Explore the development and use of a number concept

Ontario Grade 9 Mathematics

Number

Ontario MTH1W — Strand B: Number, Expectation B1.1

Numbers are so familiar that it is easy to forget they were invented — slowly, over thousands of years — to solve real problems. Early humans needed to count livestock; later, merchants needed to record debts; later still, builders discovered lengths that no fraction could describe exactly. Each new problem forced mathematicians to widen their idea of what a number actually is. This lesson traces that journey and gives you the precise language and classification skills you will use throughout the MTH1W course and beyond. By the end, you will be able to look at any number, decide exactly which "neighbourhood" of the number system it belongs to, and explain why that classification makes sense.

What you will learn

Trace how the concept of number expanded historically from natural numbers through to real numbers, and explain why each expansion was necessary.
Identify and classify numbers within the sets $\mathbb{N}$ , $\mathbb{W}$ , $\mathbb{Z}$ , $\mathbb{Q}$ , and $\mathbb{R}$ , and describe the subset relationships among them.
Apply number-set knowledge to determine the most precise classification for a given number and justify the choice.
Recognise that irrational numbers exist and cannot be written as a ratio of two integers, giving common examples such as $\sqrt{2}$ and $\pi$ .

The Story of Number: A Historical Expansion

The oldest number concept is counting. Ancient civilisations used tally marks, tokens, and eventually symbols to represent quantities of objects: one, two, three, and so on. These are the natural numbers, and they were enough as long as every answer to every question was a positive whole amount.

Problems arose quickly. What happens when you sell everything you own and have nothing left? Zero was a surprisingly late and controversial addition — many cultures resisted it because 'nothing' felt like it should not be a number. Once zero was accepted, the whole numbers were born: all the natural numbers plus zero.

Debt and temperatures below freezing created the next crisis. If you owe someone

5

coins and you only have

3

, the result of removing

5

from

3

is not a whole number — it is

-2

. Negative numbers, although met with suspicion for centuries, were eventually formalised, giving us the integers.

Sharing and measuring demanded even finer numbers. Splitting a loaf into

3

equal pieces produces a piece of size

\frac{1}{3}

, which is not an integer. Fractions — and more generally ratios of integers — extended the system to the rational numbers.

The ancient Greeks made a shocking discovery: the diagonal of a

1 \times 1

square has length

\sqrt{2}

, which cannot be written as any fraction. Numbers like

\sqrt{2}

and

\pi

are irrational. Together, rationals and irrationals fill out the real number line completely, giving us the real numbers.

Number systems were invented to solve new practical and theoretical problems.
Each new system includes all of the previous one — the sets are nested.
Zero and negative numbers were historically controversial but mathematically necessary.
Irrational numbers were discovered through geometry, not arithmetic.

The Five Key Number Sets and Their Symbols

Modern mathematics gives each number set a name and a symbol. Understanding both lets you communicate precisely. The five sets you need at this level form a chain of inclusions: each set listed below contains every set above it.

The Natural Numbers, written

\mathbb{N}

, are the counting numbers:

\mathbb{N} = \{1, 2, 3, 4, \ldots\}

. Some definitions include zero, but in the Ontario Grade 9 curriculum

\mathbb{N}

typically starts at

1

.

The Whole Numbers, written

\mathbb{W}

, add zero to the naturals:

\mathbb{W} = \{0, 1, 2, 3, \ldots\}

. Every natural number is also a whole number, but

0

is a whole number that is not a natural number.

The Integers, written

\mathbb{Z}

(from the German word Zahlen, meaning 'numbers'), include all whole numbers and their negatives:

\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}

.

The Rational Numbers, written

\mathbb{Q}

(for 'quotient'), contain every number that can be expressed as

\frac{a}{b}

where

a

and

b

are integers and

b \neq 0

. This includes all integers (since

5 = \frac{5}{1}

), all terminating decimals (since

0.75 = \frac{3}{4}

), and all repeating decimals (since

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

). The Irrational Numbers are all real numbers that are NOT rational — their decimal expansions never terminate and never repeat. The Real Numbers, written

\mathbb{R}

, are the union of all rationals and all irrationals; they correspond to every point on the number line.

