This section is arranged by the number of lines in the symbol, by which I mean the minimum number of straight lines which are required to draw it. e.g. both $+$ and $\lt $ are in the Two Lines section.

This is called the ‘prime’ symbol. ‘Prime’ means first, and the symbol often denotes the first division of some unit, or the first member of some series.

For example, $6’$ might mean six minutes (the first division of an hour), or six arcminutes (the first division of a degree), or six feet. When doubled, the prime symbol denotes the second division of a unit, e.g. $10’’$ might mean ten seconds (the second division of an hour), ten arcseconds (the second division of a degree), or ten inches.

The prime symbol is also commonly used after a variable to give a new variable related in some way to the original (see Section 7.1.3: Letters).

In calculus, it indicates a derivative, e.g. $f’$ is the first derivative of $f$, $f’’$ is the second derivative, and so on.

In probability, the prime symbol may indicate a complementary event, and in set theory it may indicate the complement of a set (Section 8.1: Sets).

In linear algebra, it can indicate the transpose of a matrix (Section 12.2: Matrices).

For two of these symbols surrounding an expression, e.g. $\left|x\right|$, see Section b.5: Brackets.

In set-builder notation (Section 8.1.4: Set-Builder Notation), this symbol means ‘such that’.

In probability, it means ‘given’, e.g. $P(A|B)$ means the probability of $A$ given $B$.

It can mean ‘divides’, e.g. $5|130$ means ‘five divides $130$’.

When followed by a subscript, it can mean ‘evaluated at’, e.g. $x^2|_{x=3}$ is $3^2$. The symbol is used in a similar way when integrating, e.g. $f(x)|_a^b$ means $f(b) - f(a)$ (Section 15.4.2: Fundamental Theorem of Calculus).

This is the symbol for subtraction (Chapter 3: Adding and Subtracting), e.g. $5-2$ means five minus two. It is also used to show negative numbers (Section 3.3: Negative Numbers), e.g. $-3$ means negative three.

In limits, it means ‘from the left’ (from the negative side), e.g. $\lim_{x\rightarrow 0^-} f(x)$ is the limit of $f(x)$ as $x$ approaches zero from the left. See Section 15.2: Limits.

When superscripted after a set (Section 8.1: Sets), it restricts the set to negative numbers, e.g. $\mathbb{R}^-$ means the set of negative real numbers.

Like the obelus, $\div$, the slash symbol usually means ‘divided by’.

In set theory, $\mathbb{Z}/n\mathbb{Z}$ or $\mathbb{Z}/n$ is the set of integers modulo $n$.

A slash over a relation negates it, e.g. $\ne$ means ‘not equal to’, and $\notin$ means ‘is not a member of’.

An underline is often used when bold font is not available, especially in handwriting. It can indicate a vector (Section 11.6: Vectors) or matrix (Section 12.2: Matrices).

The ‘vinculum’ or ‘overline’ was used to group expressions before mathematicians started using brackets, e.g. they used to write $\overline{2+3}\times5$ instead of $(2+3)\times5$. It is still used in this way for square roots, e.g. $\sqrt{2+3}$ means $\sqrt{}(2+3)$.

The vinculum can indicate a complex conjugate (Section 16.4: Complex Conjugates).

It can indicate logical negation (Section 9.1: Logic).

It can indicate the complement of a set (Section 8.1: Sets).

It can indicate a line segment (Section 10.2: Points and Lines), e.g. $\overline{AB}$ is the line segment from the point $A$ to the point $B$.

In statistics, it can indicate a mean (Section 6.2.1: Mean).

It goes over the repeating digits in a repeating decimal (Section 4.2.5: Fractions), e.g. $0.1\overline{23}$ means $0.1232323\ldots$.

$A\setminus B$ is the relative complement of $A$ in $B$, i.e. the set of things in $B$, but not in $A$. See Section 8.1: Sets.

Fractions are covered in Section 4.2.5: Fractions.

An ‘underscore’ can be used instead of a subscript, e.g. $a\_b$ means $a_b$. This is common when only ASCII characters are available on a computer.

The symbols $\Lambda$ and $\Gamma$ are covered in Section b.8: The Greek Alphabet.

This symbol can be read as ‘equals’ or ‘is equal to’. See Section 7.3: The Rules of Algebra for the properties of equality.

Sometimes the symbol is used in an asymmetric way, similar to the English word ‘is’, especially in O notation (Section 15.2.3: Big O Notation).

This is the symbol for addition (Chapter 3: Adding and Subtracting), usually read as ‘plus’.

It can be used to emphasise that a number is positive (Section 3.3: Negative Numbers), e.g. $+2$ is positive two.

In calculus, it can indicate that a limit is ‘from the right’ (from the positive side), e.g. $\lim_{x\rightarrow 0^+}f(x)$ is the limit of $f(x)$ as $x$ approaches $0$ from the right. See Section 15.2: Limits.

When superscripted after a set (Section 8.1: Sets), it restricts the set to positive numbers, e.g. $\mathbb{R}^+$ means the set of positive real numbers.

This is the symbol for multiplication (Chapter 4: Multiplying and Dividing), usually read as ‘times’.

It is also used for the Cartesian product of two sets (Section 8.1.6: Cartesian Product), and the cross product of two vectors (Section 11.6.2: Cross Product).

$\lt $ means ‘is less than’, and $\gt $ means ‘is greater than’. See Section 7.6: Solving Inequalities to find out how to work with them in algebra.

