In our number system the value (or digit) is important and so is its place within the number.

Both 425 and 263 feature the digit '2' but the positions within the number alter its meaning. In 425, the 2 represents 2 'tens' - ie. twenty; the 2 in 263 represents 2 'hundreds' - ie two hundred.

This rule applies to whole numbers and to decimal numbers. A place value table can help you determine the size of the number (see below). This is only a part of the table and it continues in both directions. To the left, thousands, ten thousands, hundred thousands, millions, etc. To the right, ten thousandths, hundredthousandths, millionths etc.

Place value table:

The table shows 425 and 263 in terms of the number of hundreds, tens and units they represent. It then shows some examples of decimal numbers.

The decimal point shows the reader where the whole numbers finish and the 'parts of whole numbers' start. It fixes a numbers position within the table.

If you see 28.375 written down (see table) you know, because we use the decimal system; that it represents 2 tens, 8 units, 3 tenths, 7 hundredths and 5 thousandths.

Likewise 17.5 (the current percentage rate of VAT) reads 1 ten, 7 units and 5 tenths.

Numbers that are less than one.

A number that is less than one can be written using the decimal system - eg 0.5. There are no whole numbers - ie no numbers to the left of the decimal point - but there are 5 tenths. This represents 5 lots of ¹/₁₀ which is the same as ⁵/₁₀, which can be simplified to ¹/₂ (a half). Therefore in decimal terms, 0.5 represents a half.

If you need more help with multiplying and simplifying fractions see the sub topic ‘Fractions’ beneath the title for this section or in the menu to the left of the screen.

NB 0.5 could be written just as '.5' - ie point five. However, convention recommends that the 0 before the decimal point is shown to make the number clearer to the reader.

Which is bigger:
0.5 or 0.26?

Sometimes when people read decimal numbers they think in terms of what the number says rather than what it actually represents (eg. 'nought point five' – which is correct, but then 'nought point twenty-six' – which is not). The 2 and the 6 represent 2 tenths and 6 hundredths not twenty-six. It should be read 'nought point two six'. This may seem like a small point but it can often lead to confusion when looking at the size of decimals.

Which is bigger: 0.5 or 0.26?
If the number is read incorrectly, as above, 'nought point twenty-six' may sound bigger than 'nought point five' because twenty-six is bigger than five. However, if you look at them in a place value table:

0.5 represents 5 tenths and no hundredths, whereas 0.26 represents only 2 tenths and then 6 hundredths. So 0.5 is larger than 0.26.

Where must you put a nought in a decimal number?

We have already noted that convention recommends a nought before a decimal point for numbers less than one to make it clearer to the reader.

You should use 'making it clear' as a guide as to when you need a nought in other places within decimal numbers. Looking at the place value table:

0.5 represents 5 tenths, but no hundredths or thousandths.

We could have written the number 0.500; it would still mean the same thing. We do not show those noughts as it is accepted that no number in that 'place' means nought.

Because numbers can continue in both directions to infinity we could show it as 0000000000.50000000000 and there would still be places on either side where further noughts could be inserted.

All the information about this number that the reader requires is in bold print - ie 0.5 is enough to make it clear to the reader what the number represents.

Look at 0.001 in the table (above) and look at where the noughts are included here.Why are they there?

If no noughts were shown (other than the one to the left of the decimal point), the number would look like 0. 1. Depending upon the spacing used by the writer this could represent 1 tenth (0.1), 1 hundredth (0.01), or 1 thousandth (0.001) and so on.

To make it clearer for the reader mark every empty space between the decimal point and the digit with noughts.

This makes it clear that 0.001 represents 0 tenths, 0 hundredths and 1 thousandth.

The same principle applies with 0.406; it would be misleading to the reader if it were written 0.4 6.

To make it clearer for the reader mark every empty space between two digits with noughts.

Insert a nought into any empty spaces between the decimal point and the digit.

