# The Order of Operations

Suppose you were asked to write this written statement into a numerical expression:

Subtract $3$ from $9$, double your answer, and then add $1$.

Answer: $13$

How can we translate this into numbers, while still maintaining the priority sequence revealed in the written statement?

Unlike in English, numerical expressions are not necessarily read from left to right. Therefore, it isn’t enough to simply write $9 – 3$ first in the numerical expression, followed by times 2 plus 1:

$$9 – 3 \times 2 +1$$

In fact, plugging this sequence of numbers into any scientific calculator would result in an output of $4$, rather than $13$: Using a scientific calculator to compute $9-3\times 2 +1$

What set of rules are being used by these calculators that lead to this conclusion? How can we manipulate our numerical expression so that the priority suggested from the written statement is maintained?

## I. Order of operations

To get the correct answer, conventions have been agreed upon. When $9-3\times 2 +1$ was inputted into the calculator, the calculator’s software began by multiplying $-3\times 2$ first, which resulted into $6$. This lead to the expression $9-6+1$, at which time the calculator computes the rest from left to right.

This suggests that when calculating numerical expressions, multiplication takes priority over addition and subtraction; the same can be said about division. A numerical expression containing only addition and subtraction (or only multiplication and division), is computed from left to right.

## II. Calculating with parentheses

If there are parentheses, start by doing the calculations inside the parentheses. Complete the calculations, following the priorities defined in section I.

Going back to our original statement, we were expected to subtract $3$ from $9$ first. To show this using numerically, we embed $9-3$ in (parentheses) or [brackets]:

$$(9-3) \times 2 + 1$$ $$6 \times 2 + 1 = \boxed{13}$$

This tells the reader to start calculating $9-3$ before anything else. Notice how we didn’t rewrite the parentheses after evaluating $9-3$. This is because what now remains within the parentheses (the $6$) cannot be simplified any further, hence the parentheses are no longer useful.

#### Note

Consider keeping the parentheses after evaluating if $(1)$ there happens to be a factor outside the parentheses or $(2)$ there exists an exponent. For example:

$$(1)\;\;\color{red}2\color{black}(5-1) \Rightarrow \color{red}2\color{black}(4)$$

$$(2)\;\;(5-1)\color{red}^3 \color{black} \Rightarrow (4)\color{red}^3$$

This will ensure calculations are performed correctly and consistently.

This convention also gives the freedom to write an expression without having to maintain the order in which it was written. For example, the variation below would also give $13$:

$$1 + 2\times(9-3) = \boxed{13}$$

To ensure that arithmetic calculations are performed consistently, this handy acronym summarizes the order of operations:

### Example 1.

Use the order of operations to calculate $C=9-(3\times2+1)$.

Start by calculating $3 \times 2 + 1$, which is in parentheses. To do this, first work out the multiplication, which takes priority:

$$C= 9 – (6 + 1)$$

Then complete the calculation inside the parentheses: $C = 9 – 7 = \boxed{2}$.

### Example 2.

Calculate $G=3+\frac{6+4}{7-2}$.

This expression can be written in the following form:

$$G=3+(6+4) \div (7-2)$$

Notice how the numbers found in numerator and denominator were written within parentheses. Any fraction whose numerator and/or denominator contain more than a single term must be written within parentheses.

Therefore, start by working out $6 + 4 = 10$ and $7 – 2 = 5$:

$$G = 3 + 10 \div 5$$

Then do the division ($10 \div 5 = 2$), which takes priority over the addition:

$$G = 3 + 2 = \boxed{5}$$

### Example 3.

Write two different numerical expressions that represent the written statement “Add $2$ to $1$, triple your answer, then subtract it from $9$.”

• $9-(2+1)\times 3$
• $-3(2+1)+9$