Kinds of Numbers#
Nested sets from counting to the complex plane, plus bases and computer formats#
Natural → integer → rational → real → complex is the main ladder. Each step adds numbers the previous step could not hold.
| Kind | Symbol (usual) | One-line idea | Example |
|---|---|---|---|
| Natural | $\mathbb{N}$ | Counting sizes | $1, 2, 3, \ldots$ |
| Whole | — | Natural plus zero | $0, 1, 2, \ldots$ |
| Integer | $\mathbb{Z}$ | Whole numbers and negatives | $-3, 0, 7$ |
| Rational | $\mathbb{Q}$ | Ratio of two integers | $\tfrac{3}{4}$, $-2$, $0.125$ |
| Irrational | (inside $\mathbb{R}$) | Real, not any exact fraction | $\sqrt{2}$, $\pi$ |
| Real | $\mathbb{R}$ | Points on the number line | $-1.5$, $e$, $\sqrt{5}$ |
| Imaginary | — | Real multiple of $i$ | $3i$, $-i$ |
| Complex | $\mathbb{C}$ | Real part plus imaginary part | $2 + 3i$ |
Two phrases sound alike but mean different things:
- Kinds of numbers — which set a value belongs to ($\mathbb{Q}$, $\mathbb{R}$, …).
- Positional number system — how you write a value in a base (decimal, binary, hex).
This article maps the sets first, then points at bases and floating-point formats data work uses every day.
For symbols and operators used with these numbers, see Grammar of Mathematics: Operators, Functions, Relations, and Symbols.
1. The nesting picture#
Think of Russian dolls: each larger set keeps everything inside the smaller one and adds more.
ℂ complex numbers
╱ ╲
╱ ╲ (real + imaginary parts)
╱ ╲
ℝ real numbers ········· number line
╱ ╲
╱ ╲ irrationals live here too (not in ℚ)
╱ ╲
ℚ rational numbers
╱ ╲
╱ ╲
ℤ integers
╱ ╲
╱ ╲
ℕ natural numbers (counting)
Inclusion chain (most common teaching order):
\[ \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} \]Some authors put $0$ inside $\mathbb{N}$; others use $\mathbb{N}={1,2,3,\ldots}$ and call ${0,1,2,\ldots}$ the whole numbers. The math is the same; only the label on the smallest set changes.
%%{init: {"flowchart": {"rankSpacing": 18, "wrappingWidth": 480}}}%%
flowchart TB
N["Natural N
count 1 2 3"]
Z["Integer Z
add 0 and negatives"]
Q["Rational Q
fractions a/b"]
R["Real R
full number line"]
C["Complex C
a + bi"]
N --> Z
Z --> Q
Q --> R
R --> C
The arrows mean “extend to include more values,” not “every number is bigger than the last.” $-1$ is an integer but not a natural number.
2. Core sets — one or two lines each#
Natural numbers ($\mathbb{N}$)#
Used to count discrete objects: $1, 2, 3, \ldots$.
Order matters; between two consecutive naturals there is no other natural.
Whole numbers#
Natural numbers together with zero $0$.
Zero means “none” in counting contexts.
Integers ($\mathbb{Z}$)#
Whole numbers, their negatives, and zero: $\ldots, -2, -1, 0, 1, 2, \ldots$.
Subtraction always stays inside $\mathbb{Z}$ (unlike naturals alone).
Rational numbers ($\mathbb{Q}$)#
Any number equal to $\tfrac{p}{q}$ with integers $p$, $q$, and $q \neq 0$.
Includes integers (take $q=1$) and terminating or repeating decimals.
Irrational numbers#
Real numbers that are not rational — no exact fraction form.
Examples: $\sqrt{2}$, $\pi$, $e$; decimal expansion never settles into a repeat block.
Real numbers ($\mathbb{R}$)#
All points on the continuous number line: rationals and irrationals together.
Every decimal expansion (finite or infinite) describes a real number.
Imaginary numbers#
Numbers of the form $bi$ where $b$ is real and $i^2 = -1$.
They measure “vertical” size on the complex plane, not on the real line.
Complex numbers ($\mathbb{C}$)#
Sums $a + bi$ with real $a$ and real $b$.
The plane with horizontal real axis and vertical imaginary axis models $\mathbb{C}$.
