Juxtaposition (putting two symbols next to each other) is the usual way to express multiplication in algebra, e.g. $ab$ means $a\times b$ (see Section 7.2: Notation). This notation is also used to represent scalar multiplication of a vector (Section 11.6: Vectors), multiplication with matrices (Section 12.2: Matrices), and multiplication with complex numbers (Section 16.3: Multiplication).

Confusingly, proper fractions also use this notation, e.g. $3\frac{1}{2}$ means $3 + \frac{1}{2}$, not $3\times \frac{1}{2}$ (see Section 4.2.5: Fractions).

When a quantity has units, it is usually written as a number followed by some letters, e.g. $3.6\,\mathrm{km}$ means $3.6$ kilometres. See Appendix A: Units.

Subscripted variables are explained in Section 7.1.3: Subscripts.

Most commonly, this arrangement indicates exponentiation, e.g. $2^3$ means $2\times 2\times 2$. See Chapter 5: Exponents and Logarithms.

Lagrange’s notation in calculus uses $f’’$ or $f^{(2)}$ to mean the second derivative of $f$, for example. See Section 15.3.1: Instantaneous Rate of Change and Section 15.3.5: Higher Derivatives.

Inverse functions can use this notation, e.g. $f^{-1}$ can mean the inverse of the function $f$. See Section 8.2.4: Inverse Functions.

This arrangement can be used similarly to a subscript, as an ‘upper index’, especially in tensor mathematics, which is not covered in this book.

This is usually just a subscripted variable raised to some power, e.g. $x_1^2$ means $(x_1)^2$.

$\mathrm{C}^\clubsuit_\clubsuit$ and $\mathrm{P}^\clubsuit_\clubsuit$ are used to indicate a combination (Section 13.5: Combinations) or permutation (Section 13.4: Permutations).

Combinations (Section 13.5: Combinations) and permutations (Section 13.4: Permutations) can also be expressed like this.

Some symbols have numbers above and below them, usually indicating that some variable ranges from the number on the bottom to the number on the top. Look up the symbol in the middle for specific examples – $\sum$ and $\prod$ can be found in Section b.8: The Greek Alphabet, and $\int$ can be found in Section b.7: Curved Lines.