Computer Science

Floating point binary to decimal

Floating point binary is a method that allows for hte storage of very large and very small numbers by taking advantage of a system very similar to scientific notation where a number is stored in two parts.

First a mantissa which is a number between -1 and 1 and second an exponent which tells us by what power of 2 we need to multiply that to form our number.

Example

For our example let's consider an 8-bit floating point number that stores the mantissa in 4 bits and the exponent in 4 bits both of them in two's compliment. We will convert the binary number 10010101 to denary.

This means the number is -0.875 x 2^5 = -28

If we change the number of bits allocated to the mantissa and the exponent we affect the range and accuracy with which we can represent numbers. The more bits we use for the exponent the more greater the range of numbers we can represent. The more bits we use for the mantissa the ore accurately we can represent fractional numbers.

With 4 bits for each of mantissa and exponent the smallest number we can make is -1 x 2^7 = -128 and the biggest number is 0.875 x 2^7 = 112.

If we use 5 bits for the mantissa and 3 bits for the exponent the smallest number we can make is -1 x 2^3 = -8 and the biggest number is 0.875 x 2^3 = 7.

It is also possible to calculate the denary number by putting the mantissa out in two's compliment fixed point binary and shifting the bits according to the exponent. To the left if the exponent is positive and to the right if the exponent is negative.

Let's see that with our example of 10010101.

The exponent is plus 5 so we need to move the number 5 places to the left.

Reading off the shifted number you can see we still get -28.

8-bit two's compliment floating point binary with 4 bits for the mantissa and 4 the exponent to denary practice.

Click the button to generate a binary number to convert