2. For each of the three different encoding schemes listed below, please specify
ID: 3878358 • Letter: 2
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2. For each of the three different encoding schemes listed below, please specify each one of the different numbers that could be exactly represented, and characterize the numbers that could not be exactly represent ed (Example) 3-bit unsigned integers Each 0 or 1 represents a binary digit. Three binary digits abc (where a -0 or 1,b -0 or 1, and c = 0 or 1) encode the number: a x 22 + b x 21 + c x 20. A total of 8 different numbers can be represented using this encoding. They are 0002-0; 0012-1; 0102-2; 0112-3; 1002-4; 1012=5; 1102-6; 1112-7 Numbers less than zero would not be representable due to underflow; numbers greater than 7 would be out-of-range due to overflow; real numbers in the range (0-, (1-2), etc., would be within range, but not exactly representable using this encoding a) (3 points) 3-bit signed integers b) (4 points) 3-bit unsigned fractional numbers x, where 0 sxExplanation / Answer
a.
3-bit signed integers:
In the 3 bits, one bit(MSB Most Significant Bit) is used to determine the sign of the number. For example if the MSB is 0 then the number is positive and vice versa.
The maximum number that can be represented with this number system is 3 and the minimum number is -3. All possible numbers in this system is 000, 001, 010, 011, 100, 101, 110, 111.
000 represents 0
1. 3-bit signed integers:
In the 3 bits, one bit(MSB Most Significant Bit) is used to determine the sign of the number. For example if the MSB is 0 then the number is positive and vice versa.
The maximum number that can be represented with this number system is 3 and the minimum number is -3. All possible numbers in this system is 000, 001, 010, 011, 100, 101, 110, 111.
000 represents 0
001 represents 1
010 represents 2
011 represents 3
100 represents -0
101 represents -1
110 represents -2
111 represents -3
b. 3 bit unsigned fractional numbers (0 x < 1 ):
It is clearly mentioned that the number is greater than or equal to 0 and less than 1. So all the 3 bits represents the fractional part of the number.
The minimum number is 0 and the maximum number is 0.875.
All possible numbers in the system are 000, 001, 010, 011, 100, 101, 110, 111. For the decimal number conversion in fractional part each bit is multipiled by 2-n ( where n is the bit position)
000 represents 0
001 represents 0.125
010 represents 0.25
011 represents 0.375
100 represents 0.5
101 represents 0.625
110 represents 0.75
111 represents 0.875
c. 5-bit fixed point fractional number:
It is mentioned that 2 bits represent integer part and 3 bits represent fractional part. This number system is the combination first two systems. The minimum number that can be represented is 0 and the maximum number is 3.875.
For example let the number be 00000, which denotes 0 in decimal. Let the number be 11001, The first two bits represents intger part. which is equal to 3 and the next three bits represents fractional part which is equal to 0.125 and the number is 3.125
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