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

$\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$ — each set is contained in the next.
A rational number has the form $\frac{a}{b}$ , $b \neq 0$ , with a, b ∈ $\mathbb{Z}$ .
Terminating and repeating decimals are always rational.
Irrational numbers produce non-terminating, non-repeating decimals.
$\mathbb{R}$ = rational numbers $\cup$ irrational numbers.

Classifying Numbers: Rules and Reasoning

When you are asked to classify a number, the goal is to name the smallest (most specific) set it belongs to, then list all sets it belongs to using the subset chain. Think of it like postal addresses: your city is part of your province, which is part of Canada — but your city is the most precise description of where you live.

Start with the most restrictive check: Is the number a natural number (a positive integer with no decimal or fraction)? If yes, it automatically belongs to

\mathbb{W}

,

\mathbb{Z}

,

\mathbb{Q}

, and

\mathbb{R}

as well. If not, move outward — check whole, then integer, then rational, then real.

For decimals, the key question is: Does the decimal terminate or repeat? A terminating decimal such as

1.6

equals

\frac{8}{5}

, so it is rational. A repeating decimal such as

0.\overline{142857}

also equals a fraction (

\frac{1}{7}

), so it is rational. A decimal that goes on forever without any repeating block — such as

3.14159265\ldots

(

\pi

) — is irrational.

Square roots need special care.

\sqrt{9} = 3

exactly, so

\sqrt{9}

is a natural number. But

\sqrt{7}

cannot be simplified to a fraction; its decimal is

2.6457513\ldots

with no repeating pattern, so it is irrational. A useful rule of thumb:

\sqrt{n}

is irrational whenever

n

is a positive integer that is not a perfect square.

x ∈

\mathbb{Q}

\iff x =

\frac{a}{b}

,\ a,b ∈

\mathbb{Z}

,\ b ≠ 0

Classify to the most specific set first, then list all sets that apply.
Terminating and repeating decimals $\Rightarrow$ rational.
Non-terminating, non-repeating decimals $\Rightarrow$ irrational.
$\sqrt{n}$ is rational only when $n$ is a perfect square (for positive integers $n$ ).

Why the Real Number Line Matters

One of the most powerful ideas in mathematics is that every real number corresponds to exactly one point on a continuous straight line, and every point on that line corresponds to exactly one real number. This is the real number line.

Rational numbers are dense — between any two rational numbers you can always find another rational number. For example, between

\frac{1}{2}

and

\frac{3}{4}

lies

\frac{5}{8}

. Yet despite being dense, the rationals still leave 'gaps' in the number line — the irrational points. Adding irrationals fills every gap and makes the line complete.

This completeness is not just a theoretical nicety. In geometry, when you calculate the exact side length of a square whose area is

5\ \text{cm}^2

, you get

\sqrt{5}

cm — an irrational number that is a perfectly real, measurable length. In science and technology, constants like

\pi

(the ratio of a circle's circumference to its diameter) are irrational but arise naturally from the physical world.

Understanding that the number system is layered — and that each layer was added to solve a specific kind of problem — helps you choose the right type of number for any situation. When counting people, you use

\mathbb{N}

. When recording a bank balance that might be negative, you need at least

\mathbb{Z}

. When measuring or working with geometry, you often need the full power of

\mathbb{R}

.

Circumference = \pi d, \pi \approx 3.14159\ldots ∈

\mathbb{R}

\setminus

\mathbb{Q}

Every real number is a unique point on the number line.
Rational numbers are dense but still leave gaps; irrationals fill those gaps.
Real-world measurements (lengths, areas, circumferences) may produce irrational results.
Choosing the right number set depends on the context of the problem.

Putting It All Together: A Classification Checklist

Here is a step-by-step strategy you can apply to any number you encounter. Work through the questions in order and stop at the first 'yes' to find the most specific set; then remember that every earlier set also applies.

