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B1.1 · Explore the development and use of a number concept
Learn explore the development and use of a number concept through clear examples and targeted practice.
B 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.
Written lesson
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.
Written lesson
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 $a, b \in \mathbb{Z}$.
- Terminating and repeating decimals are always rational.
- Irrational numbers produce non-terminating, non-repeating decimals.
- $\mathbb{R}$ = rational numbers $\cup$ irrational numbers.
Written lesson
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.
$$x \in \mathbb{Q} \iff x = \frac{a}{b},\ a,b \in \mathbb{Z},\ b \neq 0$$
- 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 $n$).
Written lesson
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 $\mathbb{R}$.
$$\text{Circumference} = \pi d,\quad \pi \approx 3.14159\ldots \in \mathbb{R} \setminus \mathbb{Q}$$
- 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.
Written lesson
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.
Apply the idea
Worked examples
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) $\dfrac{3}{8}$ (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. — $$17 \in \mathbb{N} \Rightarrow 17 \in \mathbb{W},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R}$$
- 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. — $$0 \notin \mathbb{N};\quad 0 \in \mathbb{W} \Rightarrow 0 \in \mathbb{Z},\ \mathbb{Q},\ \mathbb{R}$$
- 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. — $$-6 \notin \mathbb{N},\ \mathbb{W};\quad -6 \in \mathbb{Z} \Rightarrow -6 \in \mathbb{Q},\ \mathbb{R}$$
- 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 $\mathbb{Q}$. — $$\frac{3}{8} = 0.375 \in \mathbb{Q} \Rightarrow \frac{3}{8} \in \mathbb{R};\quad \frac{3}{8} \notin \mathbb{Z},\ \mathbb{W},\ \mathbb{N}$$
- 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. — $$x = 0.\overline{6},\quad 10x = 6.\overline{6},\quad 9x = 6,\quad x = \frac{2}{3} \in \mathbb{Q} \subset \mathbb{R}$$
- 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. — $$\sqrt{11} \approx 3.31662\ldots \notin \mathbb{Q};\quad \sqrt{11} \in \mathbb{R} \text{ (irrational)}$$
Answer: (a) $17 \in \mathbb{N}, \mathbb{W}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}$. (b) $0 \in \mathbb{W}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}$. (c) $-6 \in \mathbb{Z}, \mathbb{Q}, \mathbb{R}$. (d) $\frac{3}{8} \in \mathbb{Q}, \mathbb{R}$. (e) $0.\overline{6} = \frac{2}{3} \in \mathbb{Q}, \mathbb{R}$. (f) $\sqrt{11} \in \mathbb{R}$ only (irrational).
Review
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 ($\in \mathbb{Q}$) if and only if it can be written as $\frac{a}{b}$ with $a, b \in \mathbb{Z}$ 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.