4 Decimal numbers and decimal places

We introduced fractions in the previous section, and a fraction like can also be written as 0.5. So , 50% and 0.5 all mean the same thing and the number 0.5 is an example of a decimal number. Decimal numbers are important as calculators use them in any calculation involving not just whole numbers. They are used throughout science, and you need to become proficient at adding, subtracting, multiplying and dividing decimal numbers. Fortunately, your calculator will take the pain out of the calculations, so you can concentrate on understanding what the numbers mean.

Decimal numbers consist of two parts separated by what is called a decimal point. When printed, a ‘full stop’ is used for the decimal point. Here are four examples, with words in brackets indicating how you say the numbers: 0.5 (‘nought point five’), 2.34 (‘two point three four’), 45.875 (‘forty-five point eight seven five’) and 234.76 (‘two hundred and thirty-four point seven six’). Note that the part of the number before the decimal point is spoken as a whole number, and the part after the point is spoken as a series of individual digits. It’s also worth noting that in parts of Europe outside the UK, a comma is used instead of a full stop in decimal numbers.

What do these numbers mean? Well, the part of the number before the decimal point represents a whole number, and the part after the decimal point represents the fraction, something between nought and one, that has to be added on to the whole number. Thus if you divide 13 by 2 you get if you use fractions, but 6.5 if you use a calculator; the 0.5 is equivalent to the half. Note that when there is no whole number, i.e. the number is less than one, it is usual to print or write a zero in front of the decimal point, otherwise the decimal point might be overlooked. (Your calculator, however, may not always show the zero.) If you divide 13 by 4, then with fractions you get and with a calculator you get 3.25, so a quarter is the same as 0.25.

Conversion of any fraction to a decimal number is straightforward with a calculator. All you have to do is divide the number on the top of the fraction by the number on the bottom. Try this for yourself: with the fraction , you should obtain the decimal number 0.375.

Now just as each digit that comes to the left of the decimal point has a precise meaning that depends on where it comes in the order, so also does each digit that comes after the decimal point. These meanings are summarised in Table 4.1 for the number 7 654.321.

The 4 immediately before the decimal point means 4 units (or 4 ones), which is simply 4; the 5 signifies 5 tens, or 50; the 6 signifies 6 hundreds, or 600; and the 7 signifies 7 thousands, or 7 000. So 7 654 means 7 000 + 600 + 50 + 4.

In a similar way, the 3 after the decimal point means 3 tenths, or , the 2 means 2 hundredths, or , and the 1 means 1 thousandth, or . And, just as 7 654 means 7 thousands plus 6 hundreds plus 5 tens plus 4 units, so 0.321 means 3 tenths plus 2 hundredths plus 1 thousandth. So

Now, to add fractions, we first have to convert them to equivalent fractions with the same number on the bottom. In this case, we shall convert the first two fractions to equivalent fractions with 1 000 on the bottom.

Since is an equivalent fraction to , and is equivalent to , then

Here, we have added the numbers on the tops of the fractions together to get the total number of ‘thousandths’, but we don’t add the numbers on the bottoms of the fractions since these just tell us that we are adding ‘thousandths’ in each case.

This shows that converting a decimal number to a fraction is really quite straightforward; you just take the numbers after the decimal point (321 in the example above) and divide by 1 followed by the same number of zeros as there were digits after the decimal point (three in this case), so

A calculator does arithmetic with decimal numbers in the same way as it does with whole numbers, including carrying out operations in the right order. The only difference is that you have to key in the decimal point, using the decimal point key on the calculator, at the appropriate place in decimal numbers.

As an example, try multiplying 2.36 and 43.7. The result, 103.132, should appear in the display.