Notation - Maths: an open, verified university mathematics reference

Notation is the same across the whole site: every subject uses the same symbol for the same thing. Where conventions differ between countries, the alternative is mentioned once on this page and never mixed into the content.

Shorthand used throughout¶

These commands are defined once for the whole site. If you write for the site, use them wherever they apply, so that the notation stays consistent.

Meaning	Typed as	Renders as
real numbers	`\R`	$\R$
natural numbers $\{0, 1, 2, \dots\}$	`\N`	$\N$
integers	`\Z`	$\Z$
rational numbers	`\Q`	$\Q$
complex numbers	`\C`	$\C$
the differential in an integral	`\int_0^1 x^2 \dd x`	$\int_0^1 x^2 \dd x$
a derivative in Leibniz notation	`\dv{y}{x}`	$\dv{y}{x}$
a higher derivative	`\dvn{2}{y}{x}`	$\dvn{2}{y}{x}$
a partial derivative (multivariable subjects)	`\pdv{f}{x}`	$\pdv{f}{x}$
absolute value	`\abs{x - a}`	$\abs{x - a}$
norm of a vector	`\norm{\vb{v}}`	$\norm{\vb{v}}$
a vector	`\vb{v}`	$\vb{v}$
epsilon (as in ε–δ)	`\eps`	$\eps$
sign function	`\sgn x`	$\sgn x$
inverse hyperbolic sine	`\arsinh x`	$\arsinh x$
domain and range of a function	`\dom f`, `\ran f`	$\dom f$ , $\ran f$

The same commands in displayed formulas, where their sizes adapt to what they contain:

\dv{}{x}\bigl(x^3\bigr) = 3x^2, \qquad \dvn{2}{}{x}\bigl(\sin x\bigr) = -\sin x, \qquad \pdv{}{x}\bigl(x^2 y\bigr) = 2xy,

(1)

\abs{\frac{x^2 - 4}{x - 2} - 4} = \abs{x - 2} < \eps \quad\text{whenever}\quad 0 < \abs{x - 2} < \eps, \qquad \norm{\vb{u} + \vb{v}} \le \norm{\vb{u}} + \norm{\vb{v}},

(2)

\int_1^{e} \frac{1}{x} \dd x = 1, \qquad \sgn x = \frac{x}{\abs{x}} \text{ for } x \ne 0, \qquad \arsinh x = \ln\Bigl(x + \sqrt{x^2 + 1}\Bigr),

(3)

\N \subset \Z \subset \Q \subset \R \subset \C, \qquad f\colon \dom f \to \R, \qquad \ran f = \{f(x) : x \in \dom f\}.

(4)

Notation table¶

Topic	We write	We do not write	Notes
Natural numbers	$\N = \{0, 1, 2, \dots\}$ ; positive integers $\Z_{>0}$ or “ $n \ge 1$ ”	$\N$ meaning $\{1, 2, \dots\}$	This follows ISO 80000-2. In statements we prefer an explicit range (“for $n \ge 1$ ”).
Intervals	$[a, b]$ , $(a, b)$ , $[a, b)$ , $(a, \infty)$	$]a, b[$ , $]a, b]$	See the note on other countries below.
Natural logarithm	$\ln x$	$\log x$ for the natural logarithm
Other logarithms	$\log_{10} x$ , $\log_2 x$ , $\log_b x$	a bare $\log x$	The base is always written.
Exponential	$e^{x}$ , or $\exp(x)$ for large exponents	$\mathrm{e}^x$	An italic $e$ , as in the major calculus textbooks.
Inverse trigonometric functions	$\arcsin x$ , $\arccos x$ , $\arctan x$	$\sin^{-1} x$	The page on inverse trigonometric functions mentions $\sin^{-1} x$ as an alternative and explains why it can be confused with $1/\sin x$ .
Powers of trigonometric functions	$\sin^2 x = (\sin x)^2$	$\sin x^2$ when $(\sin x)^2$ is meant
Reciprocal trigonometric functions	$\sec x$ , $\csc x$ , $\cot x$		Defined once, in the preliminaries.
Angles	radians	degrees in calculus statements
Derivatives	$f'(x)$ , $\dv{y}{x}$ , $\dv{}{x}\bigl(f(x)\bigr)$ ; higher derivatives $f''$ , $f^{(n)}$	$\dot y$	$\dot y$ is reserved for time derivatives and is introduced in the chapter on differential equations. Leibniz notation is used for the chain rule and related rates, prime notation for rules about functions.
Differentials	$\dd x$ in integrals	an italic $dx$	A thin space and an upright d.
Integrals	$\int_a^b f(x) \dd x$ ; an antiderivative $\int f(x) \dd x = F(x) + C$	leaving out $+ C$	Each page says once that $C \in \R$ is an arbitrary constant.
Limits	$\lim_{x \to a} f(x)$ ; one-sided $x \to a^{+}$ , $x \to a^{-}$ ; $\lim_{n \to \infty} a_n$	$\lim_{x \to a+}$
Infinite limits	“ $\lim_{x \to a} f(x) = \infty$ ”, with the remark that the limit does not exist as a real number
Functions	$f\colon A \to B$ , $x \mapsto x^2$ ; “the function $f$ ” and “the value $f(x)$ ”	“the function $f(x)$ ” in definitions	Allowed informally in examples.
Composition and inverse	$f \circ g$ ; $f^{-1}$ for the inverse function; $1/f$ or $f(x)^{-1}$ for the reciprocal
Monotone functions	increasing: $x_1 < x_2$ implies $f(x_1) \le f(x_2)$ ; strictly increasing: $x_1 < x_2$ implies $f(x_1) < f(x_2)$ ; likewise decreasing ( $\ge$ ) and strictly decreasing ( $>$ )	“non-decreasing” for increasing, or “increasing” when strictly increasing is meant	“Increasing” allows the function to stay level; a result that needs $<$ says “strictly”. Defined in Functions and Their Graphs.
Definitions	$:=$ when defining in a display; “is called” in prose	$\equiv$
Approximation	$\approx$ , with the precision stated (“to 4 decimal places”)
Sequences	$(a_n)_{n \ge 1}$ or $(a_n)$ ; terms $a_n$	$\{a_n\}$	Set braces would suggest a set, which forgets order and repetition.
Series	$\sum_{k=1}^{\infty} a_k$ ; partial sums $s_n = \sum_{k=1}^{n} a_k$		The summation index is $k$ , the sequence index $n$ .
Vectors (later subjects)	$\vb{v}$ (bold upright), components $(v_1, \dots, v_n)$ ; column vectors in linear algebra	$\vec{v}$	Fixed now so that linear algebra and multivariable calculus agree.
Absolute value and norm	$\abs{x}$ , $\norm{\vb{v}}$	plain vertical bars around tall expressions in displays	Plain bars do not grow with their contents; they are fine inline.
Logic	“if and only if” and “implies” in prose; $\implies$ , $\iff$ in displays	“iff”
Quantifiers	words in the core text (“for every $\eps > 0$ there exists $\delta > 0$ ”); the symbols $\forall$ , $\exists$ only in the rigorous track
Decimal mark	a point: 3.14	a comma
Set-builder notation	$\{x \in \R : x > 0\}$	$\{x \mid x > 0\}$

Intervals in other countries¶

In some countries, for example France and Sweden, an open interval is written with reversed brackets: $]a, b[$ for what we write as $(a, b)$ , and $]a, b]$ for $(a, b]$ . This site always uses round brackets, which can be confused with an ordered pair $(a, b)$ ; the context makes clear which is meant.

For contributors¶

Answers to exercises¶

Exercise answers are checked automatically against the computer algebra system, so the final answer of an exercise is written in this subset of LaTeX:

numbers, \frac{}{}, \sqrt{}, \sqrt[n]{}, powers ^{}, \pi, e, \infty, -\infty;
\sin, \cos, \tan, \arcsin, \arccos, \arctan, \ln, \log_{b}, and \abs{};
+C for an antiderivative (the checker differentiates the answer instead of comparing it);
several answers as a comma-separated list; intervals as (a, b) or [a, b] when the exercise asks for a set (otherwise (a, b) is a point);
not \mathrm{e} and not \dfrac.

Answers that are not expressions (proofs, sketches, “does not exist”) are marked as manual and are checked by a person.

Writing style¶

Spelling: British English (normalise, behaviour), one spelling throughout.
Voice: “we” for reasoning that we do together (“we now show”), “you” for instructions (“try dragging $\eps$ ”). Present tense.
Sentences: short, with one idea per sentence in definitions and statements of results.
Emphasis: bold only for a term being defined; italics for emphasis, sparingly.
Headings: sentence case (“Common mistakes”); page titles in title case.
Displayed formulas are punctuated as part of the sentence.
No forward references in proofs. A “looking ahead” note may mention later topics.
No “clearly” or “obviously”. If a step is easy, a short reason costs nothing.
Units: applied examples state their units, and the final answer has units.
Accessibility: every figure has a text alternative, and every interactive figure has a caption describing what it shows. Colour is never the only carrier of meaning.
Translation: no wordplay or idioms in the core text, and no text inside images.