Given 32-bit binary number, please demonstrate the range it could represent is 3

Question

Given 32-bit binary number, please demonstrate the range it could represent is 3 different cases. (a) case 1: unsigned integer (b) case 2: signed integer (c) case 3: single-presion IEEE 754 floating-Point representation (please show the reseasoning). (d) Then please briefly explain the benefits of floating point representation. For Floating-Point representation, there is an event "underflow" besides "overflow". Please briefly what is underflow and overflow. You could use single-presion IEEE 754 floating-Point representation as an example to show underflow and overflow.

Explanation / Answer

Ans (6.3)

For 32 bit binary system.

(a) Range for Unsigned Integer => 0 to 4,294,967,295 (232 -1).

minimum value is 00000........0000 and maximum value 11111.......11111 till 32 bit.

(b) Range for Signed Integer => -2,147,483,648 (-231) to 2,147,483,647 (231 - 1).

(c)single precision ieee 754 floating point number represented as +/- mantissa*2exponent

1 represent - sign , and 0 represent + sign.

exponent represented using 8 bit. and mantissa represented using 23 bits.

floating point numbers are symmetric for both positive and negative numbers.so considering only positive case.

for max value matissa is maximum 1.111111111111111111111112 and exponent is 127.

so max value 1.111111111111111111111112 × 2127 = (2 223) × 2127

  For minimum value ,mantissa is 0.000000000000000000000012 and for exponent is 126.

so min value 0.000000000000000000000012× 2126 = 223 × 2126 .

(d) Benifits of floating point representation :

The main benifit of floating point number is the domain or range of numbers which is very . high.It can represent values between integers that's why they have high range.

Floating numbers are useful when the numbers are need be expressed across varying scales.And because of the scaling factor, they can represent a much greater range of values.

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