3. Important subsets (still one glance each)#
| Name | Lives in | One-line idea | Example |
|---|---|---|---|
| Prime | $\mathbb{N}$ (usually $>1$) | Only divisors are $1$ and itself | $2, 3, 5, 7, 11$ |
| Composite | $\mathbb{N}$ | Natural greater than $1$, not prime | $4, 6, 15$ |
| Even / odd | $\mathbb{Z}$ | Divisible by $2$ or not | $-4$ even; $7$ odd |
| Perfect square | $\mathbb{Z}_{\ge 0}$ | $n = k^2$ for some integer $k$ | $0, 1, 9, 16$ |
| Algebraic | $\mathbb{C}$ | Root of a polynomial with integer coeffs | $\sqrt{2}$, $i$, $\tfrac{1}{2}$ |
| Transcendental | $\mathbb{R}$ | Real, not algebraic | $\pi$, $e$ |
| Unit complex | $\mathbb{C}$ | $ | z |
Algebraic vs transcendental: almost every real number is transcendental, yet daily formulas mostly show algebraic ones ($\sqrt{2}$, cube roots) and a few famous transcendentals ($\pi$, $e$).
4. Where familiar symbols sit#
−3 −1/2 0 1 3/2 √2 π 2+3i
| | | | | | | |
Z Q Z N Q R R C
- Negative counts and debts → need $\mathbb{Z}$.
- Shares and ratios → $\mathbb{Q}$.
- Lengths of diagonals (often) → irrationals in $\mathbb{R}$.
- Phase and oscillation in physics and signal processing → $\mathbb{C}$.
5. Positional number systems (writing, not kind)#
Base (radix) tells how place value scales: decimal is base $10$, binary base $2$, hexadecimal base $16$.
| Base | Uses digits | Same value written |
|---|---|---|
| 10 | $0$–$9$ | $13$ |
| 2 | $0$, $1$ | $1101_2 = 13$ |
| 16 | $0$–$9$, A–F | $\mathrm{D}_{16} = 13$ |
The kind of number (rational, integer, …) does not change when you rewrite it in another base — only the symbols on the page change.
6. How computers store numbers (data science angle)#
| Format | Idea | Watch out |
|---|---|---|
Integer (int32, int64) | Fixed-width whole numbers | Overflow wraps or errors at the max |
Floating-point (float32, float64) | Sign, exponent, mantissa in binary | Not every decimal fraction is exact; compare with care |
| Boolean | One bit of truth | Not a subset of $\mathbb{R}$; logic, not measurement |
Machine floats are a finite set of rational-like values. They behave like reals in many formulas but fail exact equality tests such as 0.1 + 0.2 == 0.3 in common languages.
7. Beyond the complex plane (optional extensions)#
| System | One line |
|---|---|
| Quaternions | Extension with three imaginary units; good for 3D rotations, not commutative. |
| Octonions | Further extension; used in advanced physics, even less familiar rules. |
| Cardinal numbers | Measure size of infinite sets (how many), not position on a line. |
| Ordinal numbers | Measure order type of well-ordered lists, including infinite ordinals. |
You do not need these for first-year linear algebra or introductory quantum computing — but they show the ladder does not stop at $\mathbb{C}$.
8. Quick self-check#
- Is $-7$ natural? No — it is integer, not natural (in the usual $1,2,3,\ldots$ convention).
- Is $\tfrac{22}{7}$ rational? Yes — explicit ratio of integers (it is only an approximation to $\pi$).
- Is $\pi$ rational? No — it is irrational (in fact transcendental).
- Is $3i$ real? No — imaginary; $0 + 3i$ is complex with zero real part.
- Does binary change whether a value is rational? No — base is notation.
Related reading#
- Grammar of Mathematics: Operators, Functions, Relations, and Symbols — notation on top of these sets
- Anatomy of a Circle — where $\pi$ and radians live geometrically
- From Eigenvalues to Singular Value Decomposition — real and complex numbers inside matrices
- Start Here — Data Science & AI on dasarpAI — hub for learning paths
Hashtags#
#NumberSystems #NaturalNumbers #Integers #RationalNumbers #RealNumbers #ComplexNumbers #IrrationalNumbers #MathematicsForML #DataScienceFoundations #MathEducation
Comments: