8 Order of operations
The order of operations when working with numbers is sometimes taught using the BODMAS, BEDMAS or BOMDAS.
These acrostics help us remember what to do an in what order.
Equally ordered operations
Acrostics like BEDMAS, BODMAS or BOMDAS are sometimes avoided by teachers today because they can raise confusion for equally ordered operations.
How should we approach a string of operations such as [latex]6÷3×2÷8×6[/latex]?
We should simply do them in the order they are presented (the ‘answer’ to the sum above is, then, 3). The same goes for addition and subtraction.
Let’s look through each of these operations. Download a copy of the Order of operations if it helps.
When we evaluate any expression, we should proceed in the following order:
Brackets
Brackets – evaluate anything in brackets (using the order of operations if there are complicated expressions within the brackets) and then it’s easiest to replace the bracketed part with the result when we rewrite the expression.
For example,
[latex]\begin{align} (34+2) & \times 2+1 \\ = 36 & \times 2+1 \end{align}[/latex]
Powers/indices
Even though powers can sometimes be thought of as repeated multiplication, we apply this operation before multiplying or adding. If some value is raised to a power which is an expression, the power should be evaluated first. So,
[latex]\begin{align} & \ \ \ \ \ 4 \times 3^{2+1} \\ &= 4 \times 3^{3} \\ &= 4 \times 27 \end{align}[/latex]
Multiplication/Division
Multiplications and divisions are also performed before addition and subtraction, completed in the order they are given.
So you should treat
[latex]3+4 \times 6-1 \div 4[/latex]
as
[latex]3+(4 \times 6)-(1 \div 4)[/latex]
which becomes
[latex]3+24-0.25[/latex]
Addition / Subtraction
Finally, we evaluate the addition and subtraction (in the order they are given). For example, suppose we wanted to evaluate
[latex]3 − (9 − 2) × 3^2+ 10 ÷ 5 × 2[/latex]
We would first evaluate the brackets
[latex]3 − 7 × 3^2 + 10 ÷ 5 × 2[/latex]
Next, the powers:
[latex]3 − 7 × 9 + 10 ÷ 5 × 2[/latex]
Then, multiplication/division
[latex]\begin{align} & 3-63+2 \times 2 \\ &= 3-63+4 \end{align}[/latex]
Finally, addition/subtraction
[latex]\begin{align} & -60+4 \\ &= -56 \end{align}[/latex]
Algebra
Be careful of implied brackets, especially when working with algebraic statements.
In the case of algebraic statements, we sometimes leave out some of the brackets, multiplication and division symbols. Instead of writing division symbols, division is usually indicated by writing the expressions using the vinculum (i.e. similar to fractions). Multiplication is assumed if there is a number or variable next to another variable or expression and sometimes brackets will be left out if we are performing divisions or using powers raised to an expression.
Here are some examples to illustrate what we mean
| The expression below | Would usually be written as |
| [latex]3 \times x \times y[/latex] | [latex]3xy[/latex] |
| [latex]4 × x ÷y + 1[/latex] | [latex]\frac{4x}{y}+1[/latex] |
| [latex](3 + x) ÷ (2 + x)[/latex] | [latex]\frac{3+x}{2+x}[/latex] |
| [latex]y\times \frac{3+a}{2-x}+4^{(x-1)}[/latex] | [latex]\frac{y(3+a)}{2-x}+4^{x-1}[/latex] |
These are just the conventions that are usually used. In some cases, we may decide to leave the brackets or symbols in if we are worried about the expressions being misinterpreted.
