Secret Numbers

The symbols chosen for digits need not be the usual symbols. Here is a system for base four:

zero  == @     one == !     two == #      three == $

Here is a number represented in that system:     !$@#

Here is what that means in positional notation:

   !$@# == !×(!@)$ + $×(!@)# + @×(!@)! + #×(!@)@

The base in this system is written !@ which uses the digits of the system to mean one times the first power of the base plus zero times the zero^th power of the base.

This example illustrates that positional notation can be used without the usual digits 0, 1, 2, ..., 9. If the base is B, then B digit-like symbols are needed.