If you work through this section you should be able to:
The decimal system is an extension of our ordinary number system. Fractions of numbers are tenths, hundredths, thousandths, ten thousandths, and so on.
You may also view the text version of the decimal system illustration.
Decimal numbers, then, represent fractions with denominators of 10, 100, 1000 and so on.
6.666 is a decimal number
6.666 = 6 Units
When there are no whole numbers (units) it is usual to insert a zero in front of the decimal point, e.g. is 0.3
Place value is vital, so a zero is used if there is nothing in a column.
3 tenths and 4 thousandths (no hundredths)
0.304
Extra zeros at the end of decimal places make no difference though, e.g. 0.25 = 0.250000
Adding and subtracting decimals involves the same process as adding or subtracting whole numbers.
To avoid confusion when performing calculations, write them in columns with the decimal points lined up underneath each other. This makes adding or subtracting the decimals easy because you can just add up each column, making sure you keep the decimal point in the same place.
It is helpful to put zeros in any 'empty' columns and always do this for subtraction. Remember to put a zero in the units column if there are no whole numbers!
207.3 + 93 + 0.643 + 6.8
Answer: 307.743
42.3 − 8.63
Answer: 33.67
This is done in the same way as for whole numbers, and then we decide where to put the decimal point.
Ignore the decimal point at first, and multiply the two numbers together.
Then, when you have this answer, put the decimal point back in. To decide where the decimal point should be placed, count how many places of decimals there are in total in the numbers being multiplied.
12.86 × 3.6
Now we need to count the correct number of decimal places - the number of columns with numbers (not trailing zeros) after the decimal points. There are two decimal places in 12.86 and one in 3.6 so there should be three in the answer. So we place the point before the last three numbers.
Answer: 46.296
Calculate 0.2 × 0.4
0.2 × 0.4 (2 decimal places in total)
2 × 4 = 8
To get 2 decimal places, we must add one zero before the 8, thus: 0.08
Calculate 0.03 × 0.05
0.03 × 0.05 (4 decimal places)
3 × 5 = 15
To get 4 decimal places, we must add two zeroes before the 15, thus: 0.0015
When dividing decimal numbers you need to pay attention to the decimal points in the two numbers.
If the divisor (the number you are dividing by) has a decimal point, convert this to a whole number by moving the point however many places to the right are required:
150.45 - need to move it two places to the right.
0.012 - need to move it three places to the right.
Calculate 12.636 ÷ 0.6
Make the divisor a whole number.
In this case, make the divisor (0.6) into a whole number by moving the decimal point 1 place.
06, that is, the whole number 6.
Move the point in the dividend.
Move the point in the dividend (12.636) the same number of places as you did for the divisor, i.e. one place to the right.
126.36
Divide the new divisor into the new dividend.
Remember: place the decimal point in the same place, and put a zero in if necessary.
You may need to add zeroes at the end of the dividend to be able to move the right number of decimal places.
Calculate 2.8 ÷ 0.07
2.8 ÷ 0.07 = 280 ÷ 7 = 40
Also remember to add zeroes at the end of the dividend when doing the division.
Calculate 0.0037 ÷ 0.04
0.0037 ÷ 0.04 = 0.37 ÷ 4
Put the "units" zero in and because 4 does not go into 3 remember to put a zero in that column.
Complete the division by adding zeros at the end of the dividend.
This gives the result 0.0925.
Some calculations can lead to answers with an inappropriately large number of decimal places.
For example:
5.6 ÷ 1.2 = 4.6666666666666666666666666666667
Working out VAT at 17.5% on £7.90 gives an answer of £1.3825, but obviously we only require 2 decimal places as we don't deal in fractions of a penny!
Similarly, expressing as a decimal leads to an infinitely long decimal 0.3333…
There will be a level of precision which will be sufficient for a given purpose. Your calculator, for instance, will only display infinite decimals to a fixed number of figures.
What we do in these cases is to round the number. This topic will demonstrate two methods of consistently rounding off decimal numbers:
A shorthand for rounding to decimal places is to write x d.p. or sometimes xD.
