An **operation** is a mathematical process such as addition, subtraction, multiplication and division.

A **variable** (usually denoted by a lower case letter or a symbol) is a quantity which can take on different values. We use variables to express numbers which we don't know the value of (see equations). For example, in economics we typically denote profit by the Greek letter $\pi$ (pronounced “pie”) and price by $p$.

We can also add superscripts and subscripts to the letters denoting variables. For example, if we are interested in the annual profit made by a company over $10$ years, we might denote the profit made in the first year by $\pi_1$, that made in the second year by $\pi_2$, and so on.

See Variable for more information about variables.

An (algebraic) **expression** is a mathematical phrase which can contain numbers, operators, and variables. If a variable is multiplied by a number, this number is called a **coefficient**. A number on its own is called a **constant**.

For example $a+2b$, $a-2b+5$ and $\frac{a}{b}$ are all expressions containing variables $a$ and $b$. The $2$ in $a+2b$ is a coefficient and the $5$ in $a-2b+5$ is a constant.

An **equation** (sometimes referred to as a **formula**) is a mathematical statement which says that two expressions are equal. It consists of $2$ expressions, one on each side of an $=$ sign.

For example, the demand equation $Q = 10 – P$ says that the quantity demanded at a given price is equal to $10$ minus that price. Equivalently, we can say that the equation is true for all combinations of values for $P$ and $Q$ which sum to give $10$. This is easier to see if we rearrange the equation so that both of the variables are on the same side of the $=$ sign: \[10=Q+P\] For instance substituting $P=3$ and $Q=7$ into the equation clearly makes the left hand side equal to $10$.

**Solving** an equation means finding the value(s) of the variables in the equation which make the equation true (both sides equal).

An **identity** is a type of equation which that is true for *all* values of the variables involved so the $=$ sign becomes an $\equiv$ sign. Identities are used to simplify mathematical expressions and to prove that other identities are true. Since both sides of an identity are equal, we can always replace one side by the other in an expression.

For example, $a\times1=a$ is true for any value of the variable $a$ since multiplying any number by $1$ does not change the number. $a\times1=a$ is therefore an identity and we can write $a\times1\equiv a$.

A **function** is like a machine that takes a number (an **input**), changes this number (by adding another number to it, multiplying it by another number etc.) and gives back the changed number (an **output**). The output is always related to the input in some way so the same input will always give the same output. This output is *unique*, which means that each output can only be produced from a single input (there is a **one-to-one mapping** between the output and the input).

We typically denote the input (or **independent variable**) by **$x$** and the output corresponding to $x$ (or the **dependent variable**) by **$y$** or **$f(x)$**. $f(x)$ says that $f$ *is a function of* $x$. For example, $x$ might be a person's income and $f(x)$ might be the amount they spend on food each week.

**Note**: An equation can be called a function when the equation contains only two variables and for each value of one variable there can only be one possible value of the other.

**Note**: Although the output of a function is commonly denoted by $f(x)$, $f$ can also be replaced by other letters; for example we could use $g(x)$.

The function \[y=5x^2+1\] says that for any input $x$, we obtain the corresponding output by first squaring $x$, then multiplying by $5$, and finally adding $1$. If we chose $x=3$, then the output would be: \begin{align} y&=5\times(3^2)+1\\ &=5\times9+1\\ &=45+1\\ &=46 \end{align}

**Note**: If there is not a one-to-one mapping between the output and the input (more than one output can be produced from the same input) then we have a **relation**.