## Rounding

When you use a calculator to perform calculations, you will often get lots and lots of decimal places.  In exams or on assignments your teachers probably don’t expect you to write down the complete number.  There are two ways you can round off a number so that you don’t have to write all those digits after the decimal point.

### Rounding to a number of decimal places

This is the easiest type of rounding to do.  Say your teacher told you to find the answer to the following question and give the answer to 3 decimal places:

Well, you can calculate this using your calculator.  You should get something like:

0.777777777

showing up on your calculator screen. The teacher has asked for the answer to be given to 3 decimal places. To write the answer with the correct number of decimal places, follow this simple procedure:

Write the number down, but only to one less than the asked for number of decimal places.  So if you’ve been asked for 3 decimal places, only write down two decimal places.

Now find the ‘special’ digit.  The ‘special’ digit is the digit located one more decimal place to the right than the number of decimal places you’ve been asked for.  So say you’ve been asked for 3 decimal places, look at the value of the digit in the 4th decimal place – this is the ‘special’ digit.

If the ‘special’ digit is 0, 1, 2, 3, or 4, then just write your last decimal place in as it is.

If the ‘special’ digit is a 5 or larger, you need to add one to your last decimal place.

When you’re checking the value of the ‘special’ digit – you’re basically checking whether it is larger than 4 or not.  If it’s not larger than 4, then you can write your last decimal place in without changing it.  But if the ‘special’ digit is larger than 4, you need to increase the value of your last decimal place by one.

So back to our example, we’ve been asked for 3 decimal places, so first we write the number with only 2 decimal places:

0.77

Now, we’re not quite finished yet.  We still have to write the 3rd decimal place down in our answer.  Before we do this, look at the value of the ‘special’ digit – the fourth digit to the right of the decimal place.  If this digit is a 0, 1, 2, 3 or 4, then we just go ahead and write the 3rd decimal place.  However, in this case, the fourth digit is a ‘7’, so what we have to do is increase the value of the 3rd decimal place by one.  So our answer becomes:

0.778

Whenever you round a number to a certain number of decimal places, you need to check to see whether the last digit will need to be rounded up.  So if you’re rounding to 3 decimal places, you check the fourth digit (the ‘special’ digit) to the right of the decimal place, and if this digit is larger than ‘4’, you need to round up your last decimal place.

Here’s another rounding question:

 Calculate 12 ÷ 7 and give the answer to 4 decimal places. Solution Well, we can type this into our calculator.  You should get something like:                                                            1.7142857 showing up on your screen.  Since we’ve been asked to give our answer to 4 decimal places, we rewrite our answer, but only writing the first three digits to the right of the decimal point:                                                               1.714 One more thing to do – we need to write down the fourth decimal place.  Before we do this, we need to look at the value of the ‘special’ digit – the 5th digit to the right of the decimal point.  In this case, the number is an ‘8’, so we need to round up our fourth decimal place. So, instead of writing 1.7142, we round up the fourth decimal place to get:                                                               1.7143

Say we’re given the number 7.896, and told to round it to 2 decimal places.  First, we rewrite the number with only 1 digit to the right of the decimal point:

7.8

Now we need to write down the 2nd digit to the right of the decimal point.  In our original number, this is a ‘9’.  But before we write that down, we need to check whether it has to be rounded up or not.  To do this, we check the value of the 3rd digit to the right of the decimal place (the ‘special’ digit) – in this case the digit is a ‘6’.  Since 6 is larger than 4, we need to round up the 2nd digit.  But the 2nd digit is a 9 already, which means the 2nd digit becomes a zero and we round up the digit to the left of it.  Here is the whole process:

7.896

First we write it with only 1 decimal place:

7.8

Then we check the value of the 3rd decimal place, the ‘special’ digit:

3rd digit is a ‘6’

So have to round up the 2nd decimal place, but it’s already a 9.

7.89

So we change the 2nd decimal place to a zero, and round up the decimal place one to the left of it. In this case that means changing the ‘8’ to a ‘9’:

7.90

And there we have our answer, 7.90.  This is an example of when the rounding process is a bit tricky.  To cope with situations like this however is easy, you just need to remember how rounding works.

