The binary numbers below all represent 2.5

• 101.0 x 2 ^ -1
• 0.101 x 2 ^ 2
• 10100.0 x 2 ^ -3

In order to store the above number in as few bits as possible we would have to store

• The left of the decimal point
• The right of the decimal point
• The exponent

This is wastefull. If we move the decimal point so it is always at the front we can cut down the amount of information which needs to be stored. This is known as normalising.

Consider the binary number 11.11 x 2 ^ 1 . (or 111.1 which is 7.5). If we write this as 0.1111 we would have to move the decimal point 2 places to the left (2^2). We then add the exponents together to get the new exponent.

0.1111 x 2 ^ 3

Finally we can then save the mantissa and exponent.

• mantissa - 01111
• Exponent - 011

Note - The mantissa and exponent always have a set size and are stored in 2's complement.

You can also get rid of trailing zeros. For example the number below

0.0011 001

This has 2 zeros which can be removed by moving the decimal point twice to the right. This will subtract 2 from the exponent so the new floating point number will be -

0.11000 111