Some facts

The mantissa represents a positive integer between 0 and

> convert(111111111111111111111111,decimal,binary);

The exponent varies between 0 and

> convert(1111111,decimal,binary);

This representation is not unique. Dividing the Mantissa by 16 and adding 1 to the exponent will yield the same number. The effect of such a division on the binary notation of the mantissa is to shift it four places to the right:

> Mantissa/16;

> convert(%,binary);

To obtain a unique representation, we demand that the 24 bit mantissa has a 1 in the four leftmost bits.

The smallest number that can be represented is then given by 0 0000000 000100000000000000000000, or

> Mantissa:=convert(000100000000000000000000,decimal,binary);

> number:=evalf(Mantissa*16^(-64));

The largest numer is then given by 0 1111111 111111111111111111111111 or

> Mantissa:=convert(111111111111111111111111,decimal,binary);

> number:=evalf(Mantissa*16^63);

>

Using a number smaller then the smallest number a computer can represent results in underflow . Using a number larger then the largest number a computer can represent leads to overflow .