Step 1 — Is it a positive whole number (no decimal, no fraction, not zero)? If yes: it is in

\mathbb{N}

, and therefore also in

\mathbb{W}

,

\mathbb{Z}

,

\mathbb{Q}

, and

\mathbb{R}

. Step 2 — Is it zero? If yes: it is in

\mathbb{W}

,

\mathbb{Z}

,

\mathbb{Q}

, and

\mathbb{R}

, but NOT in

\mathbb{N}

. Step 3 — Is it a negative whole number? If yes: it is in

\mathbb{Z}

,

\mathbb{Q}

, and

\mathbb{R}

, but NOT in

\mathbb{W}

or

\mathbb{N}

. Step 4 — Can it be written as

\frac{a}{b}

(including terminating/repeating decimals)? If yes: it is in

\mathbb{Q}

and

\mathbb{R}

only. Step 5 — Does its decimal never terminate or repeat (e.g.,

\sqrt{5}

,

\pi

)? If yes: it is in

\mathbb{R}

only (it is irrational).

Always write your classification as a full sentence: state the most specific set and list all sets that contain it. This shows your mathematical reasoning clearly.

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

Work from the most specific set ( $\mathbb{N}$ ) outward to $\mathbb{R}$ .
Every number belongs to $\mathbb{R}$ ; the interesting question is which smaller sets also apply.
A fraction whose numerator and denominator are both integers is rational — even if it looks complicated.
Write classifications as complete statements to demonstrate reasoning.

The Five Number Sets at a Glance

Set	Symbol	Members Include	Example Members	Not Included
Natural Numbers	$\mathbb{N}$	Positive counting numbers	CAD 1, 2, 100, 3045	$0$ , negatives, fractions
Whole Numbers	$\mathbb{W}$	Natural numbers + zero	CAD 0, 1, 2, 99	Negatives, fractions
Integers	$\mathbb{Z}$	Whole numbers + negatives	$-5, 0, 4, -200$	Non-integer fractions
Rational Numbers	$\mathbb{Q}$	All ratios $\frac{a}{b}$ , $b\neq 0$	$\frac{2}{3},\ -1.4,\ 0.\overline{3},\ 7$	Irrationals like $\sqrt{2}$
Real Numbers	$\mathbb{R}$	All rationals + irrationals	$\pi,\ \sqrt{5},\ -\frac{7}{2},\ 0$	Nothing on the number line

Worked example

Classifying a Set of Six Numbers

For each of the following numbers, state the most specific number set it belongs to, and list every set from the chain

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

that contains it. Justify each answer.

(a)

17

(b)

0

(c)

-6

(d)

\frac{3}{8}

(e)

0.\overline{6}

(f)

\sqrt{11}

Classify 17
$17$ is a positive whole number with no decimal or fractional part. Positive whole numbers are natural numbers by definition.
17 ∈ $\mathbb{N}$ \Rightarrow 17 ∈ $\mathbb{W}$ ,\ $\mathbb{Z}$ ,\ $\mathbb{Q}$ ,\ $\mathbb{R}$
Classify 0
$0$ is not positive, so it does not belong to $\mathbb{N}$ . It is, however, a whole number by definition — $\mathbb{W}$ is exactly $\mathbb{N}$ plus zero.
0 ∉ $\mathbb{N}$ ; 0 ∈ $\mathbb{W}$ \Rightarrow 0 ∈ $\mathbb{Z}$ ,\ $\mathbb{Q}$ ,\ $\mathbb{R}$
Classify −6
$-6$ is negative, so it cannot be a natural or whole number. It is a negative integer: no fraction, no decimal, just a whole negative value.
-6 ∉ $\mathbb{N}$ ,\ $\mathbb{W}$ ; -6 ∈ $\mathbb{Z}$ \Rightarrow -6 ∈ $\mathbb{Q}$ ,\ $\mathbb{R}$
Classify 3/8
$\frac{3}{8}$ is already written in the form $\frac{a}{b}$ with $a = 3$ , $b = 8$ , both integers, and $b \neq 0$ . As a decimal, $3 \div 8 = 0.375$ , which terminates. It is not an integer, so the most specific set is $\mathbb{Q}$ .
$\frac{3}{8}$ = 0.375 ∈ $\mathbb{Q}$ \Rightarrow $\frac{3}{8}$ ∈ $\mathbb{R}$ ; $\frac{3}{8}$ ∉ $\mathbb{Z}$ ,\ $\mathbb{W}$ ,\ $\mathbb{N}$
Classify 0.6̄ (0.666…)
$0.\overline{6}$ is a repeating decimal. To confirm it is rational, convert it: let $x = 0.\overline{6}$ , so $10x = 6.\overline{6}$ . Subtracting: $10x - x = 6$ , giving $9x = 6$ , so $x = \frac{6}{9} = \frac{2}{3}$ . Since it equals a ratio of integers, it is rational.
x = 0. $\overline{6}$ , 10x = 6. $\overline{6}$ , 9x = 6, x = $\frac{2}{3}$ ∈ $\mathbb{Q}$ ⊂ $\mathbb{R}$
Classify √11
$11$ is not a perfect square (since $3^2 = 9$ and $4^2 = 16$ , and no integer squares to $11$ ). Therefore $\sqrt{11}$ cannot be expressed as a ratio of integers. Its decimal expansion is $3.31662\ldots$ , which continues without any repeating block. It is irrational — real, but not rational.
$\sqrt{11}$ \approx 3.31662\ldots ∉ $\mathbb{Q}$ ; $\sqrt{11}$ ∈ $\mathbb{R}$ (irrational)

