Describe and compare subsets of a number system

Ontario Grade 9 Mathematics

Number

Expectation B1.2 — How Number Sets Nest Inside Each Other

Numbers are everywhere, but not all numbers are alike. A counting number like

7

, a negative temperature like

-3

, a share of a pizza written as

\frac{3}{4}

, and the diagonal of a unit square written as

\sqrt{2}

all feel different — because they belong to different families, or subsets, of the real number system. In this lesson you will learn the names and defining properties of each major subset, discover how they nest inside one another like a set of Russian dolls, and practise sorting any number into the right family (or families). Understanding these subsets is the foundation for every calculation you will meet in Grade 9 and beyond.

What you will learn

Identify and describe the major subsets of the real number system: natural numbers, whole numbers, integers, rational numbers, and irrational numbers.
Explain in your own words why each subset is contained within the next larger set.
Classify any given number as belonging to one or more subsets.
Compare subsets by stating what each one has that the others do not.
Recognise that every real number is either rational or irrational, but not both.

The Five Key Subsets — Definitions and Symbols

The real number system is the collection of all numbers that can be placed on a number line. Within it, mathematicians have identified five subsets that you need to know for this course, each with its own symbol and personality.

The Natural Numbers, written

\mathbb{N}

, are the counting numbers:

\{1, 2, 3, 4, 5, \ldots\}

. They start at

1

and keep going forever. Some textbooks include

0

in this set; the Ontario curriculum treats

0

as separate, so watch for context.

The Whole Numbers, written

\mathbb{W}

, add exactly one new member to the natural numbers — zero:

\{0, 1, 2, 3, 4, \ldots\}

. Every natural number is automatically a whole number, but

0

is a whole number that is not a natural number.

The Integers, written

\mathbb{Z}

, extend the whole numbers by including all the negative counting numbers:

\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}

. The word integer comes from the Latin for 'untouched' or 'whole' — integers never have a fractional part.

The Rational Numbers, written

\mathbb{Q}

, are all numbers that can be expressed as a fraction

\frac{a}{b}

where

a

and

b

are integers and

b \neq 0

. This set is much larger than it first appears: every integer qualifies (since

5 = \frac{5}{1}

), and every terminating or repeating decimal qualifies (since

0.\overline{3} = \frac{1}{3}

and

1.75 = \frac{7}{4}

). The Irrational Numbers have no standard letter symbol, but they are real numbers that cannot be written as any such fraction. Their decimals go on forever without repeating — for example,

\sqrt{2} \approx 1.41421356\ldots

and

\pi \approx 3.14159265\ldots

.

Rational:

\frac{a}{b}

, where a, b ∈

\mathbb{Z}

and b ≠ 0

$\mathbb{N} = \{1, 2, 3, \ldots\}$ — counting numbers only.
$\mathbb{W} = \{0, 1, 2, 3, \ldots\}$ — whole numbers add zero.
$\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}$ — integers include negatives.
$\mathbb{Q}$ — rationals include all fractions $\frac{a}{b}$ , terminating decimals, and repeating decimals.
Irrationals are real numbers whose decimal expansions never terminate and never repeat.

How the Subsets Nest — The Inclusion Relationship

The five subsets do not sit side-by-side as equals; they nest inside each other in a strict chain. Knowing this chain lets you make instant comparisons. The relationship is: every natural number is a whole number, every whole number is an integer, every integer is a rational number, and every rational number is a real number. Written with the subset symbol:

\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}

.

The irrational numbers sit alongside the rationals inside

\mathbb{R}

, but they share no members with

\mathbb{Q}

. Together,

\mathbb{Q}

and the irrationals make up all of

\mathbb{R}

with no overlap. Think of

\mathbb{R}

as a pie divided into exactly two slices: rationals and irrationals.

This nesting has a practical consequence: when a number belongs to a smaller subset (say, the integers), it automatically belongs to every larger subset too. The number

-8

, for example, is simultaneously an integer, a rational number, and a real number — but it is not a natural number or a whole number. A number like

\frac{5}{2} = 2.5

is rational and real, but not an integer, whole number, or natural number.

