Consider the two floating$-$point systems defined below. They are similar to the one

presented in the lecture, however, have different lengths for mantissa and exponent.

Both have $12$ bit length in total. Their structure is as follows:

System $1$: $1$ sign bit, $6$ exponent bits $($excess $32$ system$),5$ mantissa bits $($non$-$

normalized form$)2^6=64=>-32$dots$+31$

System $2$: $1$ sign bit, $5$ exponent bits $($excess $16$ system$),6$ mantissa bits $($non$-$

normalized form$)$

a$)$ Which of the systems has the largest number range?

b$)$ Which of the systems has the higher accuracy$/$precision$?$

c$)$ What is the largest positive number in System $1($in binary floating$-$point code$)?$

d$)$ What is the smallest positive number in System $1>0($in binary floating$-$point

code$)?$

Hint: The sign bit is $"0"$ for positive numbers, the exponent is encoded in the excess

system which has a sign bit $"1"$ for positive numbers. The smallest value for the

mantissa needs to have a $"1"$ as least significant bit.