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B1.2 · Describe and compare subsets of a number system
Learn describe and compare subsets of a number system through clear examples and targeted practice.
B 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.
Written lesson
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 $\pi \approx 3.14159265\ldots$.
$$\text{Rational: } \frac{a}{b}, \text{ where } a, b \in \mathbb{Z} \text{ and } b \neq 0$$
- $\mathbb{N} = \{1, 2, 3, \ldots\}$ — 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.
Written lesson
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.
Written lesson
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: $\sqrt{49} = 7 \in \mathbb{Z}$. 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.
$$0.\overline{d_1 d_2 \ldots} \in \mathbb{Q} \quad \text{and} \quad \sqrt{p} \notin \mathbb{Q} \text{ when } p \text{ is not a perfect square}$$
- 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.
Written lesson
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 $\mathbb{R}$.
$$\mathbb{Q} \cup \{\text{irrationals}\} = \mathbb{R} \quad \text{and} \quad \mathbb{Q} \cap \{\text{irrationals}\} = \emptyset$$
- $\mathbb{W}$ 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 $\mathbb{R}$.
Apply the idea
Worked examples
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. — $$6 \in \mathbb{N},\ \mathbb{W},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R}$$
- 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. — $$0 \in \mathbb{W},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R} \quad (0 \notin \mathbb{N})$$
- 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. — $$-9 \in \mathbb{Z},\ \mathbb{Q},\ \mathbb{R} \quad (-9 \notin \mathbb{N},\ \mathbb{W})$$
- 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. — $$-\frac{9}{4} \in \mathbb{Q},\ \mathbb{R} \quad \left(-\frac{9}{4} \notin \mathbb{N},\ \mathbb{W},\ \mathbb{Z}\right)$$
- 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. — $$\sqrt{11} \in \mathbb{R} \text{ (irrational)} \quad (\sqrt{11} \notin \mathbb{Q})$$
- 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. — $$0.\overline{45} = \frac{5}{11} \in \mathbb{Q},\ \mathbb{R} \quad (0.\overline{45} \notin \mathbb{N},\ \mathbb{W},\ \mathbb{Z})$$
- 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. — $$-9 \in \{\mathbb{Z},\mathbb{Q},\mathbb{R}\} \quad \text{vs.} \quad -\frac{9}{4} \in \{\mathbb{Q},\mathbb{R}\}$$
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). $\quad$ $0.\overline{45}$: $\mathbb{Q}, \mathbb{R}$.
Review
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.