Comparing subsets means stating both what they share and what makes them distinct. Whole numbers and integers both contain

\{0, 1, 2, 3, \ldots\}

, but integers go further by also containing

\{-1, -2, -3, \ldots\}

. Integers and rationals both contain every whole number and every negative integer, but rationals go further by also containing numbers like

\frac{2}{3}

that cannot be written without a fraction.

\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}

The nesting chain is $\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$ .
Irrational numbers and rational numbers together fill $\mathbb{R}$ with no overlap.
A number in a smaller subset automatically belongs to every larger subset.
Comparing two subsets: identify shared members, then identify what one has that the other does not.

Classifying Any Given Number — A Step-by-Step Strategy

To classify a number, work from the innermost subset outward and ask a sequence of yes/no questions. Start with: Is it a positive whole number with no decimal or fraction? If yes, it is a natural number (and therefore also whole, integer, rational, and real). Next: Is it zero? If yes, it is a whole number but not a natural number. Next: Is it a negative whole number (no decimal or fraction)? If yes, it is an integer but not whole or natural. Next: Can it be written exactly as a fraction of two integers? If yes, it is rational. Finally: Is it a non-terminating, non-repeating decimal? If yes, it is irrational.

A few tricky cases deserve special attention. The number

0.\overline{6}

(zero point six repeating) looks like it might be irrational, but repeating decimals are always rational — in this case

0.\overline{6} = \frac{2}{3}

. Square roots of perfect squares are integers:

\sqrt{49}

= 7 ∈

\mathbb{Z}

. Square roots of non-perfect squares are irrational:

\sqrt{50}

cannot be simplified to a neat fraction. Negative fractions like

-\frac{7}{2}

are rational and real, but they belong to none of the three innermost sets.

Decimals that terminate (stop) are always rational. For example,

3.125 = \frac{25}{8}

. You can convert any terminating decimal by counting the decimal places and writing the appropriate power of ten in the denominator, then simplifying. This confirms terminating decimal

\Rightarrow

rational.

0.

\overline{d_1 d_2 \ldots}

∈

\mathbb{Q}

and

\sqrt{p}

∉

\mathbb{Q}

when p is not a perfect square

Work from the innermost subset outward when classifying.
Repeating decimals ( $0.\overline{3}$ , $1.\overline{27}$ , etc.) are always rational.
Terminating decimals are always rational.
$\sqrt{n}$ is rational only when $n$ is a perfect square.
A number can belong to several subsets simultaneously.

Comparing Subsets — Similarities and Differences

Being able to compare subsets means more than just reciting definitions — it means articulating what one group has in common with another and where they diverge. Consider

\mathbb{W}

and

\mathbb{Z}

. Similarity: both sets contain infinitely many numbers; both include

0

and all positive counting numbers; both sets are closed under addition (adding two members always gives another member of that set). Difference:

\mathbb{Z}

contains negative numbers (

-1, -2, -3, \ldots

), which

\mathbb{W}

does not. So

\mathbb{Z}

is strictly larger.

Now compare

\mathbb{Z}

and

\mathbb{Q}

. Similarity: both include all the integers; both extend infinitely in the positive and negative directions. Difference:

\mathbb{Q}

contains numbers between every pair of integers — numbers like

\frac{1}{2}

,

-\frac{3}{4}

, and

2.37

— whereas

\mathbb{Z}

has no members between consecutive integers. In fact, between any two rational numbers you can always find another rational number, a property called density, which the integers do not have.

Finally, compare

\mathbb{Q}

and the irrationals. Similarity: both are infinite subsets of

\mathbb{R}

; both contain numbers spread across the entire number line. Key difference:

\mathbb{Q}

and the irrationals share no members at all — a number is one or the other, never both. Together they cover all of

\mathbb{R}

.

