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Mathematics for science and technology
Mathematics for science and technology

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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:

multiline equation row 1 four plus two multiplication seven equals six multiplication seven row 2 equals 42

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

multiline equation row 1 four plus two multiplication seven equals four plus 14 row 2 equals 18

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

Timing: Allow about 10 minutes

Simplify:

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.

equation left hand side 12 times t plus 13 times t squared minus four minus six times t plus three times t squared equals right hand side sum with, 3 , summands 13 times t squared plus three times t squared plus 12 times t minus six times t minus four

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

multiline equation row 1 sum with, 3 , summands 13 times t squared plus three times t squared plus 12 times t minus six times t minus four equals 16 times t squared plus six times t minus four row 2

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

Timing: Allow about 10 minutes

Simplify the following:

  1. two times t squared plus four times t squared
  2. six times y minus eight times y squared plus four times y squared minus two times y
  3. four times c cubed plus two times c squared times d minus four minus c squared times d
  4. 12 plus two times r squared minus four times r times s minus three times r squared plus five times r times s
  5. sum with, 3 , summands a squared divided by two plus three times a squared plus a minus a squared plus a divided by four

Answer

  1.       equation left hand side two times t squared plus four times t squared equals right hand side six times t squared
  2.  
    multiline equation row 1 six times y minus eight times y squared plus four times y squared minus two times y equals sum with, 3 , summands negative eight times y squared plus four times y squared plus six times y minus two times y row 2 equals negative four times y squared plus four times y
  3.  
    multiline equation row 1 four times c cubed plus two times c squared times d minus four minus c squared times d equals four times c cubed plus two times c squared times d minus c squared times d minus four row 2 equals four times c cubed plus c squared times d minus four
  4.  
    multiline equation row 1 12 plus two times r squared minus four times r times s minus three times r squared plus five times r times s equals sum with, 3 , summands two times r squared minus three times r squared minus four times r times s plus five times r times s plus 12 row 2 equals sum with, 3 , summands negative r squared plus r times s plus 12
  5.  
    multiline equation row 1 sum with, 3 , summands a squared divided by two plus three times a squared plus a minus a squared plus a divided by four equals sum with, 4 , summands three times a squared minus a squared plus a squared divided by two plus a plus a divided by four row 2 equals five times a squared divided by two plus five times a divided by four

Note: equation left hand side five times a squared divided by two equals right hand side a squared and equation left hand side five times a divided by four equals right hand side a

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