If the floating-point number storage on a certain system has a sign bit, a 3-bit exponent, and a 4-bit significand:

问题

(Assume no bits are implied, there is no biasing, exponents use two’s complement notation, and exponents of all zeros and all ones are allowed.) I am trying to find the largest and smallest number that can be represented if the system is normalized. I thought that the largest number would be:

.1111 x 2^4 = 0 100 1111 = 15

and the smallest:

1.0 x 2^-4 = 0 000 0001 = 0.0625

But the answers that I saw were:

Largest: .1111 x 2^3 = 111.1 = 7.5

Smallest: 0.1 x 2^-4 = .00001 = 0.03125

I do not understand how either answer would even be possible in a 4-bit significand.

回答1:

There're a few things to decipher here.

• Exponent: 3-bits, 2's complement, no bias. This means the exponent can represent values in the range -4 (corresponding to 100) to 3 (corresponding to 011).

• Normalized without implied bit: This means significand always starts with a 1.

When you put these together, the maximum number you can write is:

 0 011 1111 = 2^3 * (2^-1 + 2^-2 + 2^-3 + 2^-4) = 7.5

Since floating point values are symmetic around 0, the minumum value you can write is -7.5 by flipping the sign bit above. But I guess your teacher is going for minumum strictly positive (i.e., non-zero) value. In that case, we pick the exponent to be as small as possible, and just keep the first bit of the significand to satisfy the normalized requirement. We get:

 0 100 1000 = 2^-4 * 2^-1 = 2^-5 = 0.03125

Hope that makes sense!

来源：https://stackoverflow.com/questions/59914695/if-the-floating-point-number-storage-on-a-certain-system-has-a-sign-bit-a-3-bit