Numbers and Notations (Preview)

Number is a concept. To talk about, record, and work with numbers, we use representations of numbers, called a notation. In everyday life, we use the notation based on decimal digits (0 – 9). But the decimal system is only one of many possible notations that can be used to represent numbers. Another notation is the binary system that TCP/IP hosts use in which 0s and 1s are used.

 

Positional Notation

In positional notation digits have a special meaning based on the position the digit occupies in the sequence of numbers.

Figure Decimal Positional Notation

 

Radix: The number base.

Position in #: The position in the sequence that is occupied by the digit, from right to left. If you look at the second row in the above table, starting with, from right to left, 0 (1st position), 1 (2nd position), 2 (3rd position), 3 (4th position), 4 (5th position), …. n [(n-1) position]. Also these numbers represent the exponential values that will be used to calculate the positional value in the 4th row.

Calculate: This is how we calculate the positional value. The calculation is performed by (radix) ^ (Position in #).

Positional Value: This is the value of the position of a digit in a number series. For example, in the figure above, if a digit occupies the third position then it’s in place value would be 1 x (10 ^ 2) = 100.

 

Note: n ^ 0 is always 1.

 

Applying the Decimal Positional Notation

Let’s look at the positional notation in action with respect to decimal number 1329.

 

Steps

  1. Firstly, match a given number to its positional value. In our example, starting with, from right to left, 9 (1st position), 2 (2nd position), 3 (3rd position), 1 (4th position).
  2. Then calculate the in place value. (9 x 1), (2 x 10), (3 x 100), (1 x 1000)
  3. Finally Add them up. 1000 + 300 + 20 + 9 = 1329.

Figure Applying the Decimal Positional Notation

The result is 1000 + 300 + 20 + 9 = 1329.

 

Binary Positional Notation

This is the language of TCP/IP hosts.

Figure Binary Positional Notation

 

 

Applying the Binary Positional Notation

In this example, we are going to look at how 10001010 corresponds to 138 in positional notation when using the binary numbering system.

The steps are as same as with applying decimal positional notation above.

 

Figure Applying the Binary Positional Notation

 

The result is 128 + 0 + 0 + 0 + 8 + 0 + 2 + 0 = 138

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