11. Converting to Normalised form

Example

A binary number is presented in this format, a 5 bit mantissa and 3 bit exponent

00111 011

Q: Is this in normalised form ?

Answer: No, because the first two bits are the same sign.

Q: Convert this into normalised form

Answer:

You want the first two bits to change sign and yet it must be the same number, so you must do two things.

a) Shift the mantissa to the left until you see a 01 pattern, pad out the right with zeros as you do so

b) For every shift to the left, decrease the exponent by 1.

Doing this will force the binary point to stay in the same place.

In its original form the number is 0.0111 with exponent 3, so the full number is

11.1

which is +3.5

Now if you normalise it, you shift the mantissa once to the left and becomes 0.1110 then decrease the exponent by one so 011 becomes 010

Expanding this to its full number again we get

11.10

Note the greater precision because there are now 2 bits after the binary point