\sqrt{2}

. From least to greatest:

-\sqrt{2}<-1.3<0

.

Check: The estimate

-1.41

is less than

-1.3

, and both numbers are less than zero.

Worked example

Order close values

Order

\sqrt{5}

,

2.2

, and

\sqrt{6}

from least to greatest. Describe each as rational or irrational.

Comparing close values

From left to right: $2.2$, $\sqrt{5}\approx2.24$, and $\sqrt{6}\approx2.45$.

Describe each number
$2.2$ is rational because its decimal ends. The values $\sqrt{5}$ and $\sqrt{6}$ are irrational.
Use approximations to compare
Write $2.2$ as $2.20$ to compare hundredths. The estimates show that $\sqrt{5}$ is just above $2.20$ , while $\sqrt{6}$ is farther to the right.
$2.2=2.20,\quad \sqrt{5}\approx2.24,\quad \sqrt{6}\approx2.45$
State the order
The approximations place the three values in increasing order.
$2.2<\sqrt{5}<\sqrt{6}$

Answer:

2.2

is rational;

\sqrt{5}

and

\sqrt{6}

are irrational. From least to greatest:

2.2<\sqrt{5}<\sqrt{6}

.

Check: The estimates

2.20

, about

2.24

, and about

2.45

increase from left to right.

Common mistakes and how to avoid them

Thinking every decimal is irrational because it has many digits.

Correction: A decimal that ends or repeats is rational. An irrational decimal continues without ending or repeating.

Treating a decimal approximation as the exact irrational number.

Correction: Use

\approx

for an approximation and keep the exact form, such as

\sqrt{2}

, when naming the number.

Ordering negative numbers by ignoring their negative signs.

Correction: Use their positions on a number line. For example,

-1.5

is less than

-1.2

because it is farther left.

Lesson summary

Rational numbers can be written as fractions of integers. Terminating and repeating decimals are rational.
Irrational numbers cannot be written as fractions of integers. Their decimals continue without ending or repeating.
Use decimal approximations to compare values while keeping exact forms for irrational numbers.
On a number line, values increase from left to right.

Check your understanding

Question 1

Which list orders

\sqrt{3}

,

1.7

, and

\sqrt{2}

from least to greatest?

$\sqrt{2}<\sqrt{3}<1.7$
$\sqrt{2}<1.7<\sqrt{3}$
$1.7<\sqrt{2}<\sqrt{3}$
correctIndex`???

Show answer and explanation

\sqrt{2}<1.7<\sqrt{3}

\sqrt{2}\approx1.41

and

\sqrt{3}\approx1.73

. Therefore,

\sqrt{2}<1.7<\sqrt{3}

.

Question 2

How would you describe

0.272727\ldots

?

It is rational because a group of digits repeats.
It is irrational because it does not end.
It is irrational because it is less than $1$ .
correctIndex`???

Show answer and explanation

It is rational because a group of digits repeats.

The group of digits

27

repeats, so the decimal is rational.

Question 3

Which order is correct from least to greatest?

$-1.2<-\sqrt{2}<0$
$-\sqrt{2}<-1.2<0$
$0<-1.2<-\sqrt{2}$
correctIndex`???

Show answer and explanation

-\sqrt{2}<-1.2<0

-\sqrt{2}\approx-1.41

, which is less than

-1.2

. Both are less than zero.

Key terms

Integer: A whole number, its opposite, or zero.
Rational number: A number that can be written as a fraction of two integers, with a denominator other than zero.
Irrational number: A number that cannot be written as a fraction of two integers; its decimal continues without ending or repeating.
Terminating decimal: A decimal that ends.
Repeating decimal: A decimal in which a digit or group of digits repeats forever.
Approximation: A value close to a number, but not necessarily its exact value.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 Mathematics, expectation B1.2. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

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B1.2 · Describe and order rational and irrational numbers

Ontario Grade 8 Mathematics

Ontario Grade 8 Mathematics — B1.2

What you will learn

1. Describe the numbers

2. Use a number line

3. A method for ordering a mixed set

4. Try the ideas

Order two rational numbers and an irrational number

Order negative and positive numbers

Order close values

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 8 Math

About this lesson and its review