Binary And Decimal Conversion | Binary To Decimal And Decimal To Binary Conversion
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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 in the first lecture.
n 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-nonzero 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.
Two methods are used to Convert Decimal fractions to Binary. The Sum-of-Weights
method, which has been described and used to convert Decimal Integers into Binary
Equivalents is applied to convert Decimal fractions into Binary fractions. This method requires
mental arithmetic and is suitable for small numbers. The conversion of Decimal fraction 0.625
into Binary equivalent is illustrated in a tabular form. Table 2.3
Sum Term Highest
Weight
Binary
Number
Sum Term
= Sum Term – Highest Weight
0.625 0.500 0.100 0.125
0.125 0.125 0.101 0
Table 2.3 Converting Decimal to Binary using Sum-of-Weights Method
Repeated Multiplication-by-2 Method
An alternate to the Sum-of-Weights method used to convert Decimal fractions to
equivalent Binary fractions is the repeated multiplication by 2 method.