Humans have the special gift of counting: I have five gold rings, and
you have a dozen red roses and together we'll run hand in hand jumping
two hundred and fifty six times through the park. These quantities can
be written down in some fashion using a series of single numbers or
digits: 5 gold rings, 12 red roses and 256 leaps. But what precisely do
the digits "1", "2", "5" and "6"
mean here?
The quantities zero through nine can be written as numbers 0, 1, 2, 3,
4, 5, 6, 7, 8, and 9. What then? We go back to zero and start with one
in the next column: 10. Then we start over counting back up through to nine
in our furthest right column and then once again the next column goes up
(is
incremented) by one and the rightmost column is back to
zero: 20, on to 99 then 100, and so on and on.
This iterative process of going up to nine, then back to zero while
incrementing the next-left column goes on indefinitely and we can
express any positive (i.e. above zero) number we can imagine. Each
column represents a number ten times as big as the last column: ten is
ten times bigger than one, a hundred is ten times bigger than ten, a
thousand is ten times bigger than a hundred, and so on.
Another way of looking at these columns is that each digit is that
amount multiplied by how much the column represents, be it one, ten,
hundred, or how ever much: 256 is (2 x 10 x 10) + (5 x 10) + 6.
Powers
Each of those columns which meant ten times ten times ... can be
expressed with an
exponent or
power, or how many times
the number in question is multiplied by itself: one hundred is ten times
ten, or ten raised to the power of two, or just ten to the power of two,
or even just ten to the two.
- 10 = 101
- 100 = 102
- 1000 = 103
And as a logical extension incidently, 1 is 10
0. In fact
anything
0 is one.
Base Ten
Humans currently
* use a numbering
system that's based on the number ten as we've seen. Why ten? Why not
nine, or eleven? The conventional wisdom seems to be that it's
because we have ten fingers & and thumbs, surely not by
coincidence collectively referred to as "digits". I'm not
aware of other theories although would be interested to
hear any!
Let's explore bases some more, seeing
what happens when we use a number other than ten...
Footnote
When I said "currently" I'm imagining that we as a species,
upon wide dissemination and global discussion of the information
contained in these pages, will move to a numbering system based on
twelve in the not too distant future :-)