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.
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 4n instead of 4 × n; 4mn is 4 × m × n
- writing 4n rather than n4
- writing the product of multiplying x by itself as x2 instead of xx
- writing x rather than 1x.
You will now look at some examples of simplifying expressions.
Worked example 1
12t + 13t2 – 4 – 6t + 3t2
There are three kinds of terms in this expression. Simple numbers like 4, terms like 12t and terms like 13t2.
Two useful rules follow:
- when adding like terms, add the coefficients; for example, 2x + 5x = 7x
- don’t mix powers of the same variable – you can’t add x + 2x2 to give a single term, as they represent different numbers; for example, when x = 2, x2 = 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 t2together, and begin the statement with the terms that relate to t2.
Now, add like terms together, and simplify as far as possible.
Looking at 16t2 + 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:
16t2 + 6t – 4 = 2(8t2 + 3t – 2)
See how you get on with simplifying in this activity.
Activity 1 Simplifying expressions
Simplify the following:
Before looking in more detail at the use of brackets in mathematics the next section will revise operations involving positive and negative numbers.