- Real-world mathematics depends on being able to classify numbers and to move between different representations (fractions, decimals, roots, and symbols like $\pi$).
- In IB MYP Standard Mathematics, you will work mainly within the real number system, which includes familiar counting numbers as well as numbers like $\sqrt{2}$.
Sets Are The Language Used To Classify Numbers
To classify numbers precisely, we use set notation.
Set
A collection of objects. Each object in the set is called an element (or member) of the set.
Subset
A set $A$ is a subset of $B$ (written $A\subseteq B$) if every element of $A$ is also an element of $B$.
Complement
The set of elements not in a set. The complement of $A$ (relative to $U$) is $A'$.
When our "universal set" is the set of real numbers $\mathbb{R}$, the complement $\mathbb{Q}'$ means "all real numbers that are not rational".
- Set notation is designed to be compact and unambiguous.
- It lets you say things like "every natural number is rational" using a single statement: $\mathbb{N} \subseteq \mathbb{Q}$.
Natural Numbers, Integers, And Real Numbers Form A Nesting Structure
The number system is often shown as nested sets: each set sits inside a larger set.
Natural numbers (N)
The natural numbers $\mathbb{N}$ are the counting numbers. Depending on convention, $0$ may or may not be included.
Integers (Z)
The integers $\mathbb{Z}$ are the whole numbers and their negatives: $\{\dots,-3,-2,-1,0,1,2,3,\dots\}$.
Real numbers (R)
The real numbers $\mathbb{R}$ are all numbers that can be placed on the number line, including both rational and irrational numbers.
The most important "containment facts" are:
- $\mathbb{N} \subseteq \mathbb{Z}$ (counting numbers are integers)
- $\mathbb{Z} \subseteq \mathbb{Q}$ (integers are rational)
- $\mathbb{Q} \subseteq \mathbb{R}$ (rationals are real)
- Be careful with $0$ and $\mathbb{N}$.
- Some courses define $\mathbb{N}=\{1,2,3,\dots\}$ and others define $\mathbb{N}=\{0,1,2,3,\dots\}$.
- Always follow the convention used in your class or exam.
Rational Numbers Are Exactly The Numbers That Can Be Written As Fractions
A key idea in MYP is that rational numbers are precisely the numbers that can be expressed as a fraction.
Rational numbers (Q)
A rational number $q$ is any number that can be written as a fraction $q=\frac{a}{b}$ where $a,b\in\mathbb{Z}$ and $b\neq 0$.
This definition immediately explains why all integers are rational, because any integer $n$ can be written as $\frac{n}{1}$.
Two Common Ways To Recognize Rational Numbers
- Fraction form: If you can write the number as $\frac{a}{b}$ with integers $a$ and $b\neq 0$, it is rational.
- Decimal pattern: A rational number written as a decimal either terminates (ends) or repeats.
- $\frac{1}{5}=0.2$ (terminating, so rational)
- $\frac{1}{3}=0.333\dots$ (repeating, so rational)
- $-3=-3.0$ (terminating, so rational)
- If a decimal repeats, you can convert it to a fraction.
- For example, let $x=0.777\dots$.
- Then $10x=7.777\dots$, so $10x-x=7$, giving $9x=7$ and $x=\frac{7}{9}$.
Irrational Numbers Cannot Be Written As Fractions
Not all real numbers are rational.
Irrational numbers (Q')
An irrational number is a real number that cannot be written in the form $\frac{a}{b}$ with integers $a$ and $b\neq 0$. The set of irrationals is often written $\mathbb{Q}'$.
- Typical irrational numbers include $\pi$, $\sqrt{2}$, etc.
- Their decimal expansions do not terminate and do not repeat.
- Think of rational numbers as "numbers you can pin down with a repeating pattern" in decimal form.
- Irrational numbers are like a decimal that never settles into any repeating cycle, no matter how far you go.
Visualizing The Classification Of Real Numbers
A diagram helps you see how the sets fit together and where examples belong.
