 Crossing the boundary - analogue universe, digital worlds

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# 3.7 How we work with numbers

Most civilisations have had to face the problem of counting and recording numbers. Our own culture has adopted the so-called Arabic system of numbers. This system is now used more or less worldwide. In this section I will look very briefly at some of its key features.

We have an infinity of numbers at our disposal. If we start counting from 1, we can in theory go on for ever. But although there is an infinity of numbers, we only have a very small, fixed number of digits to play with – the figures 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. This is no accident: we have ten fingers. But how, then, do we get beyond nine? This is something we learned at school. We represent the number after 9 as

10

which we can read as one group of ten plus zero, in other words ten. We create a new column to the left and use it to count groups of ten. So now we can represent numbers above ten, such as

37

which is three groups of ten (thirty) plus seven, i.e. thirty-seven.

Obviously, two columns only take us as far as ninety-nine, after which we run out of digits again. But we can reuse the same idea: we create a new column to the left of our tens column; this column counts groups of a hundred, so

345

is three groups of one hundred (three hundred), plus four groups of ten (forty) plus five: three hundred and forty-five.

We can add a new column to the left every time we run out of columns and digits. So

4621

is four groups of one thousand, plus six groups of one hundred, plus two groups of ten plus one.

By now you should see a pattern starting to emerge. Each column counts groups that are ten times bigger than the groups counted by the column immediately to its right, with the rightmost column counting ones. This can be represented as follows.

Groups of 10000    Groups of 1000    Groups of 100    Groups of 10    Ones
10×10×10×10×1 10×10×10×1 10×10×1 10×1 1s

Put slightly more mathematically, the following version tells exactly the same story.

Groups of 10000    Groups of 1000    Groups of 100    Groups of 10    Ones
104 103 102 101 1s

The fact that our way of doing arithmetic

• uses ten digits (0 to 9); and

• each column counts groups ten times bigger than those counted in the column to its right;

leads to it being called a base 10 arithmetic, or a decimal system, from the Latin decima meaning ‘a tenth’.

Using base 10, we can count to, and write down, any number we want.

## Exercise 3

Computer scientists sometimes use an octal (base 8) system? What digits would we need for that and what would the columns represent?

### Discussion

To start with, we need only the first eight digits, 0 to 7, so we can discard 8 and 9.

The first column will count units as before. Each new column will count groups eight times the size of the groups counted by the column immediately to its right. So the table would look like this:

Groups of 4096    Groups of 512    Groups of 64    Groups of 8    Ones
84 83 82 81 1s
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