Introducing number bases

A ‘number base’ is, essentially, a group of digits that we use for counting and arithmetic. The number base we usually use for counting is 10, which is probably because we have ten fingers. We call this the decimal (or base ten) counting system and it’s used in most of the world.

Base ten

Theoretically, we could continue having a new symbol for each new number. Some number bases higher than ten – for instance base 16 (hexadecimal) use letters of the alphabet for the next digits. However, as you can imagine, it would rapidly become quite difficult to remember all the different symbols if we had a new one for every number. Arithmetic would be a nightmare.

So the inventor of the decimal system decided that we would count in groups of ten. When we pass the final digit, 9, instead of using yet another symbol, we write the next number as 10. That means one group of ten and nothing else.  The next number is written as 11 meaning one lot of ten and one more. 12 is one group of ten and two more, and so on.

When we reach the end of the digits again, ie 19, we recall that this is one group of 10 and an extra 9. Thus the next number will be two groups of ten, so we write it as 20. Two tens and an extra 1 is 21, three tens is written 30, and so on. This logic continues using the same ten digits right up to 99 which is nine groups of ten with nine over. After that we have to make a new unit, ie a hundred, which of course we write as 100.

If that sounds confusing, you might want to read the page about place value. s a reminder, the number 1 means ‘one’ when it’s on its own.  But when it’s written as 10 it means ‘ten’ (which is ten ones) and when it’s written as 100 it means ‘a hundred’ (which is ten tens).

Other number bases

Now imagine we only have 9 fingers, or that the symbol 9 doesn’t exist.  Instead of counting in tens we would count in nines.  That  means we’d have:  1, 2, 3, 4, 5, 6, 7, 8..  and then the symbol 10 would represent nine, not what we call ten. 10 would be one lot of nine, and none over.  So what we would call ‘ten’ would be  represented in base 9 as 11 – ie one lot of nine and one over.  Note that when using bases other than ten, it’s usual to talk about the symbol 11 as ‘one-one’ rather than ‘eleven’, to save confusion.

As you can probably imagine, we can choose any number as the ‘base’.  In base 6, for instance, the number we know of as 6 would be represented as 10 (one-zero).  The base two (or binary) system used in computers has only two digits:  0 and 1.  10 represents two, 11 represents three, 100 represents four, and so on. You can read about this in more detail on the page about the binary system.

If the base is higher than ten, letters of the alphabet are typically used.  Base sixteen (also known as hexadecimal, another system used by computers) uses the letters A for eleven, B for twelve, C for thirteen, D for fourteen and E for fifteen.  The number written as 10 in hexadecimal is sixteen.

Arithmetic using number bases

Arithmetic in other bases can be laid out much as regular arithmetic in base ten, but you must always remember which base you are in.  3+3=6 in all bases from seven upwards.  But in base six, where 10 represents six, 3+3=10.  We usually write this with a little superscript showing what base we are in, to avoid confusion.  In other words, 36 + 36 = 106.  In base five, 35 + 35 >= 115. (one lot of five, and one over).

For more sites looking at arithmetic in different number bases, see: NRich on number bases or  Purplemath: base 4 and base 7.

Number bases in everyday life

Understanding different number bases may seem esoteric and unnecessary to anyone not involved in programming computers. But if you consider for a moment, we do use other number bases in ordinary life. An hour, for instance, is divided into 60 minutes.  We don’t use extra letters for telling the time – we would need far too many! –  but we do understand that  120 minutes is two hours, 180 minutes is three hours, and so on.

Until the 1970s when the UK adopted the metric system, we used to use various number bases for weighing and measuring, too.  In the USA, this still happens.  Our parents and grandparents were well aware that twelve inches was a foot, and sixteen ounces was a pound, and fourteen pounds one stone, and so on.   You can sometimes still find these kinds of measurements in older cookery books.   They seem very complex to us now, used as we are to having everything in tens, hundreds and thousands in the metric system, but prior to 1971, people were competent at figuring out complex arithmetic using the old ‘imperial’ system.

Number bases in the imperial system

Our grandparents might have had to answer questions such as:  ‘if you have one plank of wood that measures three feet and four inches, and another measuring two feet and nine inches, how much will they measure when placed end to end?’   The logical answer might seem to be five feet and thirteen inches, thinking in base ten.  But because inches work in base twelve, thirteen inches is actually one foot and one inch.  So the answer would be six foot and one inch.

Today, we might ask a question such as:  ‘if this DVD is two hours and forty-five minutes long, and the other one is one hour and fifty minutes, how much time would it take to watch them both, one after another?’

Again, we might think that the answer is three hours and ninety-five minutes, adding hours and minutes separately.  But minutes work in base 60, so we must remember that ninety-five minutes is one hour and thirty-five minutes, so the correct answer is four hours and thirty-five minutes.

Other maths pages you might find useful:

Understanding angles
Simultaneous equations
Prime numbers and factors