Insert a nought into any empty spaces between two digits.

You do not have to show any noughts to the right of the last digit.

Decimal numbers can be very long; you may need to round them to make them more 'manageable' to read and use.

For example. here is pi ( ) shown to 22 decimal places;

3.1415926535897932384626

and it does, in fact, continue beyond this to an infinite number of places.

In the place value table (below) it is shown to 3 decimal places (ie 3 digits after the decimal point). By rounding it to 3 decimal places, it has lost some of its accuracy but has been made into a more 'manageable' number to read and use.

NB This number pi ( ) is used when calculating the area, and the circumference of a circle. If you have a scientific calculator you will have a '' button on it.

When working with decimals you may need to decide how accurate your answer needs to be.

As a guide:

• When performing calculations try to keep the decimals in their 'longest form' (ie as long as your calculator will allow) until the very end of the calculation as this will make your final answer more accurate.
• When you have your final answer in 'long form', round the answer to a size appropriate for the work you are doing.
• If you want your answer to be very accurate then 4 decimal places (ie 4 digits after the decimal point) are usually sufficient. Unless specifically asked to, do not give an answer with more than 4 decimal places- eg 3.1416.
• If you want your answer to be clear to read and only reasonably accurate, then 2 decimal places (ie 2 digits after the decimal point) are often enough - eg 3.14.
• Make your answer realistic, if you have worked out that the average number of people attending a series of concerts is 132.2245, do not leave this answer with 4 decimal places. An average of 132 people is clearer and more realistic.

These examples all use pi () . Here it is again shown to 22dp (decimal places):
3.1415926535897932384626

Look at the first 3 decimal places only (ie one more than the number of places you want to round to)

3.141

Is the 3^rd decimal place more or less than 5?

The 3^rd decimal place is 1 (less than 5) so we round down.

Therefore to 2 dp = 3.14

Look at the first 5 decimal places only (ie one more than you want to round to):

3.14159

Is the 5^th decimal place more or less than 5?

The 5^th decimal place is 9 (more than 5) so we round up.

Therefore to 4 dp = 3.1416

Look at the first 4 decimal places only (ie one more than you want to round to):

3.1415

Is the 4^th decimal place more or less than 5?

The 4^th decimal place is exactly 5. In this case we round up.

Therefore to 3 dp = 3.142

If the 'next' number (ie. the one after the one you want to round to) is:

0, 1, 2, 3, 4 then round down.

If the 'next' number (ie the one after the one you want to round to) is:

5, 6, 7, 8, 9 then round up.

Activity 1

If you want to practise rounding decimals, have a go at the questions below:

1. 2.784 to 2dp
2. 0.839152 to 3dp
3. 0.33815 to 4dp
4. 18.246001 to 3dp
5. 0.5011515 to 4dp

Are you right? If not, have another look at the examples earlier in this topic or have a look at some of the 'Resources You Can Use' for further help with this topic.

Some decimal numbers have an infinite number of decimal places but have digits that repeat, or re occur, in a pattern; these are called 'recurring' decimals.

Eg ¹/₃ as a decimal is 0.3333333333333. This can be rounded to 0.3 (to 1 dp) although, as seen earlier this will lose some accuracy. (3 x ¹/₃ = 1, but 3 x 0.3 = 0.9) It is possible with recurring decimals to maintain their accuracy by writing them:

The dot over the 3 tells the reader that the 3 is repeating or recurring.

Sometimes a decimal may be recurring when it is not just one number repeated but a sequence or pattern of numbers:

eg 7 13 = 0.538461538461

Here we have a sequence of numbers repeating:
0.538461538461538461

So it could be written:

The dots are placed over the first and last number in the sequence. This tells the reader that the sequence of numbers between the dots is repeated.

For more help with converting decimals into percentages and fractions, or vice versa, see the sub topic ‘Converting Decimals, Fractions and Percentages’ beneath the title for this topic or in the menu to the left of the screen.