Question

Exercise 4.4 Find the decimal value of the DP number for each of the bit patterns (1) 001111111111 000...000 (2) 0 01111111111 000...001 (3) 0 10000110011 000...000 (3) 0 10000110011 000...001

Answer 1

Answer:

1)
0 01111111111 0000000000000000000000000000000000000000000000000000
sign bit is 0(+ve)
exp bits are 01111111111
Converting 01111111111 to decimal
   01111111111
   => 0x2^10+1x2^9+1x2^8+1x2^7+1x2^6+1x2^5+1x2^4+1x2^3+1x2^2+1x2^1+1x2^0
   => 0x1024+1x512+1x256+1x128+1x64+1x32+1x16+1x8+1x4+1x2+1x1
   => 0+512+256+128+64+32+16+8+4+2+1
   => 1023
in decimal it is 1023
so, exponent/bias is 1023-1023 = 0
frac bits are 

IEEE-754 Decimal value is 1.frac * 2^exponent
IEEE-754 Decimal value is 1. * 2^0
1. in decimal is 1
   => 1.
   => 1x2^0
   => 1x1
   => 1
   => 1
so, 1 * 2^0 in decimal is 1
so, 0011111111110000000000000000000000000000000000000000000000000000 in 64-bit format is 1
Answer: 1

2)
0 01111111111 0000000000000000000000000000000000000000000000000001
sign bit is 0(+ve)
exp bits are 01111111111
Converting 01111111111 to decimal
   01111111111
   => 0x2^10+1x2^9+1x2^8+1x2^7+1x2^6+1x2^5+1x2^4+1x2^3+1x2^2+1x2^1+1x2^0
   => 0x1024+1x512+1x256+1x128+1x64+1x32+1x16+1x8+1x4+1x2+1x1
   => 0+512+256+128+64+32+16+8+4+2+1
   => 1023
in decimal it is 1023
so, exponent/bias is 1023-1023 = 0
frac bits are 0000000000000000000000000000000000000000000000000001

IEEE-754 Decimal value is 1.frac * 2^exponent
IEEE-754 Decimal value is 1.0000000000000000000000000000000000000000000000000001 * 2^0
1.0000000000000000000000000000000000000000000000000001 in decimal is 1.0000000000000002
   => 1.0000000000000000000000000000000000000000000000000001
   => 1x2^0+0x2^-1+0x2^-2+0x2^-3+0x2^-4+0x2^-5+0x2^-6+0x2^-7+0x2^-8+0x2^-9+0x2^-10+0x2^-11+0x2^-12+0x2^-13+0x2^-14+0x2^-15+0x2^-16+0x2^-17+0x2^-18+0x2^-19+0x2^-20+0x2^-21+0x2^-22+0x2^-23+0x2^-24+0x2^-25+0x2^-26+0x2^-27+0x2^-28+0x2^-29+0x2^-30+0x2^-31+0x2^-32+0x2^-33+0x2^-34+0x2^-35+0x2^-36+0x2^-37+0x2^-38+0x2^-39+0x2^-40+0x2^-41+0x2^-42+0x2^-43+0x2^-44+0x2^-45+0x2^-46+0x2^-47+0x2^-48+0x2^-49+0x2^-50+0x2^-51+1x2^-52
   => 1x1+0x0.5+0x0.25+0x0.125+0x0.0625+0x0.03125+0x0.015625+0x0.0078125+0x0.00390625+0x0.001953125+0x0.0009765625+0x0.00048828125+0x0.000244140625+0x0.0001220703125+0x6.103515625e-05+0x3.0517578125e-05+0x1.52587890625e-05+0x7.62939453125e-06+0x3.814697265625e-06+0x1.9073486328125e-06+0x9.5367431640625e-07+0x4.76837158203125e-07+0x2.384185791015625e-07+0x1.1920928955078125e-07+0x5.960464477539063e-08+0x2.9802322387695312e-08+0x1.4901161193847656e-08+0x7.450580596923828e-09+0x3.725290298461914e-09+0x1.862645149230957e-09+0x9.313225746154785e-10+0x4.656612873077393e-10+0x2.3283064365386963e-10+0x1.1641532182693481e-10+0x5.820766091346741e-11+0x2.9103830456733704e-11+0x1.4551915228366852e-11+0x7.275957614183426e-12+0x3.637978807091713e-12+0x1.8189894035458565e-12+0x9.094947017729282e-13+0x4.547473508864641e-13+0x2.2737367544323206e-13+0x1.1368683772161603e-13+0x5.684341886080802e-14+0x2.842170943040401e-14+0x1.4210854715202004e-14+0x7.105427357601002e-15+0x3.552713678800501e-15+0x1.7763568394002505e-15+0x8.881784197001252e-16+0x4.440892098500626e-16+1x2.220446049250313e-16
   => 1+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+2.220446049250313e-16
   => 1.0000000000000002
so, 1.0000000000000002 * 2^0 in decimal is 1.0000000000000002
so, 0011111111110000000000000000000000000000000000000000000000000001 in 64-bit format is 1.0000000000000002
Answer: 1.0000000000000002