\mathbb{Q}

\cup \{irrationals\} =

\mathbb{R}

and

\mathbb{Q}

\cap \{irrationals\} = \emptyset

$\mathbb{W}$ and $\mathbb{Z}$ share all non-negative integers; $\mathbb{Z}$ adds all negative integers.
$\mathbb{Z}$ and $\mathbb{Q}$ share all integers; $\mathbb{Q}$ additionally contains all fractions and mixed decimals.
$\mathbb{Q}$ is dense — there is always another rational number between any two rationals.
$\mathbb{Q}$ and the irrationals are completely disjoint subsets that together form $\mathbb{R}$ .

Subset Membership at a Glance

Number	$\mathbb{N}$	$\mathbb{W}$	$\mathbb{Z}$	$\mathbb{Q}$
$8$	✓	✓	✓	✓
$0$	✗	✓	✓	✓
$-5$	✗	✗	✓	✓
$\frac{3}{7}$	✗	✗	✗	✓
$\sqrt{3}$	✗	✗	✗	✗ (irrational)

Worked example

Classifying and Comparing a Set of Six Numbers

For each of the following six numbers, state every subset of the real number system to which it belongs. Then write one sentence comparing the subset memberships of

-9

and

-\frac{9}{4}

. The numbers are:

6

,

0

,

-9

,

-\frac{9}{4}

,

\sqrt{11}

,

0.\overline{45}

.

Classify $6$
The number $6$ is a positive counting number with no decimal or fractional part, so it qualifies for every subset in the nesting chain.
6 ∈ $\mathbb{N}$ ,\ $\mathbb{W}$ ,\ $\mathbb{Z}$ ,\ $\mathbb{Q}$ ,\ $\mathbb{R}$
Classify $0$
Zero is not a counting number (it is not in $\mathbb{N}$ under the Ontario convention), but it is the starting point of the whole numbers, and it is an integer, rational ( $0 = \frac{0}{1}$ ), and real.
0 ∈ $\mathbb{W}$ ,\ $\mathbb{Z}$ ,\ $\mathbb{Q}$ ,\ $\mathbb{R}$ (0 ∉ $\mathbb{N}$ )
Classify $-9$
The number $-9$ is negative, so it is not natural or whole. It has no fractional part, so it is an integer. Every integer is rational ( $-9 = \frac{-9}{1}$ ) and real.
-9 ∈ $\mathbb{Z}$ ,\ $\mathbb{Q}$ ,\ $\mathbb{R}$ (-9 ∉ $\mathbb{N}$ ,\ $\mathbb{W}$ )
Classify $-\frac{9}{4}$
The fraction $-\frac{9}{4} = -2.25$ has a numerator and denominator that are both integers, and the denominator is not zero, so it is rational by definition. It is not an integer because $-2.25$ has a non-zero decimal part. It is real.
- $\frac{9}{4}$ ∈ $\mathbb{Q}$ ,\ $\mathbb{R}$ (- $\frac{9}{4}$ ∉ $\mathbb{N}$ ,\ $\mathbb{W}$ ,\ $\mathbb{Z}$ )
Classify $\sqrt{11}$
Since $11$ is not a perfect square (the perfect squares near $11$ are $9$ and $16$ ), the square root cannot be simplified to a ratio of integers. Its decimal expansion is $\sqrt{11} \approx 3.31662\ldots$ , which neither terminates nor repeats. Therefore $\sqrt{11}$ is irrational and real only.
$\sqrt{11}$ ∈ $\mathbb{R}$ (irrational) ( $\sqrt{11}$ ∉ $\mathbb{Q}$ )
Classify $0.\overline{45}$
The bar over $45$ means the digits $45$ repeat forever: $0.454545\ldots$ . Any repeating decimal can be converted to a fraction. Let $x = 0.\overline{45}$ . Then $100x = 45.\overline{45}$ , so $100x - x = 45$ , giving $99x = 45$ and $x = \frac{45}{99} = \frac{5}{11}$ . This is a fraction of two integers, confirming it is rational.
0. $\overline{45}$ = $\frac{5}{11}$ ∈ $\mathbb{Q}$ ,\ $\mathbb{R}$ (0. $\overline{45}$ ∉ $\mathbb{N}$ ,\ $\mathbb{W}$ ,\ $\mathbb{Z}$ )
Compare $-9$ and $-\frac{9}{4}$
Both numbers are negative, both are rational and real, but $-9$ is also an integer while $-\frac{9}{4}$ is not — $-9$ belongs to one more subset ( $\mathbb{Z}$ ) than $-\frac{9}{4}$ does.
-9 ∈ \{ $\mathbb{Z}$ , $\mathbb{Q}$ , $\mathbb{R}$ \} vs. - $\frac{9}{4}$ ∈ \{ $\mathbb{Q}$ , $\mathbb{R}$ \}