From the diagram (and the definitions):
- $35 \in \mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{R}$
- $0 \in \mathbb{Z}, \mathbb{Q}, \mathbb{R}$ (and sometimes $\mathbb{N}$)
- $-3 \in \mathbb{Z}, \mathbb{Q}, \mathbb{R}$ but $-3 \notin \mathbb{N}$
- $\frac{1}{5} \in \mathbb{Q}, \mathbb{R}$ but $\frac{1}{5} \notin \mathbb{Z}$
- $\pi,\ \sqrt{2} \in \mathbb{Q}'$ and also $\in \mathbb{R}$
Using Set-Builder Notation To Describe Number Sets
- Sometimes we define a set by describing a rule for its elements. This is called set-builder notation.
- For rational numbers, a set-builder description is: $$\mathbb{Q}=\left\{\left.\frac{a}{b}\ \right|\ a\in\mathbb{Z},\ b\in\mathbb{Z},\ b\neq 0\right\}$$
- Read this as: "the set of all fractions $\frac{a}{b}$ where $a$ and $b$ are integers and $b$ is not zero."
- Set-builder notation is powerful because it shows the defining property of the set.
- For $\mathbb{Q}$, the defining property is "can be written as a fraction of integers".
Interpreting Common Statements About Number Sets
- In classwork and exams you are often asked whether a statement is always true, sometimes true, or never true.
- The key is to unpack the definitions.
- $0 \in \mathbb{Z}$: always true (0 is an integer).
- $0 \in \mathbb{Q}$: always true because $0=\frac{0}{1}$.
- $\mathbb{N} \subseteq \mathbb{Q}$: always true because every natural number $n$ can be written as $\frac{n}{1}$.
- $\mathbb{Q} \subseteq \mathbb{N}$: never true because numbers like $\frac{1}{5}$ are rational but not natural.
- $\mathbb{Q} \subseteq \mathbb{Q}'$: never true because $\mathbb{Q}'$ is the complement of $\mathbb{Q}$ in $\mathbb{R}$, so they do not overlap.
- Do not confuse $\subseteq$ (subset) with $\in$ (element).
- For example, $3 \in \mathbb{Z}$ is true, but $3 \subseteq \mathbb{Z}$ makes no sense because 3 is not a set.
A Useful Complement Fact
- If the universal set is $\mathbb{R}$, then:
- $\mathbb{Q}' = \mathbb{R} \setminus \mathbb{Q}$ (all real numbers that are not rational)
- Any number in $\mathbb{Q}'$ must be irrational.
- So, a good "set for $\pi$ and $\sqrt{2}$" is:
- $\{\pi,\sqrt{2}\} \subset \mathbb{Q}'$.
Zero And Negative Numbers In The Development Of Number Systems
The way we classify numbers today was built over time.
- Zero appeared much later than counting numbers.
- Across history it has been used to represent "nothingness", to show absence, and as a placeholder in place-value systems, before being treated as a number in its own right.
- Negative numbers were used in ancient China for practical contexts such as commerce and taxation, where different colored counting rods represented positive and negative quantities that could cancel.
- This history helps explain why some conventions (like whether $0$ is in $\mathbb{N}$) can vary: number systems are human-made representations designed for usefulness.
Accuracy, Exactness, And Representation Matter In Real Applications
- A number can be represented in multiple ways, and the choice affects how we communicate precision.
- $\frac{1}{5}$ is an exact value.
- $0.2$ is also exact in this case.
- But many decimals are approximations, for example $\pi \approx 3.14$.
- When a question asks for an exact answer, leave it in a form like $\frac{a}{b}$ or $\sqrt{n}$ (for non-square $n$) rather than rounding.
- When a question asks for a rounded answer, state the degree of accuracy (for example, "to 2 decimal places").
- For each number $-3,\ \pi,\ \frac{1}{5},\ \sqrt{2},\ 0,\ 35$, decide whether it belongs to $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, and $\mathbb{R}$.
- Explain in one sentence why $\sqrt{2}\notin\mathbb{Q}$.
- Write one example of a number in $\mathbb{Q}\cap (\mathbb{Z})'$ (rational but not an integer).