Answer: (a) 17 ∈

\mathbb{N}

,

\mathbb{W}

,

\mathbb{Z}

,

\mathbb{Q}

,

\mathbb{R}

. (b)

0 ∈

\mathbb{W}

,

\mathbb{Z}

,

\mathbb{Q}

,

\mathbb{R}

. (c)

-6 ∈

\mathbb{Z}

,

\mathbb{Q}

,

\mathbb{R}

. (d)

\frac{3}{8}

∈

\mathbb{Q}

,

\mathbb{R}

. (e)

0.

\overline{6}

=

\frac{2}{3}

∈

\mathbb{Q}

,

\mathbb{R}

. (f)

\sqrt{11}

∈

\mathbb{R}

only (irrational).

Check: Verify the subset chain: every number placed in

\mathbb{N}

must also appear in all larger sets. Every number placed only in

\mathbb{R}

must fail the fraction test — confirm

\sqrt{11}

cannot simplify by checking that

11

is not a perfect square:

3^2 = 9 \neq 11

and

4^2 = 16 \neq 11

. ✓ Verify

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

by dividing:

2 \div 3 = 0.6666\ldots

✓

Common mistakes and how to avoid them

Thinking that

0

is a natural number and writing 0 ∈

\mathbb{N}

.

Correction: In the Ontario MTH1W curriculum,

\mathbb{N}

starts at

1

. Zero is the element that distinguishes

\mathbb{W}

from

\mathbb{N}

. Always write 0 ∈

\mathbb{W}

, not 0 ∈

\mathbb{N}

.

Classifying

\sqrt{4}

as irrational because it contains a square-root symbol.

Correction: Always simplify first.

\sqrt{4} = 2

exactly, which is a natural number. The presence of a radical sign does not automatically make a number irrational — check whether the result is a perfect square.

Saying a repeating decimal such as

0.\overline{7}

is irrational because its decimal 'goes on forever'.

Correction: A decimal that repeats is rational. 'Going on forever' alone does not make a number irrational — the decimal must also have no repeating block.

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

, which is rational.

Listing only one set when classifying a number, e.g., writing only

\mathbb{Q}

for the number

-3

instead of

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

.

Correction: Because the sets are nested (

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

), a number belongs to its most specific set and every larger set. Always list all applicable sets.

Assuming every fraction is irrational because it is 'not a whole number'.

Correction: A fraction

\frac{a}{b}

with integer numerator and non-zero integer denominator is rational by definition. Irrational numbers cannot be expressed as such a fraction at all — not that they happen to be written differently.

Lesson summary

Number systems expanded historically from $\mathbb{N}$ to $\mathbb{W}$ to $\mathbb{Z}$ to $\mathbb{Q}$ to $\mathbb{R}$ , each step driven by a new mathematical need.
The sets are nested: $\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$ , so any number belonging to a smaller set automatically belongs to all larger sets.
A number is rational (∈ $\mathbb{Q}$ ) if and only if it can be written as $\frac{a}{b}$ with a, b ∈ $\mathbb{Z}$ and $b \neq 0$ ; terminating and repeating decimals always qualify.
A number is irrational if its decimal expansion is non-terminating and non-repeating; $\sqrt{n}$ is irrational whenever $n$ is a positive integer that is not a perfect square.
To classify any number, apply the checklist from $\mathbb{N}$ outward, stop at the most specific set, and list every set that contains it.
Choosing the appropriate number set matters in practice: context (counting, measuring, recording debt) determines which set is most suitable.