The ‘circumflex’ is used above a variable to give a new variable related to the original (similar to the use of the prime symbol). It is read as ‘hat’ in mathematics, e.g. $\hat x$ is ‘$x$ hat’.

It could indicate a unit vector (Section 11.6: Vectors), an estimate in statistics, or a Fourier transform.

This symbol means ‘angle’. In geometry, it is used to label angles, e.g. $\angle ABC$ means the angle at $B$ in triangle $ABC$ (see Section 10.3: Angles).

When followed by a point, vector, or complex number, it denotes the angle measured clockwise from the positive $x$-axis (Section 11.5: Polar Coordinates).

See $\clubsuit’$ in Section b.6.1: One Line.

This symbol can mean ‘is perpendicular to’ (Section 10.3: Angles) or ‘is independent of’ (Section 14.1: Classical Definition).

This symbol means ‘is parallel to’ (Section 10.3: Angles).

See also $\left\|\clubsuit\right\|$ in Section b.5: Brackets.

This symbol can mean ‘and’ in logic (Section 9.1: Logic), or the ‘wedge product’ or ‘exterior product’ of Euclidean vectors.

This symbol means ‘or’ in logic (Section 9.1: Logic).

This symbol means ‘not’ in logic (Section 9.1: Logic).

The symbols $\Delta$, $\Pi$, and $\Xi$ are covered in Section b.8: The Greek Alphabet.

Like the symbol $\Rightarrow$, this symbol can mean ‘implies’ (Section 9.1: Logic).

It is also used in calculus to mean ‘tends to’ (Section 15.2: Limits).

When written above a variable, e.g. $\vec v$, it is an alternative to bold font, indicating a vector (Section 11.6: Vectors).

This symbol, as the opposite of $\rightarrow$, means ‘is implied by’ or ‘if’ (Section 9.1: Logic).

These symbols mean ‘plus or minus’ and ‘minus or plus’, e.g. the solution to the equation $x^2 = 4$ is $x=\pm 2$, and if $x=\pm 2$ then $-x = \mp 2$. See Section 7.5.1: Actions.

These symbols mean ‘greater than or equal to’ and ‘less than or equal to’ (Section 7.6: Solving Inequalities).

This symbol means ‘is not equal to’.

This is the symbol for an equivalence relation (something similar to equality).

In particular, it can mean geometrical congruence (Section 10.6.5: Congruence and Similarity) or modular congruence (Section 7.9: Modular Arithmetic and Diophantine Equations), e.g. $43\equiv 3\pmod{10}$ means ‘$43$ is congruent to $3$ modulo $10$’ ($43$ and $3$ have the same remainder when divided by $10$).

It’s also used instead of $=$ in identities, i.e. equations that are always true, rather than just an equation that happens to be true for certain values. For example, $a^2-b^2\equiv (a-b)(a+b)$ is an identity, whereas $a-3=5$ is not.

This is the symbol for ‘for all’ (Section 9.1: Logic).

This symbol is sometimes drawn with three lines, but usually with four. See Section b.6.4: Four Lines.

The asterisk can be used instead of $\times$ when that symbol is not available, for example when only ASCII characters can be typed on a computer.

Sometimes two asterisks are used to indicate exponentiation (Chapter 5: Exponents and Logarithms), e.g. $2**3$ means $2^3$.

This symbol can indicate a complex conjugate, e.g. $z^*$ is the complex conjugate of $z$ (Section 16.4: Complex Conjugates).

The asterisk can indicate convolution (e.g. $f*g$), the multiplicative group of a ring (e.g. $\mathbb{C}^*$), or the dual space of a vector space (e.g. $V^*$), which are not covered in this book.

This upside-down version of delta is called ‘nabla’, and represents the del operator in vector calculus, which is not covered in this book. The Laplace operator can be represented as $\nabla^2$ or $\nabla\cdot\nabla$.

The symbol $\Sigma$ is covered in Section b.8: The Greek Alphabet.

This symbol means ‘maps to’, e.g. $f:x\mapsto x^2$ means the same as $f(x) = x^2$: that $f$ maps $x$ to $x^2$. See Section 8.2: Functions.

Like the symbol $\rightarrow$, this means ‘implies’ (Section 9.1: Logic).

As the reverse of $\implies$, this symbol means ‘is implied by’ or ‘if’ (Section 9.1: Logic).

These symbols mean ‘much less than’ and ‘much greater than’.

These symbols are used instead of $\le$ (less than or equal to) and $\ge$ (greater than or equal to) when those symbols aren’t available.

This symbol is known as the ‘number sign’, ‘hash’, or ‘pound sign’. It can denote set cardinality (Section 8.1.5: Cardinality).

This is the symbol for ‘square root’ (Section 5.1.2: Roots).

This symbol means ‘there exists’ (Section 9.1: Logic).

This symbol is used at the end of a proof to indicate that the proof is finished.

The letter $z$ is often written with a crossbar when mathematics is handwritten, so that $2$ and $z$ can’t be confused. See Section 7.1.2: Handwriting.

This symbol is a combination of $\leftarrow$ and $\rightarrow$ and means ‘if and only if’ (Section 9.1: Logic).

The asterisk may be drawn with three lines or five. See Section b.6.3: Three Lines.

This symbol means ‘is defined to be’.

This symbol is a combination of $\Leftarrow$ and $\Rightarrow$ and means ‘if and only if’ (Section 9.1: Logic).