Positional Number System

A positional Number system is a system which represents numbers by an ordered set of numerals symbols called digits,  in which the value of numbers depends on its position. So This system gives a different meaning to the same symbol depending on its position.  Our daily life number system, decimal numbering system (base 10) is an example of a Positional Number System. The position in which the digit appears affects the value of that digit.

For example, the number 4321 has a very different value than the number 1234; although the same digits are used in both numbers. In a positional number system, the value of each digit is determined by which place it appears in the full number. The lowest place value is the rightmost position, and each successive position to the left has a higher place value. So, the rightmost position represents the "ones" column, the next position represents the "tens" column, the next position represents "hundreds" and the next position represents the “thousand” etc. Therefore, the number 4321 represents 4 thousand 3 hundred and 2 tens and 1 one, whereas the number 1234 represents 1 thousand 2 hundred and 3 tens and 4 ones.  Any “base” system, base 2 (binary), octal (base 8) and hexadecimal (base 16) is an example of a Positional Number System.

Decimal Positional Notation

The decimal positional notation system work as described in Figure 1. Description for each row is given below the table. To use the positional system, match a given number to its positional value. The example in Figure 2 and figure 3 illustrates how positional notation is used with the decimal number 1234 and 4321.

Radix

The identifies the number base.  The decimal notation system is based on 10, therefore the radix is 10.

Position in

This describes the position of the decimal number. The position in is starting from right to left. So 0 is the 1st position 1 is the 2nd position, 2 is the 3rd position, 3 is the 4th position and so on. These numbers also represent the exponential value that will be used to calculate the positional value of the 4th row.

Calculate

The 3rd row is used to calculates the positional value by taking the radix and raising it by the exponential value of its position. Important is that n⁰ is always = 1.

Position Value

The first row of the table shows the number base or radix. So the value listed, from left to right, represents units of thousands, hundreds, tens, and ones.

Binary Positional Notation

Similarly, the binary positional notation operates as shown in figure 4. Binary has also a Radix, Position, which is from right to left; Calculation and positional value as describe for decimal.The Radix for Binary is 2. The Figure 5 illustrates how a binary number 11101001 convert to the decimal number 233.