Binary to decimal conversion

Most real world quantities are represented in Decimal Number System. Digital Systems on the other hand are based on the Binary Number System.

Therefore, when converting from the Digital Domain to the real-world, Binary numbers have to be represented in terms of their Decimal equivalents.

The method used to convert from Binary to Decimal is the Sum-of-Weights method. The Sum-of-Weights method has been used to represent the Caveman numbers ∆↑, ∆Ω↑∑ and the Binary numbers 10011 and 1011.101.

Sum-of-Weights Method

Sum-of-weights as the name indicates sums the weights of the Binary Digits (bits) of a Binary Number which is to be represented in Decimal. The Sum-of-Weights method can be used to convert a Binary number of any magnitude to its equivalent Decimal representation.

In the Sum-of-Weights method an extended expression is written in terms of the Binary Base Number 2 and the weights of the Binary number to be converted.

The weights correspond to each of the binary bits which are multiplied by the corresponding binary value. Binary bits having the value 0 do not contribute any value towards the final sum expression.

The Binary number 101102  is therefore written in the form of an expression having weights 2⁰, 2¹, 2², 2³ and 2⁴ corresponding to the bits 0, 1, 1, 0 and 1 respectively. Weights 2⁰ and 2³ do not contribute in the final sum as the binary bits corresponding to these weights have the value 0.

101102         = 1 x 2⁴ + 0 x 2³ + 1 x 2² + 1 x 2¹ + 0 x 2⁰

= 16 + 0 + 4 + 2 + 0

= 22

Sum-of-non-zero terms

In the Sum-of-Weights method, the Binary bits 0 do not contribute towards the final sum representing the decimal equivalent. Secondly, the weight of each binary bit increases by a factor of 2 starting with a weight of 1 for the least significant bit. For example, the Binary number 101102 has weights 2⁰=1, 2¹=2, 2²=4, 2³=8 and 2⁴=16 corresponding to the bits 0, 1, 1, 0 and 1 respectively.

The Sum-of-non-zero terms method is a quicker method to determine decimal equivalents of binary numbers without resorting to writing an expression.

In the Sum-of-non- zero terms method the weights of non-zero binary bits are summed, as the weights of zero binary bits do not contribute towards the final sum representing the decimal equivalent.

The weights of binary bits starting from the right most least significant bit is 1, The next significant bit on the left has the weight 2, followed by 4, 8, 16, 32 etc. corresponding to higher significant bits.

In binary number system the weights of successive bits increase by an order of 2 towards the left side and decrease by an order of 2 towards the right side.

Thus a binary number can be quickly converted into its decimal equivalent by adding weights of non-zero terms which increase by a factor of 2.

Binary numbers having an integer and a fraction part can similarly be converted into their decimal equivalents by applying the same method.

A quicker method is to add the weights of non-zero terms. Thus for the numbers

100112 = 16 + 2 + 1 = 19

1011.1012 = 8 + 2 + 1 + ½ + 1/8 = 11 + 5/8 = 11.625