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 |
---|---|---|---|---|
10^{4} | 10^{3} | 10^{2} | 10^{1} | 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 |
---|---|---|---|---|
8^{4} | 8^{3} | 8^{2} | 8^{1} | 1s |