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Skills for Learning

Maths & Stats - Basics

Learning Outcomes

If you work through this page you should be able to:

  • Understand the basic principles of the four main mathematical operations.
  • Know the sequence in which operations should be tackled in calculations.
  • Understand the use of 'powers' and how to work with numbers 'raised to a power'.
  • Be able to apply this knowledge of operations and their priority order to both positive and negative numbers.

There are four basic arithmetic operations:

Note: means equals, or equal to

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Some arithmetic calculations can be complicated and involve more than one operation. Before these can be performed we need to know in which order to apply operations.

For example, for the problem 6 + 5 × 2, there are two different results depending on which operation is performed first.

22 is the result if the addition is performed first.

6 + 5 = 11, 11 × 2 = 22

16 is the result if the multiplication is performed first.

5 × 2 = 10, 6 + 10 = 16

Therefore, we need conventions and rules to indicate in which order things should be done.

The convention indicating priority for arithmetic operations is:

  1. Brackets
  2. Exponents or powers (raising to a power)
  3. Division and Multiplication
  4. Addition and Subtraction

To remember this you can use the acronym BEDMAS

So the answer to 6 + 5 × 2 is 16 because multiplication comes before addition.

However, if it had been written (6 + 5) × 2 the answer would be 22 because operations in brackets should always be completed first. Therefore, brackets can be used to indicate that you want a certain operation, within a more complex calculation, to be carried out first.

An exponent or a power is the small superscripted number that you see on the top right of a number - for example, 25.

In this case, we would say that "two has been raised to the power of five" and this means that we want five twos to be multiplied together.

So, 25 = 2 × 2 × 2 × 2 × 2 = 32

106 ("ten to the power of six") = 1,000,000

Some special cases

Raising to the power of 2 is called squaring.
9 squared is 92 = 9 × 9 = 81

Raising to the power of 3 is called cubing.
4 cubed is 43 = 4 × 4 × 4 = 64

A power of 1 means just the number itself, as we have then only one of the number to multiply.
71 = 7; 311 = 31; 1021 = 102
We do not usually write a power of 1 unless we want to emphasise the fact.

Any number raised to a power of zero has the value of 1
80 = 1; 100 = 1; 520 = 1

By the BEDMAS rule, raising to a power (exponentiation) is done even before multiplication. Only if part of the expression is in brackets is it evaluated earlier.

So: 3 × 23 = 3 × 8 = 24
but (3 × 2)3 = 63 = 216

Remember

You do not multiply the number by the power - the power tells you how many of the number to multiply together.

The most important use of powers involves powers of 10, as these form the basis of our number system and this way of writing them is a very compact notation that saves writing lots of zeros (and avoids errors!) when dealing with very large numbers.

  • Units 1 = 100
  • Tens 10 = 101
  • Hundreds 100 = 10 × 10 = 102
  • Thousands 1000 = 10 × 10 × 10 = 103

and so on.

You will notice that the exponent determines the number of zeroes when we write the power of 10 out in full. So 109 can be written out in full directly as 1,000,000,000 and a 1,000,000,000,000 is 1012 in the power notation.

Remember

You do not multiply the number by the power - the power tells you how many of the number to multiply together.

For more on the use of powers of 10 to represent large and small numbers, see the 'Standard form' section.

Using a number line can help when adding or subtracting negative numbers. Some people find it useful to think of the number line in their head.

Example of a number line.

When we add we move along the line to the right.

When we subtract we move along the line to the left.

Where we end is the answer to the problem.

Example 1

To solve the problem 6 − 3 + 5

  1. Start at the number 6
  2. Move three places to the left (to 3)
  3. Move five steps to the right (to 8)
  4. The answer is 8

How the solution for example 1 appears on a number line

Example 2

To solve the problem −4 + 5 − 6

  1. Start at −4
  2. Move five steps to the right (to 1)
  3. Move six places to the left (to −5)
  4. The answer is −5

When multiplying or dividing using negative numbers there are two simple rules to remember:

  • Two like signs (two pluses or two minuses) multiplied together or divided give a positive answer
  • Two unlike signs (one plus and one minus) multiplied together or divided give a negative answer

Ignore the plus or minus signs at first: multiply or divide the numbers as if they were all positive numbers; then work out whether the answer should be a plus or minus number, depending on whether there were two like or two unlike signs.

Examples

  • 9 × (−5) = −45
  • (−5) × (−12) = 60
  • (−12) ÷ 3 = −4
  • (−21) ÷ (−7) = 3

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