Answer:

6

:

\mathbb{N}, \mathbb{W}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}

.

\quad

0

:

\mathbb{W}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}

.

\quad

-9

:

\mathbb{Z}, \mathbb{Q}, \mathbb{R}

.

\quad

-\frac{9}{4}

:

\mathbb{Q}, \mathbb{R}

.

\quad

\sqrt{11}

:

\mathbb{R}

(irrational).

\quad

0.\overline{45}

:

\mathbb{Q}, \mathbb{R}

.

Check: Verify the nesting rule: every set listed for a number should be a superset of the previous one in the chain

\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}

. For

0.\overline{45}

, confirm

\frac{5}{11}

:

\frac{5}{11} = 0.\overline{45}

✓ (divide

5 \div 11

to check). For

\sqrt{11}

, confirm non-perfect-square:

3^2 = 9 ‹ 11 ‹ 16 = 4^2

, so

\sqrt{11}

is not an integer ✓.

Common mistakes and how to avoid them

Assuming

\sqrt{n}

is always irrational. Students often label

\sqrt{36}

as irrational.

Correction:

\sqrt{36} = 6

, which is a natural number. Only square roots of non-perfect squares are irrational. Always simplify the square root first.

Thinking a number can belong to

\mathbb{Q}

and also be irrational at the same time.

Correction: Rational and irrational are mutually exclusive categories. If a number is rational, it cannot be irrational, and vice versa. There is no overlap between these two sets.

Forgetting that negative fractions like

-\frac{2}{5}

are rational numbers.

Correction: The definition of rational is

\frac{a}{b}

where a, b ∈

\mathbb{Z}

and

b \neq 0

. Integers include negatives, so

-\frac{2}{5}

qualifies with

a = -2

and

b = 5

.

Treating a repeating decimal like

0.\overline{7}

as irrational because its decimal 'goes on forever'.

Correction: A decimal that repeats a pattern is always rational. Only non-terminating AND non-repeating decimals are irrational. For example,

0.\overline{7} = \frac{7}{9}

.

Stating that integers are not rational numbers — believing fractions and integers are separate groups.

Correction: Every integer

n

can be written as

\frac{n}{1}

, which satisfies the definition of rational. Integers are a subset of the rationals, not a separate branch.

Lesson summary

The real number system contains five key subsets: natural numbers $\mathbb{N}$ , whole numbers $\mathbb{W}$ , integers $\mathbb{Z}$ , rational numbers $\mathbb{Q}$ , and irrational numbers — each with distinct defining properties.
The subsets nest in the chain $\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$ , meaning a number in any subset also belongs to all larger subsets in the chain.
Rational numbers include all fractions $\frac{a}{b}$ (with integer $a$ , $b$ and $b \neq 0$ ), all terminating decimals, and all repeating decimals.
Irrational numbers are real numbers whose decimal expansions never terminate and never repeat; $\mathbb{Q}$ and the irrationals together fill $\mathbb{R}$ with no overlap.
To classify a number, work from the innermost subset outward using the definition of each set as a yes/no test.
To compare two subsets, identify what members they share and what one set contains that the other does not.

Check your understanding

Question 1

Which of the following numbers is irrational?

$\sqrt{25}$
$0.\overline{12}$
$\sqrt{7}$
$-\frac{4}{3}$

Show answer and explanation

\sqrt{7}

\sqrt{25} = 5

, a natural number.

0.\overline{12} = \frac{12}{99} = \frac{4}{33}

, a rational number.

-\frac{4}{3}

is a rational number.

\sqrt{7}

cannot be simplified because

7

is not a perfect square, and its decimal

2.6457\ldots

never terminates or repeats, so it is irrational.

Question 2

A student says: 'The number

-13

is an integer but not a rational number.' What is wrong with this statement?

Nothing is wrong; $-13$ is only an integer.
Integers are a subset of rational numbers, so $-13$ is both an integer and a rational number.
$-13$ is not an integer because it is negative.
$-13$ is irrational because it is negative.

Show answer and explanation

Integers are a subset of rational numbers, so

-13

is both an integer and a rational number.

Every integer can be written as a fraction with denominator

1

. Since

-13 = \frac{-13}{1}

and both

-13

and

1

are integers with

1 \neq 0

,

-13

satisfies the definition of a rational number. The nesting chain confirms

\mathbb{Z} \subset \mathbb{Q}

.

Question 3

Which pair of subsets is described correctly?

Whole numbers and integers are identical sets.
Natural numbers include zero but integers do not.
Every whole number is an integer, but not every integer is a whole number.
Rational numbers and irrational numbers overlap inside $\mathbb{R}$ .

Show answer and explanation

Every whole number is an integer, but not every integer is a whole number.

The whole numbers

\{0, 1, 2, \ldots\}

are all contained within the integers

\{\ldots, -2, -1, 0, 1, 2, \ldots\}

, so

\mathbb{W} \subset \mathbb{Z}

. However, negative integers like

-4

are integers but not whole numbers, so the two sets are not equal. Rational and irrational numbers never overlap.

Question 4

How many of the following sets does the number

\frac{16}{4}

belong to:

\mathbb{N}

,

\mathbb{W}

,

\mathbb{Z}

,

\mathbb{Q}

,

\mathbb{R}

?

$1$ — it is only rational because it is written as a fraction.
$3$ — it is an integer, rational, and real.
$5$ — it belongs to all five sets.
$2$ — it is rational and real only.

Show answer and explanation

5

— it belongs to all five sets.

First simplify:

\frac{16}{4} = 4

. The number

4

is a positive counting number with no fractional part, so it is a natural number. Since

\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}

, it automatically belongs to all five sets.

Key terms

Real Numbers ( $\mathbb{R}$ ): The complete set of all numbers that can be located on a number line, including all rationals and all irrationals.
Natural Numbers ( $\mathbb{N}$ ): The positive counting numbers $\{1, 2, 3, 4, \ldots\}$ , not including zero in the Ontario Grade 9 curriculum.
Whole Numbers ( $\mathbb{W}$ ): The set $\{0, 1, 2, 3, \ldots\}$ — the natural numbers together with zero.
Integers ( $\mathbb{Z}$ ): All positive and negative whole numbers together with zero: $\{\ldots, -2, -1, 0, 1, 2, \ldots\}$ .
Rational Numbers ( $\mathbb{Q}$ ): All numbers expressible as $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$ ; equivalently, all terminating or repeating decimals.
Irrational Numbers: Real numbers that cannot be written as a fraction of two integers; their decimal expansions are non-terminating and non-repeating (e.g., $\sqrt{2}$ , $\pi$ ).
Subset: A set whose every member also belongs to a larger set. For example, $\mathbb{N}$ is a subset of $\mathbb{W}$ because every natural number is also a whole number.
Density (of rational numbers): The property that between any two distinct rational numbers there is always another rational number, meaning rationals have no 'gaps' between consecutive members.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 9 Mathematics (MTH1W), 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.

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B1.2 · Describe and compare subsets of a number system

Ontario Grade 9 Mathematics

Expectation B1.2 — How Number Sets Nest Inside Each Other

What you will learn

The Five Key Subsets — Definitions and Symbols

How the Subsets Nest — The Inclusion Relationship

Classifying Any Given Number — A Step-by-Step Strategy

Comparing Subsets — Similarities and Differences

Subset Membership at a Glance

Classifying and Comparing a Set of Six Numbers

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MTH1W

About this lesson and its review