In the earlier example, it seems natural to round the £1.3825 down to the nearest penny, so in this case that means to two decimal places. What you want, then, is only two figures after the decimal point.
The rule is very simple:
Look at the digit in the next decimal place (so if 2 d.p. are required, look at the third decimal place digit)
If it is 5 or greater round up; if it is under 5, leave it unchanged.
For example £1.3825 to 2 d.p. is £1.38
The following roundings are to 2 d.p in each case and the third place digit is highlighted.
2.47819 = 2.48
2.1349 = 2.13
5.0049 = 5.00
3.20602 = 3.21
5.795 = 5.80
Note how digits in further decimal places are ignored and how, in the final example, a "rounded-up" 9 rolls over the previous figure also.
This method is not appropriate in all cases - when considering measurements in different units for instance. A measurement of 308.1 cm, for example, is to the nearest millimetre and has one decimal place. However, expressing the measurement in metres to the same precision - again to the nearest mm - is 3.081 metres, so the concept of decimal places is not helpful in this case. What we consider instead is the number of significant figures.
To find the number of significant figures in a number we count them from left to right, except leading zeroes. Leading zeroes are not significant but, reading from left to right, the first non-zero digit is the first significant figure and after that zeroes are significant.
2.05 is 3 s.f.
0.00002 is 1 s.f.
30.00 is 4 s.f.
0.00020 is 2 s.f.
For a variety of reasons it is sometimes necessary to express a number to a certain number of significant figures.
Using an example given above, both the measurements 308.1cm and 3.081m are expressed to four significant figures. The zero is considered significant as it comes after a non-zero digit.
But consider 0.003081 (this is the same measurement in kilometres).
The first three zeroes (the units zero before the decimal point and the two leading decimal place zeroes) are not significant so this too has 4 significant figures (written for short as 4 s.f. or 4S).
You should see that this is reasonable as all three measurements are expressed to the same degree of precision - it is just that depending on the units being used the figures have different place values.
Now that we know what significant figures are, how do we round off long decimal numbers to a number of significant figures?
The rule for rounding to a certain number of significant figures is exactly the same as for rounding to decimal places:
Look at the digit in the next decimal place.
If it is 5 or greater round up the last significant figure required; if it is under 5 leave it unchanged.
For example (with the digit being inspected highlighted):
2.4567 = 2.46 (3 s.f.)
0.000415 = 0.00042 (2 s.f.)
23.00025 = 23.0003 (6 s.f.)
It is possible to express whole numbers to a certain number of significant figures - for example when a greater degree of precision is not required.
A particular measurement of the distance of the moon from Earth is given as 252,350 miles. What is this to 2 significant figures?
Obviously we cannot just write down the first two figures, so we maintain the order of magnitude by changing subsequent digits to zeroes, to give: 250,000.
Similarly, 913,738.35 is 914,000 to 3 significant figures (round the third digit up as the fourth is 7, change the whole number digits to zeroes, drop the decimal fraction digits).
14,567.43 = …
To convert decimals to fractions you must know the value of each decimal place (see 'The decimal system' at the beginning of this section). Remember: the first column after the decimal point was tenths, the second was hundredths, and so on. So look at the place value of the last non-zero figure and this gives the denominator of the fraction.
The number of decimal places in the decimal will determine how many zeroes are in the denominator, as you will see in the examples below.
Examples showing how the number of decimal places in the decimal determine how many zeroes are in the denominator
As with all fractions that you write, always check to see if they will simplify.
Examples of simplifying fractions
To find out how to convert fractions to decimals and for more on simplifying fractions, see the Fractions section.
To convert a decimal into a percentage, simply multiply by 100, which is the same as moving the decimal point two places to the right. If necessary, add a zero.
Examples of converting decimals to percentages
Remember: every 100% is actually a whole, so in the last example above (5.293) we had 5 units, therefore, we expect to have more than 500%.
To find out how to convert percentages to decimals, see the Percentages and ratios section.
You can download a version of this Decimals activity in Word format:
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