As explained on the page about number bases, we usually use ten digits for counting. This is probably because we have ten fingers. We call this the decimal counting system, and is the one used in most countries for money. We also commonly use it for measuring (unless you are in a country that does not yet use the metric system), and general arithmetic. But if there were a different number of digits, we could still do calculations. One alternative, much used in technology, is the binary system.
In the decimal system, and indeed every number base, the symbol 1 refers to a single unit. In the everyday decimal system, the symbol 10 – a one followed by a zero – refers to a ‘group’ of ten and no extra units. Likewise, the symbol 100 (one hundred) refers to ten ‘groups’ of ten. This is so ingrained in our understanding that it seems obvious.
Now imagine we want to count using not our fingers, but our hands. Our number base would then be two rather than ten. Base two is known as binary. In the binary system, there are only two digits – one and zero. That might sound very straightforward, but in practice it leads to long strings of ones and zeroes.
Using the binary system
Number one is, as always, represented by the symbol 1. But then there are no more digits other than zero. If the base is two, then the number we know of as ‘two’ has to be represented as 10 – meaning one group of two, and none over.
Then, clearly, ‘three’ is represented 11 – ie one two and one extra. And by the time we reach the number ‘four’, normally represented as 4, we have to move to three symbols in the binary system: we show it as 100.
Thinking back briefly to the decimal system, you will recall that each time a new zero is placed at the end of a sequence of numbers, it refers to a multiple of ten. So 1,000 is 100 x 10 and 1,000,000 is 100,000 x 10.
In base two, similarly, each new zero represents a multiple of two. As just shown, four (which we know as 2 x 2) is represented as 100 in binary. The next multiple of two is eight, so that will be represented as 1000 in binary. To check that this makes sense, here is a list of the first eight decimal numbers and their binary equivalents:
Converting from decimal to binary
If you have any decimal number – for instance 27 – and need to convert it to binary without using a computer, the easiest method is to think in terms of powers of 2 in the decimal system. As shown, 8 is represented as 1000, so 16 will be 10000, and 32 will be 100000. 27 falls between 16 and 32, so it will be a five digit number in binary.
When 16 is subtracted from 27, the remainer is (in decimal notation) 11. So now 11 must be turned into binary, which is a little confusing since 11 could itself refer to the binary representation of 3. It is crucial to keep track of which system you are using! The highest power of two in eleven is eight; subtracting that from eleven leaves three, and we know that three is two plus one.
So the decimal number 27 can be broken down into sixteen plus eight plus two plus one, or, including all the relevant powers of two:
(1 x 16) + (1 x 8) + (0 x 4) + (1 x 2) + (1 x 1)
Thus it is represented in binary as 11011.
Converting from binary to decimal
This is rather more straightforward, and is simply the reverse of the previous process. If presented with the binary number 1011010, for instance, recall that each of the 1s or 0s refers to a power of two. The final digit is zero, so there are no units – it must be an even number. The penultimate digit is a 1, so that represents the decimal two.
Taking the given binary number from left to right, it represents:
(1 x 64) + (0 x 32) + (1 x 16) + (1 x 8) + (0 x 4) + (1 x 2) + (0 x 1)
in other words: 64 + 16 + 8 + 2 = 90.
Addition in binary is as simple as knowing that 1 + 1 = 10, 10 + 10 = 100, and so on. You must understand place value, and be very careful not to miss any of the 1s or 0s, but the actual computation could hardly be easier. This is why early computers functioned in binary: 1 was represented by a lightbulb or circuit being ‘on’ and 0 represented by ‘off’.
Other basic maths for young children: