# 2.1 Sequence of arithmetical calculations

You may be familiar with the acronym BIDMAS (or BODMAS or BEDMAS). This is a reminder that in any sum there is standard order in which to carry out the calculations.

That is:

Brackets

Indices

Division

Multiplication

Addition

Subtraction

To show how important these rules are look at this example.

4 + 2 × 7

Carrying this out from left to right, without considering BIDMAS gives:

And now using BIDMAS (multiplication must come before addition):

Two very different answers – showing why the accepted order of operation, BIDMAS, is so vital.

To clarify the order, or change the order, that operations are carried out in you can use brackets.

Looking at 4 + 2 × 7 again. This can be written as:

4 + (2 × 7)

So, now the intended order of operation is clear.

If the brackets have been included to show (4 + 2) × 7 this would change the order of operation.

Just as with arithmetic algebra has rules and conventions in the way that it operates. In other words, there are things that you have to do (rules) and things that mean that mathematics is presented consistently (conventions). Conventionally, when there are no brackets, multiplication and division are done before addition and subtraction; most calculators will do this automatically.

Other useful conventions are:

- writing 4
*n*instead of 4 ×*n*; 4*mn*is 4 ×*m*×*n* - writing 4
*n*rather than*n*4 - writing the product of multiplying
*x*by itself as*x*^{2}instead of*xx* - writing
*x*rather than 1*x*.

You will now look at some examples of simplifying expressions.

## Worked example 1

Simplify:

12*t* + 13*t*^{2} – 4 – 6*t* + 3*t*^{2}

There are three kinds of terms in this expression. Simple numbers like 4, terms like 12*t* and terms like 13*t*^{2}.

Two useful rules follow:

- when adding like terms, add the coefficients; for example, 2
*x*+ 5*x*= 7*x* - don’t mix powers of the same variable – you can’t add
*x*+ 2*x*^{2}to give a single term, as they represent different numbers; for example, when x = 2,*x*^{2}= 4.

By convention, the terms with the highest power go at the start of the expression.

So, start by collecting all the terms that relate to *t* and *t*^{2}together, and begin the statement with the terms that relate to *t*^{2}.

Now, add like terms together, and simplify as far as possible.

Looking at 16t^{2} + 6t – 4, you should be able to see that all the coefficients are divisible exactly by 2. So, the expression can be simplified further to:

16*t*^{2} + 6*t* – 4 = 2(8*t*^{2} + 3*t* – 2)

See how you get on with simplifying in this activity.

## Activity 1 Simplifying expressions

Simplify the following:

### Answer

Note: and

Before looking in more detail at the use of brackets in mathematics the next section will revise operations involving positive and negative numbers.