If a digit is a ‘9’, and you increase the value of it by 1, then that digit becomes 0, and you increase the value of the digit just to its left by 1.

Here’s an even trickier case:

Round 9.99996 to 1 decimal place

So first we write the number with 0 decimal places:

9

Then we look at the 2nd decimal place (the ‘special’ digit) – it’s a ‘9’.  So we need to increase the value of the first decimal place by 1 – but it’s already a ‘9’.  So it becomes a 0 and we have to increase the value of the digit just to its left.  The digit just to its left is a ‘9’ too, so it becomes a 0, and we increase the value of the digit just to its left by 1.  Here’s the whole process:

Original number:

9.99996

Writing it with no decimal places:

9

The ‘special’ digit in the original number is a ‘9’:

So need to round up the first decimal place:

The first decimal place becomes a ‘0’, and we need to increase the value of the digit to its left by one:

For the next step it might make it easier if you remember that 9.0 is the same as 09.0. It’s already a ‘9’ as well, so it becomes a ‘0’, and we increase the value of the digit just to its left by one.  The digit just to its left isn’t normally written, but if you did write it, it would be a ‘0’.  So we increase it to ‘1’:

And there we have our answer, 9.99996 rounded to 1 decimal place is 10.0.  The answer looks quite different to the original number doesn’t it?  In this case this doesn’t mean we’re wrong however, it just shows how different the answer can look after rounding.

### Rounding to a number of significant figures

Rounding off to a certain number of significant figures means that the number you write only has a certain number of digits, left or right of the decimal point.

### What are significant digits?

A digit is counted as a significant digit except in the case of zeros directly after a decimal point for a number that is smaller than 1.  For instance, look at the following numbers:

·         427 has 3 significant digits

·         0.5234 has 4 significant digits

·         232.03 has 5 significant digits

0.000023 only has 2 significant digits.  We don’t count the zeros directly after the decimal point because the number is smaller than 1.  Be careful though:

23.000023 has 8 significant figures.  Although it has zeros directly after the decimal point, they are counted as significant figures in this case because the number is larger than 1.

The procedure for rounding a number to a certain number of significant figures is similar to rounding to a number of decimal places:

Write the number down, but with one less digit than you’ve been asked to round to.

Now find the ‘special’ digit.  The ‘special’ digit is the digit located just to the right of the last significant digit.

Look at the value of the ‘special’ digit.

If the ‘special’ digit is 0, 1, 2, 3, or 4, then just write your last significant digit as it is.

If the ‘special’ digit is a 5 or larger, you need to add one to your last significant figure.

Let’s do an example to show how the steps work:

 Round 1652 to two significant figures. Solution Here’s a diagram pointing out each part of the number. First we write down the number, but with only one significant figure (one less than we’ve been asked for):                                                                1652 The rest of the number has been shown in a very light colour.  Next we look at the value of the ‘special’ digit – in this case the special digit is a ‘5’.  Because 5 is larger than 4, we need to add one to the last significant digit:                                                                1752 And lastly, we can write the rest of the insignificant digits as zeros:                                                                1700 Now, if someone looks at what you’ve just written, they don’t know whether you’ve written 1700 exactly, or whether you’ve rounded some other number and it has become 1700 through rounding (like what we just did).  There is a way to tell the reader how many significant figures you have written.  What you need to do is use exponential notation.  Instead of writing 1700, you’d write:                                                             or                                                             If you haven’t done powers yet, for the moment just think of 102 as being two 10s multiplied by each other, so  is really 17 × 10 × 10 = 17 × 100 = 1700.  In the same way, is really three lots of 10s multiplied by each other, so  = 1.7 × 10 × 10 × 10 = 1.7 × 1000 = 1700. This would show the reader that there are only two significant figures in your answer – the ‘1’ and the ‘7’.  Otherwise the reader might think that you had four significant figures – the ‘1’, the ‘7’ and the two 0s.

Here’s another significant figures question:

 Round 68.5389 to 4 significant figures. Solution Well, I start by writing the number with only 3 significant figures (one less than I’ve been asked for):                                                              68.5389 Next, I look at the value of the special digit:                                                              68.5389 It’s an 8, so I need to add one to the last significant digit:                                                  68.54 (instead of 68.53)