Binary to Decimal conversion of a number having fractional part

In this tutorial we will learn binary to decimal conversion for a number with fractional part.

Before we dive into the main topic lets talk a little about Decimal and Binary Number System that we are going to work with in this tutorial.

A decimal number system consists of 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. So, any number that we use in our daily life is actually in decimal number system.

A binary number system consists of only 2 digits: 0 and 1. And it is most commmonly used in computers.

How to convert a binary number with fractional part into decimal number?

To convert a binary number having fractional part into decimal form we have to multiply the
tenths position by 2^-1
hundredths position by 2^-2
and so on…

Convert binary number 0.001_{(base 2)} into decimal form

The following table shows the places, binary number and the multipliers for the corresponding places.

= 0x20  +  0x2-1  +  0x2-2  +  1x2-3
= 0 + 0 + 0 + 0.125
= 0.125

So, the required decimal number is
0.001_{(base 2)} = 0.125_{(base 10)}
Alternatively, (0.001)₂ = (0.125)₁₀
Where, (base 10) means the number is in decimal number system and (base 2) means the number is in binary number system.

Convert binary number 1010.00101_{(base 2)} into decimal form

To convert a binary number having integer and fractional part into decimal form we have to multiply the integer part
ones place with 2⁰
tens place with 2¹
hundreds place with 2²
ans so on...
and the fractional part
tenths position by 2^-1
hundredths position by 2^-2
and so on...

The following table shows the places, binary number and the multipliers for the corresponding places.

= 1x23  + 0x22  +  1x21  +  0x20  +  0x2-1  +  0x2-2  +  1x2-3  +  0x2-4  +  1x2-5
= 8 + 0 + 2 + 0 + 0 + 0 + 0.125 + 0 + 0.03125
= 10.15625

So, the required decimal number is
1010.00101_{(base 2)} = 10.15625_{(base 10)}
Alternatively, (1010.00101)₂ = (10.15625)₁₀
or, (1010.00101)₂ = (10.16)₁₀ (approx. value)
Where, (base 10) means the number is in decimal number system and (base 2) means the number is in binary number system.