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.
Definition
Set
A collection of objects. Each object in the set is called an element (or member) of the set.
Definition
Subset
A set $A$ is a subset of $B$ (written $A\subseteq B$) if every element of $A$ is also an element of $B$.
Definition
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".
Note
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.
Definition
Natural numbers (N)
The natural numbers $\mathbb{N}$ are the counting numbers. Depending on convention, $0$ may or may not be included.
Definition
Integers (Z)
The integers $\mathbb{Z}$ are the whole numbers and their negatives: $\{\dots,-3,-2,-1,0,1,2,3,\dots\}$.
Definition
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)
Common Mistake
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.
Definition
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.
Example
$\frac{1}{5}=0.2$ (terminating, so rational)
$\frac{1}{3}=0.333\dots$ (repeating, so rational)
$-3=-3.0$ (terminating, so rational)
Tip
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.
Definition
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.
Analogy
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.
$0 \in \mathbb{Z}, \mathbb{Q}, \mathbb{R}$ (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."
Note
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.
Example
$0 \in \mathbb{Z}$: 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.
Common Mistake
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.
Case study
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 $\pi \approx 3.14$.
Exam technique
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").
Active recall
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).