2: Binary, bits and bytes (T171_PC)

1: Software and hardware — Sat, 05 Jul 2008 18:35:11+0100

To appreciate the importance of the various breakthroughs in the history of the computer industry you will need a basic knowledge of how a computer works, and in this segment you will look at how a computer represents information.

At its very lowest level a computer operates by turning on or off millions of tiny switches, called transistors. In computers these transistors can only be in one of two states; that is, on or off. Such devices are thus referred to as two-state devices. Another example of a two-state device might be a conventional light switch. It is either on or off, with no intermediate state. The states of ?on? and ?off? can be represented by the numbers 1 and 0.

In mathematics the term binary is used to refer to a number system which has only two digits, that is 1 and 0. The number system we use in everyday life has ten digits, 0 to 9, and is called denary. The binary system is the smallest number system that can be used to provide information.

Any number from our normal, denary system can be represented in binary; 0 in denary is 0 in binary. Similarly 1 in denary is 1 in binary. When you get to 2 in denary you have a problem. There are no more symbols in binary; you are restricted to only 1 and 0. So how do you represent two? This question is similar to asking how you represent ten in denary. Once you get to nine you have run out of digits, so you simply create a new column and start afresh, using 1 and 0. This is also what you do in binary, so 2 in denary becomes 10 in binary. When you move on to 3 (in denary) you proceed as before; 3 becomes 11 in binary. The table below shows how denary numbers convert to binary.

Denary number Binary equivalent
0 0
1 1
2 10
3 11
4 100
5 101
6 110
7 111
8 1000
9 1001
10 1010

It is useful to think of binary in terms of columns. The first column represents units, so a 0 here means no units, i.e. 0, and a 1 means 1 unit. The next column represents the numbers of 2s, so a 1 in this column means 2. The next column represents 4s and so on, with each column being twice as big as the previous one. This is also what we do in denary, each column being a factor of 10 bigger than the previous one. So the denary number 2902 can be interpreted as (2 x 1000) + (9 x 100) + (0 x 10) + (2 x 1). If you want to convert binary numbers to denary, this is a useful method. For instance, if I wanted to convert the numbers 1000100 and 11001 to denary I would make a set of columns as shown.

As I mentioned earlier, a computer functions by manipulating 1s and 0s. As you have seen, you can represent any denary number in binary. It is also possible to represent any letter of the alphabet, or other character, using binary by simply assigning a code to it in the computer. For instance, there is an agreed representation of text known as International Alphabet Number 5 (IA-5) in which the letter ?A? is represented by the binary pattern 1000001. When I type the letter A, this binary number will be stored in my computer. I can later retrieve it and the letter A will be displayed on screen. This will only happen if the computer has received instructions to treat 1000001 as an IA-5 character. The same pattern could be used to represent the denary number 65. The computer knows what to do with the data because it has instructions from a program, and these instructions are themselves binary representations.

It is worth examining the difference between data and instructions. The data is the current information the computer program is working with. This might be some numbers I am adding up, or some text I am typing. It will vary from instance to instance. The instructions are what the computer does with the data. This must always be consistent, for example clicking on the Save button will always save the data.

So numbers and text can be represented using the binary system. What else can? Images can be represented using a technique known as bit-mapping. This divides an image up into thousands of cells and allocates a value to each cell. If the image is in black and white, each cell will have a value of 1 (indicating it is black) or 0 (indicating it is white). Colour can be represented by allocating more information to each cell to indicate the proportion of red, green and blue (RGB) values. A wide spectrum of colours can be created by varying the relative values of red, green and blue.

What else can be represented in binary? The answer is just about anything. Sound, like images, can be divided up into different segments and each given an appropriate binary value, which can then faithfully reproduce the sound. This is what your music CD player does.

You will hear people talk about computers being ?digital?. Sound, light and other natural signals are usually analogue. The difference between digital and analogue is an important one as it underlies the advantage in using computers for many tasks. You will learn more about what is meant by analogue and digital in the next section.

So computers work by manipulating 1s and 0s. These are binary digits, or bits for short. Single bits are too small to be much use, so they are grouped together into units of 8 bits. Each 8-bit unit is called a byte. A byte is the basic unit which is passed around the computer, often in groups. Because of this the number 8 and its multiples have become important in computing. You will particularly encounter the numbers 8, 16, 32 and 64 in various contexts in computing literature, and this is usually due to the 8-bit byte being the basic building unit. The key point to appreciate is that although basing your entire system on only two digits may seem limiting, these two digits can be used to represent almost anything.

You will also hear people speak of kilobytes, megabytes and gigabytes or often just ?K?, ?meg? and ?gig? as in, ?This computer has 64 megs of RAM?, or ?This file is 45 K?. Bits, bytes, kilobytes and megabytes are merely ways of measuring the size of things computers deal with. A kilobyte is 2 to the power of 10 bytes. This is actually 1024 bytes, but is close enough to a thousand to be given the prefix kilo, meaning a thousand. Similarly, a megabyte is 2 to the power of 20 (or 1 kilobyte squared), which comes out as 1,048,576 bytes. For the sake of convenience, this is called a megabyte, meaning a million bytes. A gigabyte is 1000 megabytes.

Denary number	Binary equivalent
0	0
1	1
2	10
3	11
4	100
5	101
6	110
7	111
8	1000
9	1001
10	1010

Labspace remix test1

2: Binary, bits and bytes (T171_PC)

Denary number Binary equivalent 0 0 1 1 2 10 3 11 4 100 5 101 6 110 7 111 8 1000 9 1001 10 1010

Denary number Binary equivalent
0 0
1 1
2 10
3 11
4 100
5 101
6 110
7 111
8 1000
9 1001
10 1010