Binary to Decimal conversion of a number having fractional part

Conversion

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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.

placeonesDecimal
Point
tenthshundredthsthousandths
binary0.001
multiplier202-12-22-3

= 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.

placethousandshundredstensonesDecimal
Point
tenthshundredthsthousandthsten
thousandths
hundred
thousandths
binary1010.00101
multiplier232221202-12-22-32-42-5

= 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.