These are called parentheses or round brackets. They are most commonly used to show order of operations (Section 7.2: Notation). For example, in the expression $(1+2)\times 3$, the parentheses indicate that $1+2$ is calculated first, before the multiplication.

Parentheses go around ordered lists (or ‘tuples’), e.g. $(1,2,3)$. If there are two elements in the list, it’s called an ordered pair. Tuples can indicate coordinates (Section 11.1: Cartesian Coordinates) of points or vectors (Section 11.6: Vectors). The difference between a tuple and a set is that the order of a set doesn’t matter; $(1,2,3)$ is not the same as $(3,2,1)$, but $\{1,2,3\}$ is the same as $\{3,2,1\}$.

Parentheses surround the argument of a function (Section 8.2: Functions), e.g. $f(x)$. This is just another example of an ordered list, but with only one element; you can also find functions with multiple arguments, e.g. $f(x,y,z)$.

A pair of numbers within parentheses, e.g. $(0,1)$, can also indicate an open interval (Section 8.1.3: Intervals). Intervals can be closed at one end and open at the other, e.g. $(0,1]$ or $[0,1)$.

An array of numbers within parentheses is a matrix (Section 12.2: Matrices), e.g. \[\begin{pmatrix}1&2\\3&4\end{pmatrix}.\] If there is only one row or column, it may indicate a vector (Section 11.6: Vectors). If there are only two numbers in one column, e.g. $4\choose 2$, it could also be a combination (Section 13.5: Combinations).

A raised natural number in parentheses can indicate how many times some symbol is repeated, e.g. $f^{(4)}(x)$ is the same as $f’’’’(x)$ (Section 15.3.5: Higher Derivatives), and $40!^{(5)}$ is the same as $40!!!!!$

These are called square brackets. They have many of the same uses as round brackets – they are sometimes used to show order of operations (Section 7.2: Notation), tuples, vectors (Section 11.6: Vectors), and matrices (Section 12.2: Matrices). They can indicate closed intervals (Section 8.1.3: Intervals), e.g. $[0,1]$ means all real numbers from $0$ to $1$ inclusive.

They can indicate the floor of a number – see $[\![\clubsuit]\!]$ below. The brackets may be around the other way for the floor, e.g. $]2.1[$ instead of $[2.1]$.

In calculus, they are used when integrating (Section 15.4.2: Fundamental Theorem of Calculus).

In chemistry, they denote concentration.

Square brackets are also used to denote commutators, Lie (or Jacobi-Lie) brackets, and Iverson brackets, which are not covered in this book.

$[\![\clubsuit]\!]$ or $\lfloor\clubsuit\rfloor$ indicate the floor of a number, while $\lceil\clubsuit\rceil$ indicates the ceiling. The floor is the next integer down, and the ceiling is the next integer up, e.g. $\lfloor2.1\rfloor = 2$ and $\lceil2.1\rceil= 3$.

This usually indicates the absolute value of a number – its distance from $0$, e.g. $\left|-3\right| = 3$. Similarly, it can indicate the magnitude (length) of a Euclidean vector (Question 11.6.1) or complex number (Question 16.2.7). These are all examples of norms (see $\left\|\clubsuit\right\|$ below).

When surrounding an array of numbers, it indicates the determinant of a matrix (Section 12.4: Square Matrices).

It can also indicate the cardinality of a set (Section 8.1.5: Cardinality).

The norm of a vector in a vector space is a generalisation of the idea of size or length. It is not covered in this book.

These are called ‘braces’ or ‘curly brackets’. Like round brackets, they can be used to indicate the order of operations (Section 7.2: Notation).

They are used to describe sets (Section 8.1: Sets), e.g. $\{1,2,3\}$ and $\{x:x/2\in\mathbb{Z}\}$ are both sets.

Curly brackets can denote the fractional part of a number, e.g. $\{5.71\}$ is $0.71$.

A single opening curly bracket is used in piecewise function definitions (Section 8.2.2: Maps in General), e.g. \[\abs{x} = \left\{\begin{array}{ll} x,&\textrm{if }x\ge 0\\ -x,&\textrm{if }x\lt 0 \end{array}\right.\]

These are called ‘angle brackets’. Like round brackets (see $(\clubsuit)$ above) they can surround tuples.

They can indicate the average of a function over time.

Angle brackets are used to denote the inner product of two vectors, e.g. $\langle x,y \rangle$. Inner products are not covered in this book. Bra-ket notation (also known as Dirac notation), e.g. $\langle \phi | \psi \rangle$, is commonly used in quantum mechanics.