What is the difference between a positional number system and unary numerical system?

Answer 1

In a unary numerical system, there is only one digit: 1. Any number larger than one is represented by the symbol 1 repeated the appropriate number of times. For example, 7 (decimal) = 1111111 (unary). It is the simplest numerical system to learn but becomes quite unmanageable. Using a 10 point font the biggest number that you could write on a normal page is approx 3584: the exact value depends on the margins. At that rate you would need approximately 5 billion (yes, BILLION) pages to write out the value of the US debt (in June 2014)! And the cost of all that paper will send it up some more!

By contrast, a positional system uses a base band b different digits. The place value of the digit immediately to the left of the "decimal" point is b0 = 1, the place value of the digit to its left is b1, then to its left is b2 and so on.

In the decimal system, the US debt is approx USD 17530000000000. It took only(!) 14 digits to write it out. In the hexadecimal (base 16) system, the same number can be represented by USD 414DE9280, requiring only 9 digits.