4.1 Decimal places and rounding

So far in this section, you have met decimal numbers with one, two or three digits after the decimal point. The number of digits after the decimal point is termed the number of places of decimals. For example, we say that the number 1.735 is expressed to three decimal places. 7 is in the first decimal place, 3 in the second decimal place and so on.

Now often when you do a calculation your calculator will display an answer with perhaps 7 decimal places – for example, it will indicate that is 0.333 333 3. (Note the convention of leaving a gap after every third digit after the decimal point in the same way that a gap is left every third digit before the decimal point, counting from the decimal point in each direction.) In most cases, it is not necessary to give all of these digits. We might be happy to know that is about 0.33, and to forget about the thousandths and the ten-thousandths and so on. Or sometimes it is enough to know that is about 0.3. If we approximate in this way, we say that we are rounding the number. But rounding is a bit more complicated than just chopping off the unwanted digits.

If we wanted to round 1.264 5 to two decimal places, we would need to look at the first digit to be removed – 4 in this case. If the first digit removed is a 0, 1, 2, 3 or 4, then the last remaining digit – 6 in this case – is left unchanged. So the answer would be 1.26.

However, if the first of the digits that are removed is a 5, 6, 7, 8 or 9, then the last remaining digit is increased by one. So, for example, if 1.264 5 is rounded to one decimal place, the answer is 1.3 – the 2 is rounded up to 3 because the first digit removed was 6.

The reason for rounding up when the first digit removed is 5 or greater is clear if you bear in mind that the number that is midway between 1.2 and 1.3 is 1.25. So all numbers between 1.25 and 1.3 are closer to 1.3 than they are to 1.2. It therefore makes sense to round up the last remaining digit whenever it has been followed by a digit between 5 and 9. Note that by convention, the digit 5 is rounded up.

Sometimes, rounding a decimal number will produce a zero as the final digit; for example, both 1.803 and 1.798 become 1.80 when rounded to two decimal places. Don’t be tempted to ignore the final zero in these cases though, because it contains important information about the decimal number. Quoting a length as 1.80 metres tells you that the measurement is between 1.795 0 metres and 1.804 9 metres because numbers within this range are equal to 1.80 metres when rounded to two decimal places. Quoting the length as 1.8 metres, on the other hand, means that it is between 1.750 metres and 1.849 metres, which is a much larger range.