# Binary Number System: Numbers with a Fractional Part

## Binary Number System: Numbers with a Fractional Part

#### Understanding Fractional Binary Numbers

• Fractional binary numbers represent values that exist between whole numbers. They’re similar to decimal fractions but use a binary number system instead.
• A binary fraction is placed to the right of the binary point. For example, in the binary number ‘101.01’, ‘.01’ is the fractional part.
• Each binary fraction represents a ‘portion’ of 2, such as 1/2 (0.5 in decimal) for the first digit after the binary point, 1/4 (0.25 in decimal) for the second digit, and so on.

#### Binary to Decimal Conversion of Fractional Numbers

• Converting a fractional binary number to a decimal involves multiplying each digit after the point by 2 raised to the negative power of its position. For instance, in ‘.101’, the first ‘1’ is multiplied by 2^(-1), and the second ‘0’ by 2^(-2).
• The sum of these calculations provides the decimal equivalent of the fractional binary number.

#### Decimal to Binary Conversion of Fractional Numbers

• To convert a decimal fraction to binary, multiply the fraction by 2 and record the integer part. Repeat the process with the remaining fractional part until it becomes zero or until a desirable level of precision is reached.
• The binary representation is then composed of the integer parts obtained in each step, from top to bottom.

#### Fractional Binary Numbers in Computing

• Fractional binary numbers have a critical role in computing and digital systems. They enable the representation and operations of non-integer values in binary form.
• Common applications include graphics programming, digital signal processing, and scientific computations where precision and fractional values are frequently used.
• It is crucial to consider the precision while dealing with fractional binary numbers as binary fractions may not accurately represent all decimal fractions, which can lead to rounding errors.