Number systems have been around for thousands of years. You can see the remnants of several systems in our existing civilization. The common system is the existing system based on ten. However, our clocks have twelve hours, sixty minutes and sixty seconds. These are remnants of a Babylonian system that counted by sixties. I'm not sure where the division into 12 hours came from; however, 60 = 5 x 12, so perhaps it has roots here. Then again, maybe it came from counting the spaces in between and around the fingers? Our units of angular measurement (degrees and seconds) are also based on 60, so we have 360 degrees in a circle : 360 = 6 x 60. There are hints here of a relation to the year, since we have 365 days in a year. We also use the amount of a dozen, and there are 12 inches to a foot, and so on. Our existing methods of counting are indeed intermingled with the past, and more extensive documentation of this history can be found in books on the history of mathematics, so I won't dwell on this.
When we speak of 10, 12 and 60, these are called "bases" by mathematicians. The base 10 is the most common probably because we have 10 fingers (including the thumb as a finger). So intuitively, most ancients learned to count things in groups of ten. So the first step in the development of a number system would be to introduce a different symbol for each finger (or digit). So somewhere in the past, they agreed to use the Arabic symbols 0,1,2,3,4,5,6,7,8 and 9. This is slightly illogical, since the most logical is to introduce another symbol for "ten" and omit zero altogether, however this is another story that I address in an Alternate Number System which you can read about elsewhere on these pages.
Anyway, the first step was to select ten symbols. Then, when the counting
went above ten, the options were either to introduce more symbols, or implement
what mathematicians call a "positional number system". Such a system is
formed by grouping by tens in the case of base ten. So finally, we ended
up with numbers such as 1234. The system is called positional, because
the position (order) of the digits is important. When we "expand"
the number 1234, it becomes :
1 x 1000 + 2 x 100 + 3 x 10 + 4
So we have two systems at work here : A symbolic system of digits, and
the positional number system itself. This is a base ten positional number
system since it is based on powers of ten.
There is no reason why other bases cannot be used, and in the
world of computing, we find the use of base 2 (binary) and also bases 8 (octal)
and 16 (hexadecimal) which are derived from base 2.
In the hexadecimal system, mathematicians have introduced alphabetic symbols
for the digits ten = A, eleven = B, and so on up to fifteen = F. Once again,
it may be more logical to introduce a sixteenth symbol (G?) in base sixteen,
however that was not done, and we retained zero.
So in mathematical terms, we can generalize this idea as follows. Let b
be the base of the system, and let a(j) be the n digits of any number such that
j goes from 1 to n. So in the symbolism of mathematics, the simple
idea of a number system is represented by:
SUM (a(j)*b^(j-1))
where the SUM is over j, and 0 <= a(j) <= (b-1)
which simply means that valid digits must be between 0 and (b-1) inclusive.
So there you have it. This describes the existing positional number system. If you have followed this, then congratulations. When you have absorbed most of this, you will be ready to read the next page: An Alternate Number System.
Enjoy.