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There is a simpler explanation of number bases, along with interactive pages on binary , binary fractions , normalised floating-point binary , and hexadecimal in the Interactive section. There is also a number bases Abacus that you can use to experiment with different number bases and you can watch videos on number bases on the AdvancedICT YouTube channel. A German mathematician called Leopold Kronecker once said that "God gave us the integers, and all the rest is the work of man.
Without getting too philosophical, it helps to remember that, often, what we think of as numbers are actually symbols representing the number - just as the way you write your name isn't you, and you're still the same person if you type your name, write it in a different colour, or use a different alphabet. Number bases are different ways of writing and using the same number.
We use a system called base 10, or denary, for our arithmetic, but there are almost as many number bases as there are numbers. Many people think that we use base 10 because we have 10 fingers on which we can count. Computers, and other electronic devices, can only reliably use an electrical current, or the absence of a current, to count like having two fingers , and so they tend to use base 2 binary internally.
The important thing to remember is that number bases are just different ways of writing down numbers - just as Roman Numerals or tally charts are - but other than that the numbers behave as normal.
This doesn't mean that their arithmetic is fundamentally different - in fact, the way that the number bases behave is entirely consistent. You will be familiar with the idea of base 10, and the column headings you used to talk about at primary school - units, tens, hundreds, thousands, etc. They have columns, with headings, that are used to represent numbers, and different digits are placed in different positions, starting at the right. The base is usually written as a subscript after the number, so you can tell that 2 is 7 in binary, and not one hundred and eleven.
You can work in any number base except 1, which wouldn't really make sense , and some programming languages such as Lisp let you do that. In computing, however, you generally only come across the following four bases, and you know base 10 already. These common bases also have proper names, shown in parentheses:.
The largest digit you can have in any column is the one less than the number of the base. So for binary base 2 it's 1, then 7 for Octal base 8 , 9 for Denary base 10 , etc. When we write a number in base 10, we know its value because we multiply the individual digits by their corresponding column headings.
Other number bases work in exactly the same way. You probably won't encounter binary much these days, but it's useful to understand that this is how computers work internally, so you can understand concepts such as parallel transmission. The fact that computers use binary is why everything is a multiple of 2 - why computers come with 8Mb, 16Mb, 32Mb, 64Mb, etc. It's main use is probably in combination with the bitwise logic techniques shown on the previous page, to combine and split values stored in the same byte or word.
There are some other binary-related terms you'll need to know. Firstly, a bit is a binary digit - i. This is the smallest unit of storage you can have inside a computer. Groups of 8 bits are called bytes. A byte can be used to represent a number, or a colour, or a character e. You may also hear the term nibble , which is 4 bits. Finally, a word is the largest numbers of bits that a processor can handle in one go - for example, when we say that new computers have bit processors, we mean that the word length is bits, or 8 bytes.
The largest value that you can store using a particular numbers of bits can be determined quite easily. Using n bits, the largest value you can store is 2 n - 1 , and the number of different values you can store is 2 n from 1 to 2 n - 1 , and then 0 as well.
A bit computer can therefore handle values up to 4,,, in one clock cycle - it can obviously cope with larger numbers, but it would need to split them up first.
I've never come across anything that uses octal! I think it's probably included on exam specifications for purely academic reasons, and because it's easy to convert into binary see below. Hexadecimal is still used quite a lot - particularly for things like colours in HTML or programming languages.
It's also quite useful because representations of large numbers are relatively compact, but are easily converted to binary so that you can see the bit patterns. You've no doubt noticed that with numbers in base 10, you can move the digits left or right one place by multiplying or dividing the number by The same trick works with different number bases - you just multiply and divide by the base number e.
This can be useful for things like creating hexadecimal colour values e. The 24 bits are made up of 8 bits each for the amount of red, green and blue in the colour. So, each component is represented by 8 bits - i. If you know how much red, green and blue you want, how do you combine them to find the complete colour? We can leave the blue value as it is, but we need to move the green value along two places. Obviously, if you were just trying to work out the colour yourself, you wouldn't need to go through these steps, but if you were to create a program like my colour mixer , then this is how you'd do it.
You can convert from Octal to Binary and vice-versa by grouping this binary digits or bits into threes, and then changing them into their binary equivalents.
Finally, a number base joke! Why do programmers confuse Christmas with Halloween? Just combine the two digits: Just combine the two results: