Introduction: What is Positional Notation?
The act of counting things, and, by extension, symbols to represent quantities, are almost as old as
civilization itself. The earliest system, developed over six thousand years ago in Sumer, used small clay
tokens, which were later traded for stampings made in a tablet using those tokens, to represent different
quantities of something owned, typically livestock. This innovation was instrumental to life, and led to not
only mathematics, but also writing, economics, and, through them, virtually every facet of modern civilization.
However, this system, and many that came after it, were limited in their usefulness. Although such an additive
system works well on the small scale, it lacks any representation for zero, and as the number of tallies
increases, using this numerical representation becomes increasingly time- and space-consuming. The system also
lacks means for representing partial items, what we would call fractions.
Eventually, and through the rise and fall of many systems, a solution arose: a numerical representation in
which an individual symbol's value increased based on its placement relative to other symbols. This type of
system is known as Positional Notation, which is the type of system the world uses today. When you write "12,"
you know that the 1 is larger than the two because it is written first, and thus worth more.
Positional Notation was first used by the Babylonians, and has existed in many iterations before the systems
we commonly use now became ubiquitous. You can read more about the development of positional notation and some
of the different forms of it through history
here.
The system we are most familiar with is known as "Base-10," or "the decimal system". However, this system, though convenient, is not the only one present in our lives, nor is it
universally the best. You can learn about the advantages (and a couple drawbacks) of positional notation, and the
ways non-base-10 systems can be useful
here. Then, you can learn more about how these systems work
mathematically and how to work with them
here.