How does binary work?

Like the decimal system that we all use on a daily basis, binary notation is just another way of writing down numbers. To get the idea of how it works across, I'll start off by briefly explaining how the decimal number system works: 

How to read decimal numbers

31

The number above is 'thirty-one'. You know that because you've been brought up to count using the decimal number system. We, as a developed society, use it to a large extent not only because there are some nice, easy to remember patterns that make themselves present with a decimal number system, but because that's what our parents used, and their parents, and their parents...

How does it work?
The number system we're familiar with uses ten 'bases' - the symbols 1, 2, 3, 4, 5, 6, 7, 8, 9 and 0 - displayed in a place value system. That means that where we put them in relation to each other is important. We know that '31' means 'thirty-one' because written like that the 3 represents 'three tens' and the 1 represents 'one unit'. Put together, three tens and one unit make thirty one (of whatever we're counting) altogether.

We can represent ever bigger numbers by adding in more columns to the left- the next one, for example, allows us to say how many hundreds we want (0 - 9), the one after that describes how many thousands there are, and so on for ever and ever and ever, if we wanted to.

That's a crash course in the decimal "base ten" number system- remember that's the one you use every day. On to binary.

How to read binary numbers

11111

The number above is also 'thirty-one', or rather it represents the same amount as the number 31 does in the decimal number system.

How does it work?
The binary system works in a similar way to the decimal one, except it uses only two 'bases': 0 and 1 (i.e. there are no 2s, 3s, 4s, 5s, etc). Instead of each digit telling us how many 'ones', 'tens', 'hundreds', etc are in the number, the columns are labelled in a different way. The first column (starting from the right) is still 'units' (or 'ones'), but the next one to the left now represents how many twos we want in our number. The one after that is how many fours, and then it's how many eights. Can you see the pattern? Each column (as we move to the left) represents double the value of the one before it, so:

11111 means:

One 'sixteen', one 'eight', one 'four', one 'two' and one 'one'. Added together this makes thirty-one.

Another example, on binary day:
Today's date is the tenth of October, 2010, or 10 10 10, and is being called 'binary day' for fairly obvious reasons.

so as a binary number, 101010 means:

One 'thirty-two', no 'sixteen', one 'eight', no 'four', one 'two' and no 'one'. Add the thirty-two, eight and two together and you get:

42