Representing and manipulating data in computers

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	Time: 20 hours Level: Intermediate

	Introduction Introduction Resource Computers are designed to receive, store, manipulate and present data. This unit explains how computers do this, with reference to the examples of a PC, kitchen scales and a digital camera. In particular...

	1 Representing data in computers: introduction 1 Representing data in computers: introduction Resource A computer is designed to do the following things:

	2 Representing data in the kitchen scales 2.1 Introduction Resource Study note: You may like to click on the link below to the Numeracy Resource as you study Section 2. It offers additional explanations and extra practice on some of the topics, and you may find... 2.2 Representing numbers: positive integers Resource A very straightforward way of finding binary codes to represent positive integers is simply to use the binary number that corresponds to each integer. This is because every positive integer in the everyday... 2.2.1 Positve integers: denary numbers Resource The number system which we all use in everyday life is called the denary representation, or sometimes the decimal representation, of numbers. In this system, the ten digits 0 to 9 are used, either singly... 2.2.2 Positve integers: binary numbers Resource Just as a denary number system uses ten different digits (0, 1, 2, 3, … 9), a binary number system uses two (0, 1). 2.2.3 Positve integers: converting denary numbers to binary Resource If computers encode the denary numbers of the everyday world as binary numbers, then clearly there needs to be conversion from denary to binary and vice versa. You have just seen how to convert binary... 2.2.4 Positve integers: encoding larger integers Resource The examples and activities in this section have looked only at 8-bit numbers. They have illustrated all of the principles of encoding positive integers as binary numbers without introducing the complication... 2.3 Representing numbers: fractions Resource 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.... 2.4 Representing numbers: negative integers Resource In Section 2.2. I showed you how integers can be encoded if they are known to be positive, treating the integers in the kitchen scales as if they were known to be positive. However, if the user invokes... 2.5 Representing weights Resource A physical quantity such as weight has the property that it can take on any value, not just a finite set of values. For instance, at one time the ingredients in the scalepan could weigh 29.2569427 grams,... 2.6 Representing true/false quantities Resource Sometimes a quantity that is to be represented in a computer has only two possible values, either true or false. An example of such a true/false quantity in the kitchen scales is the one that represents... 2.7 Input and output considerations Resource So far in Section 2 I have focused on how the data is represented, or encoded, inside the weighing-scales computer. But how does it get into the computer? And how does it get out again in a form that users...

	3 Representing data in the digital camera 3.1 Introduction Resource Digital cameras need to represent still pictures digitally, and this means that I need to introduce you to how still images are represented. I shall do this in Section 3.2. 3.2 Representing still images Resource There are two basic methods of representing still images in a computer: bit maps (also sometimes called raster graphics or raster images) and vector graphics (also sometimes called geometrical-shape graphics... 3.3 Compression Resource The previous section mentioned the large file size of bit-map representations of even small pictures. Therefore just a few images use up a great deal of storage space. This can be inconvenient for PC users,... 3.4 Input and output considerations Resource CCDs are not inherently able to detect colour, only brightness. So it is necessary to rely on the fact that any colour of light can be made up from the three primary colours of light: red, blue and green....

	4 Representing data in the PC 4.1 Introduction Resource Personal computers, or PCs, are very versatile computers and can perform a huge range of tasks. So whereas the uses of the kitchen scales and the digital camera indicate clearly what types of data are... 4.2 Representing text Resource Study note: You will need to refer to the Reference Manual while you are working through this section. 4.3 Representing moving images Resource A moving image is simply a series of still images presented at sufficiently short time intervals that the eye smoothes over the change from one image to the next. In practice, this means the images must... 4.4 Representing sound Resource Sound, such as speech or music, is an analogue physical quantity that varies with time, and so the ideas you have already met in Section 2.5 about converting analogue weights to digital form are relevant... 4.5 Input and output considerations Resource In this final portion of Section 4, I shall look in outline at how text, moving pictures and sound can be input into a PC and output from it. I'll leave aside the possibility that the data has been obtained...

	5 Representing data in computers: conclusion 5 Representing data in computers: conclusion Resource Study note: You will need to refer to the Reference Manual while you are working through this section.

	6 Manipulating data in computers: introduction 6 Manipulating data in computers: introduction Resource Sections 1 to 5 of this unit have shown that in a computer all types of data are represented by binary codes, and that programmers must make sure that the programs they write treat this data appropriately...

	7 Binary arithmetic 7.1 Adding unsigned integers Resource Study note: You may like to have the Numeracy Resource (attached below) to hand as you study Section 7. It offers extra practice with the manipulations, and you may find this useful. 7.2 Adding 2's complement integers Resource The leftmost bit at the start of a 2's complement integer (which represents the presence or absence of the weighting −128) is treated in just the same way as all the other bits in the integers. So the... 7.3 Subtracting 2's complement integers Resource You will probably have carried out subtraction of denary numbers using rules for subtraction that include the process of ‘borrowing’ whenever you need to subtract a larger digit from a smaller one. It... 7.4 Multiplying 2's complement integers Resource Multiplication can be thought of as repeated addition. For instance, in denary arithmetic 7.5 Dividing 2's complement integers Resource Just as multiplication can be turned into repeated additions, so division can be turned into repeated subtractions. And just as shifting a binary integer one place to the left equates to multiplying by... 7.6 Arithmetic with binary fractions Resource My final point in the preceding section brings home the fact that integer arithmetic is not really suitable when divisions are to be performed. It is also not suitable where some or all of the values involved...

	8 Logic operations 8.1 Introduction Resource Study note: You may like to have the Numeracy Resource to hand as you study Section 15. It offers extra practice with the logic operations, and you may find this useful. 8.2 The NOT operation Resource The NOT operation (note that, as with all logic operators, NOT is always written in capital letters) acts bit by bit on a single binary word according the following rules: 8.3 The AND operation Resource The AND operation combines two binary words bit by bit according to the rules 8.4 The OR operation Resource The OR operation (occasionally called the inclusive-OR operation to distinguish it more clearly from the exclusive-OR operation which I shall be introducing shortly) combines binary words bit by bit according... 8.5 The exclusive-OR operation Resource The exclusive-OR operation (usually abbreviated to XOR, pronounced ‘ex-or’) combines two binary words, bit by bit, according to the rules: 8.6 Summary Resource The logic operations introduced here are summarised in Table 1, which is an example of what is known as a ‘truth table’. It shows what the result (‘output’) of each logic operation is for all possible...

	9 Conclusion 9 Conclusion Resource This unit started with the idea that computers have become an important part of everyday life, especially when all the ‘invisible’ computers that surround us are taken into account – those embedded in...

	References and Acknowledgements Acknowledgements Resource