Question

Give the decimal number represented by the 32-bit word 1011 0011 1101 0000 0000 0000 0000 0000 if a. An unsigned representation has been used b. A sign magnitude representation has been used c. A signed 2’s complement representation has been used d. A 754 IEEE representation has been usedGive the decimal number represented by the 32-bit word 1011 0011 1101 0000 0000 0000 0000 0000 if a. An unsigned representation has been used b. A sign magnitude representation has been used c. A signed 2’s complement representation has been used d. A 754 IEEE representation has been used.

Answer 1

a)
1011001111010000000000000000
=> 1x2^27+0x2^26+1x2^25+1x2^24+0x2^23+0x2^22+1x2^21+1x2^20+1x2^19+1x2^18+0x2^17+1x2^16+0x2^15+0x2^14+0x2^13+0x2^12+0x2^11+0x2^10+0x2^9+0x2^8+0x2^7+0x2^6+0x2^5+0x2^4+0x2^3+0x2^2+0x2^1+0x2^0
=> 1x134217728+0x67108864+1x33554432+1x16777216+0x8388608+0x4194304+1x2097152+1x1048576+1x524288+1x262144+0x131072+1x65536+0x32768+0x16384+0x8192+0x4096+0x2048+0x1024+0x512+0x256+0x128+0x64+0x32+0x16+0x8+0x4+0x2+0x1
=> 134217728+0+33554432+16777216+0+0+2097152+1048576+524288+262144+0+65536+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0
=> 188547072
Answer: 188547072

b)
Left most bit is 1. so, it's negative.
Now, let's convert rest of the bits 011001111010000000000000000 to decimal
=> 011001111010000000000000000
=> 0x2^26+1x2^25+1x2^24+0x2^23+0x2^22+1x2^21+1x2^20+1x2^19+1x2^18+0x2^17+1x2^16+0x2^15+0x2^14+0x2^13+0x2^12+0x2^11+0x2^10+0x2^9+0x2^8+0x2^7+0x2^6+0x2^5+0x2^4+0x2^3+0x2^2+0x2^1+0x2^0
=> 0x67108864+1x33554432+1x16777216+0x8388608+0x4194304+1x2097152+1x1048576+1x524288+1x262144+0x131072+1x65536+0x32768+0x16384+0x8192+0x4096+0x2048+0x1024+0x512+0x256+0x128+0x64+0x32+0x16+0x8+0x4+0x2+0x1
=> 0+33554432+16777216+0+0+2097152+1048576+524288+262144+0+65536+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0
=> 54329344
so, 1011001111010000000000000000 from sign-magnitude to decimal is -54329344
Answer: -54329344

c)
since left most bit is 1, this number is negative number.
so, follow these steps below to convert this into a decimal value.
I. first flip all the bits. Flip all 0's to 1 and all 1's to 0.
   11111011001111010000000000000000 is flipped to 00000100110000101111111111111111
II. Add 1 to above result
00000100110000101111111111111111 + 1 = 00000100110000110000000000000000
III. Now convert this result to decimal value
=> 100110000110000000000000000
=> 1x2^26+0x2^25+0x2^24+1x2^23+1x2^22+0x2^21+0x2^20+0x2^19+0x2^18+1x2^17+1x2^16+0x2^15+0x2^14+0x2^13+0x2^12+0x2^11+0x2^10+0x2^9+0x2^8+0x2^7+0x2^6+0x2^5+0x2^4+0x2^3+0x2^2+0x2^1+0x2^0
=> 1x67108864+0x33554432+0x16777216+1x8388608+1x4194304+0x2097152+0x1048576+0x524288+0x262144+1x131072+1x65536+0x32768+0x16384+0x8192+0x4096+0x2048+0x1024+0x512+0x256+0x128+0x64+0x32+0x16+0x8+0x4+0x2+0x1
=> 67108864+0+0+8388608+4194304+0+0+0+0+131072+65536+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0
=> 79888384
Answer: -79888384

d)
1 01100111 1010000000000000000
sign bit is 1(-ve)
exp bits are 01100111
   => 01100111
   => 0x2^7+1x2^6+1x2^5+0x2^4+0x2^3+1x2^2+1x2^1+1x2^0
   => 0x128+1x64+1x32+0x16+0x8+1x4+1x2+1x1
   => 0+64+32+0+0+4+2+1
   => 103
in decimal it is 103
so, exponent/bias is 103-127 = -24
frac bits are 101

IEEE-754 Decimal value is 1.frac * 2^exponent
IEEE-754 Decimal value is 1.101 * 2^-24
1.101 in decimal is 1.625
   => 1.101
   => 1x2^0+1x2^-1+0x2^-2+1x2^-3
   => 1x1+1x0.5+0x0.25+1x0.125
   => 1+0.5+0.0+0.125
   => 1.625
so, 1.625 * 2^-24 in decimal is 9.685754776000977e-08
so, 1011001111010000000000000000 in IEEE-754 single precision format is -9.685754776000977e-08
Answer: -9.685754776000977e-08