Question

How to find the largest denormalized number of 16 bit? The answer I found is 0x00FF. However, I just dont get it. Can someone explain please

Also why is that in 16 bit, a value of negative infinity is 0xFF00?

Answer 1

When you are using 16 bit number as per the format, leftmost single bit is used to decide positive or negative number. When this leftmost bit is 1, it indicates negative number and setting this bit to 0 will represent positive number. After this bit, next 7 bits are used for exponent and then rightmost 8 bits are used for mantissa.

Denomalized number is used when the digit to be represented is so low that it cannot be fit into normalized representation with correct precision because in normalized representation, we write the number as 1.f x 2^e , where f is the fractional part and e is the exponential part. However normalized representation requires 1 to be present before decimal. However if number is so low that it cannot be represent precisely by keeping 1 before decimal point without loosing precision then denomalized representation is used which represent the number in format 0.f x 2^e.

Therefore the highest number that is used for denomalized representation is when fraction part f contains all 1 and exponent e is zero. Any increment to this number will use normalized representation.

So in 16 bit representation, where we have 8 bits for mantissa, we will set all these bits to 1 to represent fractional part and then 7 bit exponent will be 0 and also since it is a positive number, so leftmost bit will be 0. Hence the largest denomalized number will be 0x00FF.

Infinity is represented by keeping every bits in exponent field set to 1 and every bits in mantissa set to 0. If it is a positive infinity then leftmost bit will be 0 and it it is a negative infinity then leftmost bit will be 1.

Hence positive infinity will be represented as 0x7F00 and negative infinity will be represented as 0xFF00.

Please comment for any clarification