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Introduction

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In the end, everything comes down to maths! Even when your computer is playing music or viewing a web site, it's doing maths.

Normally we use a base-10 system, also called a decimal or denary system, to represent numbers. This uses the ten digits 0 to 9, with the value of each digit indicated by its place value. So the number 369 represents 300 + 60 + 9. The values of the digits increase by powers of 10 (the base).

The CPUs of computers execute instructions by performing millions of calculations per second using transistors operating as switches. These switches have only two states: they are either off or on. They either transmit an electric current or they do not.
Computers therefore use a base-2 or binary system using the two states to represent the digits 0 and 1. These are called bits (short for binary digits). And that's all there is to it! All program instructions and data are, fundamentally, long strings of 1s and 0s.

As we are not familiar with talking about and remembering large binary numbers, and it's important to know what's happening inside the computer, it's obviously very useful to be able to convert between the two systems.

We also have to be able to do binary maths, such as adding, multiplying and dividing binary numbers; there's a special trick for multiplying or dividing by powers of two.

As humans find it very difficult to remember large binary numbers, computer scientists also use the hexadecimal system: a base-16 number system using 16 digits with values from 0 to 15. The digits 0 to 9 are used as normal; the values 10 to 15 are represented by the letters A, B, C, D, E and F.

The hexadecimal system is used just for our benefit – computers don't use it, they're quite happy with binary!
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