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.

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).

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 $\nabla\cdot\nabla$.

See Section b.5: Brackets

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).