Number#

A number is an abstract object that obeys precise rules of combination — addition, multiplication, and their inverses — and is used to count, order, or measure. Numbers are not physical things; they are positions within a structure. What makes something a number is not what it is but how it behaves under those operations.

The word "number" covers several overlapping systems, built up in stages. Each new system is constructed from the previous one to make some operation always work that did not work before. Almost every other idea in mathematics and the physical sciences is expressed through them.

What numbers do#

Numbers fulfill three distinct roles, often conflated:

• Cardinal — answering how many? Three apples.
• Ordinal — answering which position? The third runner.
• Measure — answering how much? A length of 3.14 metres.

A fourth common use — labels such as phone or jersey numbers — is not a mathematical use at all, since the operations of arithmetic do not apply.

Zero#

Zero is the number representing the absence of quantity. It is also the additive identity: $a + 0 = a$ for every number $a$.

Zero looks obvious only in retrospect. It was the last digit to be accepted in most numeral systems and the foundation of positional notation, which makes writing arbitrarily large numbers efficient. Without zero, the digits in $307$ cannot be distinguished from those in $37$. Its history as a placeholder (Babylonian, Mayan) is older than its history as a number (India, c. 7th century CE).

Whether the natural numbers start at $0$ or $1$ is a convention. This entry uses $0$.

Constructing the number systems#

The systems below are not given; each is constructed from the previous by a precise procedure. The clean progression is logical reconstruction, not history: negatives and complex numbers were both received with hostility for centuries before being formalised.

Natural numbers#

The natural numbers are the counting numbers:

Formally, $\mathbb{N}$ is characterised by the Peano axioms: there is a starting element $0$ and a successor function $S$ such that every natural is either $0$ or the successor of a unique predecessor, and induction holds. In set theory, one common implementation is $0 = \emptyset$ and $n + 1 = n \cup \{n\}$.

$\mathbb{N}$ is closed under addition and multiplication, but not under subtraction: $3 - 5$ has no answer in $\mathbb{N}$.

Integers#

The integers are

They are built from $\mathbb{N}$ as equivalence classes of pairs $(a, b)$, where the pair represents the difference $a - b$ and two pairs are identified when $a_1 + b_2 = a_2 + b_1$. Subtraction now always works.

Rational numbers#

The rational numbers $\mathbb{Q}$ are the ratios of integers, constructed as equivalence classes of pairs $(a, b)$ of integers with $b \neq 0$, where $(a, b) \sim (c, d)$ when $ad = bc$. Division by nonzero rationals is closed; the result is the field of rationals.

A rational number's decimal expansion always either terminates ($\tfrac{1}{4} = 0.25$) or eventually repeats ($\tfrac{1}{7} = 0.\overline{142857}$). The converse also holds.

Real numbers#

The rationals leave gaps. The diagonal of a unit square has length $\sqrt{2}$, which the Pythagoreans proved cannot be written as a ratio of integers. The real numbers $\mathbb{R}$ fill these gaps.

Two equivalent constructions are standard. A real number is either a Dedekind cut of the rationals — a partition of $\mathbb{Q}$ into a lower and upper set with no greatest element below — or an equivalence class of Cauchy sequences of rationals. Both yield the same structure: an ordered field with the completeness property, that every bounded set of reals has a least upper bound. Completeness is the property that distinguishes $\mathbb{R}$ from $\mathbb{Q}$ and underwrites all of calculus.

Reals whose decimal expansion neither terminates nor repeats are called irrational. The irrationals include not only $\sqrt{2}$ but the transcendentals such as $\pi$ and $e$, which are not roots of any polynomial with rational coefficients.

Complex numbers#

The polynomial $x^2 + 1 = 0$ has no real root. The complex numbers $\mathbb{C}$ extend $\mathbb{R}$ by introducing a single new element $i$ with $i^2 = -1$. Formally, a complex number is an ordered pair $(a, b)$ of reals with addition componentwise and multiplication

Writing $i = (0, 1)$ recovers the familiar $a + bi$. The fundamental theorem of algebra states that every non-constant polynomial with complex coefficients has a complex root; algebraically, $\mathbb{C}$ is the place where polynomial equations always have solutions.

Geometrically, $\mathbb{C}$ is the plane: addition is translation, and multiplication combines rotation with scaling.

The chain of inclusion#

Each system contains the previous one as a subset that preserves all arithmetic:

Each inclusion is conservative: facts true in the smaller system remain true in the larger one. A natural number is also an integer, rational, real, and complex, and arithmetic gives the same answer in every system.

Beyond the complex numbers#

The extension does not have to stop at $\mathbb{C}$. The Cayley–Dickson construction repeatedly doubles the dimension, producing the quaternions $\mathbb{H}$ (dim 4), the octonions $\mathbb{O}$ (dim 8), the sedenions (dim 16), and so on.

Each step costs an algebraic property:

• $\mathbb{R} \to \mathbb{C}$ loses order (no natural notion of "less than").
• $\mathbb{C} \to \mathbb{H}$ loses commutativity: in general $ab \neq ba$.
• $\mathbb{H} \to \mathbb{O}$ loses associativity: in general $(ab)c \neq a(bc)$.
• $\mathbb{O} \to$ sedenions loses the no-zero-divisors property: nonzero elements can multiply to zero, and the structure ceases to be useful.

Hurwitz's theorem makes this precise: $\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$ are the only finite-dimensional normed division algebras over the reals. The quaternions encode rotations in three dimensions and are used in computer graphics and aerospace. The octonions appear in certain symmetries of fundamental physics.

Significance#

The number systems are the universal alphabet of quantitative thought. Counting and ordering give arithmetic and combinatorics; the rationals suffice for everyday commerce and most engineering; the reals are the language of continuous quantity, on which calculus, classical physics, and analysis rest; the complex numbers govern oscillation, waves, and quantum mechanics, where amplitudes are inherently two-dimensional. Digital computation works almost entirely in integers and finite approximations of the reals.

Understanding number is the prerequisite for every entry that uses proof, measurement, or computation.