Check your understanding

Question 1

Which of the following is the most specific number set that contains

-9

?

$\mathbb{N}$ (Natural Numbers)
$\mathbb{W}$ (Whole Numbers)
$\mathbb{Z}$ (Integers)
$\mathbb{Q}$ (Rational Numbers)

Show answer and explanation

\mathbb{Z}

(Integers)

-9

is a negative whole number. It is not natural (not positive) and not whole (not

\geq 0

), but it is an integer — a negative member of

\mathbb{Z}

. It also belongs to

\mathbb{Q}

and

\mathbb{R}

, but

\mathbb{Z}

is the most specific (smallest) set.

Question 2

Which number in the list below is irrational?

$0.\overline{12}$
$\sqrt{36}$
$-\frac{5}{7}$
$\sqrt{13}$

Show answer and explanation

\sqrt{13}

0.\overline{12}

is a repeating decimal, so it is rational.

\sqrt{36} = 6

, a natural number.

-\frac{5}{7}

is already a ratio of integers, so it is rational.

\sqrt{13}

cannot be simplified because

13

is not a perfect square (

3^2=9

,

4^2=16

), so its decimal never terminates or repeats — it is irrational.

Question 3

A student says: 'All integers are rational numbers.' Is this statement true?

No — integers and rational numbers are completely separate sets.
No — only positive integers are rational.
Yes — every integer $n$ can be written as $\frac{n}{1}$ , satisfying the definition of rational.
Yes — but only because both sets contain zero.

Show answer and explanation

Yes — every integer

n

can be written as

\frac{n}{1}

, satisfying the definition of rational.

Every integer

n

can be expressed as

\frac{n}{1}

, which has an integer numerator, an integer denominator, and a non-zero denominator. This satisfies the definition

\frac{a}{b}

,\ b ≠ 0,\ a,b ∈

\mathbb{Z}

exactly. Therefore

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

, and the student's statement is true.

Question 4

Which of the following numbers belongs to

\mathbb{W}

but NOT to

\mathbb{N}

?

$1$
$0$
$-1$
$\frac{1}{2}$

Show answer and explanation

0

\mathbb{W}

contains all of

\mathbb{N}

plus zero. The only member of

\mathbb{W}

that is not in

\mathbb{N}

is

0

itself. 1 ∈

\mathbb{N}

;

-1 \notin \mathbb{W}

;

\frac{1}{2} \notin \mathbb{W}

.

Key terms

Natural Numbers ( $\mathbb{N}$ ): The set of positive counting numbers $\{1, 2, 3, 4, \ldots\}$ , used to count discrete objects.
Whole Numbers ( $\mathbb{W}$ ): The set $\{0, 1, 2, 3, \ldots\}$ ; natural numbers together with zero.
Integers ( $\mathbb{Z}$ ): The set $\{\ldots, -2, -1, 0, 1, 2, \ldots\}$ ; all whole numbers and their negatives.
Rational Number ( $\mathbb{Q}$ ): Any number expressible as $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$ . Includes all terminating and repeating decimals.
Irrational Number: A real number whose decimal expansion is non-terminating and non-repeating; it cannot be written as $\frac{a}{b}$ for any integers $a$ , $b$ . Examples: $\sqrt{2}$ , $\pi$ .
Real Numbers ( $\mathbb{R}$ ): The complete set of all rational and irrational numbers; every point on the number line corresponds to exactly one real number.
Subset: Set $A$ is a subset of set $B$ (written $A \subset B$ ) when every element of $A$ is also an element of $B$ .
Perfect Square: A non-negative integer that is the square of another integer (e.g., $0, 1, 4, 9, 16, 25, \ldots$ ). The square root of a perfect square is always an integer.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 9 Mathematics (MTH1W), expectation B1.1. 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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B1.1 · Explore the development and use of a number concept

Ontario Grade 9 Mathematics

Ontario MTH1W — Strand B: Number, Expectation B1.1

What you will learn

The Story of Number: A Historical Expansion

The Five Key Number Sets and Their Symbols

Classifying Numbers: Rules and Reasoning

Why the Real Number Line Matters

Putting It All Together: A Classification Checklist

The Five Number Sets at a Glance

Classifying 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