Representing and manipulating data in computers
Representing and manipulating data in computers

This free course is available to start right now. Review the full course description and key learning outcomes and create an account and enrol if you want a free statement of participation.

Free course

Representing and manipulating data in computers

2.3 Representing numbers: fractions

In the denary system, a decimal point can be used to represent fractions, as in 6.5 or 24.29. One way of encoding fractions uses an exactly analogous method in binary numbers: a ‘binary point’ is inserted.

Some examples of 8-bit binary fractions are:

  • 0.0010110

  • 110.01101

  • 0101110.1

The weightings that are applied to the bits after the binary point are, reading from left to right, 1/2, 1/4, 1/8, etc. (in just the same way as in denary fractions they are 1/10, 1/100, etc.).

Now, 1/4 is the same as 1/22, and 1/8 is the same as 1/23, and there is a convenient notation for fractions like this, which is to write 1/22 as 2−2, 1/23 as 23, and so on. In other words, a negative sign in the exponent indicates a fraction.

Using this notation, the bits after the binary point in a binary fraction are weighted as follows:

2−1 2−2 2−3 2(4
(1/2 = 0.5) (1/4 = 0.25) (1/8 = 0.125) (1/16 = 0.0625)

So, for instance, 0.010 1000 in binary is equal to 1/4 + 1/16 = 5/16 (or 0.25 + 0.0625 = 0.3125) in denary.

One problem with encoding fractions like this is that there is no obvious way of representing the binary point itself within a computer word. Given the 8-bit number 0101 1001, where should the point lie? The way of solving this problem is to adopt a convention that, throughout a particular application, the binary point will be taken to lie between two specified bits. This is called the fixed-point representation. Once a convention has been adopted – for example, that the binary point lies between bits 7 and 6 – it should be adhered to throughout the application.

Another problem is that it may not be possible to represent a denary fraction exactly with a word of given length. For example, it is not possible to represent denary 0.1 exactly with an 8-bit word. The nearest to it is binary 0.000 1101 with a denary difference of 0.0015625. (Try it for yourself!) So it is just not possible to represent all fractions exactly with a binary word of fixed length. This second problem can be reduced, but not eliminated, if a multiple-length representation is used. For example, with 12 bits the denary fraction still cannot be represented exactly, but now the nearest to it is binary 0.000 1100 1101 with a much smaller denary difference of 0.00009765625.

Fortunately this problem of exact representation does not occur in the kitchen-scales example. In recipes using imperial weights it is traditional to use 1/2 oz, 1/4 oz, etc., and these are fractions which can be exactly represented by the fixed-point representation just described.

Box 3: Floating-point representation

Another way of representing binary fractions is by the floatingpoint representation. This is the preferred method in many applications as it is a very flexible method, though rather complex. It uses the same basic idea as a fixed-point fraction, but with a variable scale factor. Two groups of bits are used to represent a single quantity. The first group, called the mantissa, contains the value of a fixed-point binary fraction with the binary point in some predefined position – say, after the most-significant bit. The second group, called the exponent, contains an integer that is the scale factor.

An example is a mantissa of 0.111 1000, which evaluates to the fixed-point fraction 1/2 + 1/4 + 1/8 + 1/16 = 15/16, together with an exponent of 0000 0011, which evaluates to the denary integer 3. But what fraction does this mantissa-exponent pair represent?

The answer is that it represents (mantissa × 2exponent), which is 15/16 × 23. This works out to 15/16 × 8, which is 7.5.

Another example is a mantissa of 0.011 0000 and an exponent of 0000 0001. Here the mantissa evaluates to 1/2 + 1/8, which is 3/8. The exponent evaluates to 1. So the fraction represented here is 3/8 × 21. This works out to 3/8 × 2, which is 3/4.

Notice that the fraction being represented in the first example is a ‘mixed fraction’ – that is, a mixture of an integer and a fraction which is less than 1 – while in the second example the fraction being represented is simply less than 1. This ability to represent both types of fractions with identical encoding methods makes the floating-point representation extremely versatile.

T224_2

Take your learning further

Making the decision to study can be a big step, which is why you'll want a trusted University. The Open University has 50 years’ experience delivering flexible learning and 170,000 students are studying with us right now. Take a look at all Open University courses.

If you are new to university level study, find out more about the types of qualifications we offer, including our entry level Access courses and Certificates.

Not ready for University study then browse over 900 free courses on OpenLearn and sign up to our newsletter to hear about new free courses as they are released.

Every year, thousands of students decide to study with The Open University. With over 120 qualifications, we’ve got the right course for you.

Request an Open University prospectus