3. Binary Representation of Floating Point Numbers

Splitting the Number

To convert a floating point number into binary representation, splitting the number into integer and fractional parts should be the first step.

x = 13.375

We can get the integer part easily as follows:

integer_part = int(x)

Extracting the fractional part can be tricky for newbies:

fractional_part = x - int(x)

Converting the Integer Part

You can convert an integer into binary following the steps:

• Divide the integer by 2
• Get the remainder
• Repeat until the quotient is 0
• Reverse the order of the remainders

def convert_integer_to_binary(n):
    if n == 0:
        return "0"
    s = ""
    while n > 0:
        s += str(n % 2)
        n //= 2
    return s[::-1]

Converting the Fractional Part

You can convert an integer into binary following the steps:

• Multiply the fraction by 2
• Get the integer part of the result
• Continue with the fractional part of the result
• Repeat while the fractional part is not 0
• Use the integers in the order they were obtained

def convert_fractional_to_binary(n):
    if n == 0:
        return "0"
    s = ""
    while n > 0:
        n *= 2
        s += str(int(n))
        n -= int(n)
    return s

Combine the Parts to get the Binary Representation

Once you have the both integer and fractional parts, you can concentrate two parts and truncate or pad the result with zeros to get the desired precision.

x = 13.375
integer_part = int(x)
fractional_part = x - int(x)
binary_number = (
    convert_integer_to_binary(integer_part)
    + "."
    + convert_fractional_to_binary(fractional_part)
)

Note

Keep in mind that these steps can be applied to conversion from decimal to any base. It doesn't have to be base-2, so called binary.

Reference

NUMERICALCULATOR