Math Help - Normalised Mantissa

1. Normalised Mantissa

Here is the problem:

"Suppose you work with a 8-bit (binary) computer (that is, numbers are represented by a

binary word of 8 bits). Reserving one bit each for the sign of the normalised mantissa, the

sign of the exponent and the exponent itself, explain to how many signi cant gures you can

represent numbers (not using a hidden bit).

What is the smallest nonzero number you can represent and what is the biggest? How many

numbers can be represented in this way?

How big must a number be for this computer to overflow?"

2. Re: Normalised Mantissa

Since we have only one bit for the exponent, but allow it to be positive or negative, whatever number is given by the mantissa is multiplied by $2^0$= 1, $2^1= 2$ or $2^{-1}= 1/2$. With 5 bits for the mantissa, the largest that can be is $11111= 2^5+ 2^4+ 2^3+ 2^2+ 2^0= 2^6- 1= 63$. Since that can be multiplied by 2, the largest number that can be represented in that way is 2(63)= 126 and, of course, 127 will cause overflow. The smallest non-zero mantissa is, of course, 00001= 1. Since it can be multiplied by 1/2, the smallest positive number that can be represented in that way is 1/2. There are $2^5= 32$ different mantisas of which 31 are non-zero. Those, combined with the three possible mantissas, gives 3(31)= 93 different positive numbers. Adding the 93 different negative number, and, of course, 0, there are 187 different numbers representable in this way.

3. Re: Normalised Mantissa

Originally Posted by Shizaru
Here is the problem:

"Suppose you work with a 8-bit (binary) computer (that is, numbers are represented by a

binary word of 8 bits). Reserving one bit each for the sign of the normalised mantissa, the

sign of the exponent and the exponent itself, explain to how many signicant gures you can

represent numbers (not using a hidden bit).

What is the smallest nonzero number you can represent and what is the biggest? How many

numbers can be represented in this way?

How big must a number be for this computer to overflow?"
8 bit floating point! How many bits are you going to allocate for the mantissa?

Or to put it another way, since you are talking about floating point, you have not told us enough about the representation. Of course if we allocate all the free bits to the exponent we will get the largest possible and the smallest possible numbers that can be represented like this, but with only one significant bit (and I am pretty sure you don't mean the smallest non-zero number that can be represented, but the smallest positive number or the number with smallest absolute value).

CB