3)
0 10000110011 0000000000000000000000000000000000000000000000000000
sign bit is 0(+ve)
exp bits are 10000110011
Converting 10000110011 to decimal
   10000110011
   => 1x2^10+0x2^9+0x2^8+0x2^7+0x2^6+1x2^5+1x2^4+0x2^3+0x2^2+1x2^1+1x2^0
   => 1x1024+0x512+0x256+0x128+0x64+1x32+1x16+0x8+0x4+1x2+1x1
   => 1024+0+0+0+0+32+16+0+0+2+1
   => 1075
in decimal it is 1075
so, exponent/bias is 1075-1023 = 52
frac bits are 

IEEE-754 Decimal value is 1.frac * 2^exponent
IEEE-754 Decimal value is 1. * 2^52
1. in decimal is 1
   => 1.
   => 1x2^0
   => 1x1
   => 1
   => 1
so, 1 * 2^52 in decimal is 4503599627370496
so, 0100001100110000000000000000000000000000000000000000000000000000 in 64-bit format is 4503599627370496
Answer: 4503599627370496

4)
0 10000110011 0000000000000000000000000000000000000000000000000001
sign bit is 0(+ve)
exp bits are 10000110011
Converting 10000110011 to decimal
   10000110011
   => 1x2^10+0x2^9+0x2^8+0x2^7+0x2^6+1x2^5+1x2^4+0x2^3+0x2^2+1x2^1+1x2^0
   => 1x1024+0x512+0x256+0x128+0x64+1x32+1x16+0x8+0x4+1x2+1x1
   => 1024+0+0+0+0+32+16+0+0+2+1
   => 1075
in decimal it is 1075
so, exponent/bias is 1075-1023 = 52
frac bits are 0000000000000000000000000000000000000000000000000001

IEEE-754 Decimal value is 1.frac * 2^exponent
IEEE-754 Decimal value is 1.0000000000000000000000000000000000000000000000000001 * 2^52
1.0000000000000000000000000000000000000000000000000001 in decimal is 1.0000000000000002
   => 1.0000000000000000000000000000000000000000000000000001
   => 1x2^0+0x2^-1+0x2^-2+0x2^-3+0x2^-4+0x2^-5+0x2^-6+0x2^-7+0x2^-8+0x2^-9+0x2^-10+0x2^-11+0x2^-12+0x2^-13+0x2^-14+0x2^-15+0x2^-16+0x2^-17+0x2^-18+0x2^-19+0x2^-20+0x2^-21+0x2^-22+0x2^-23+0x2^-24+0x2^-25+0x2^-26+0x2^-27+0x2^-28+0x2^-29+0x2^-30+0x2^-31+0x2^-32+0x2^-33+0x2^-34+0x2^-35+0x2^-36+0x2^-37+0x2^-38+0x2^-39+0x2^-40+0x2^-41+0x2^-42+0x2^-43+0x2^-44+0x2^-45+0x2^-46+0x2^-47+0x2^-48+0x2^-49+0x2^-50+0x2^-51+1x2^-52
   => 1x1+0x0.5+0x0.25+0x0.125+0x0.0625+0x0.03125+0x0.015625+0x0.0078125+0x0.00390625+0x0.001953125+0x0.0009765625+0x0.00048828125+0x0.000244140625+0x0.0001220703125+0x6.103515625e-05+0x3.0517578125e-05+0x1.52587890625e-05+0x7.62939453125e-06+0x3.814697265625e-06+0x1.9073486328125e-06+0x9.5367431640625e-07+0x4.76837158203125e-07+0x2.384185791015625e-07+0x1.1920928955078125e-07+0x5.960464477539063e-08+0x2.9802322387695312e-08+0x1.4901161193847656e-08+0x7.450580596923828e-09+0x3.725290298461914e-09+0x1.862645149230957e-09+0x9.313225746154785e-10+0x4.656612873077393e-10+0x2.3283064365386963e-10+0x1.1641532182693481e-10+0x5.820766091346741e-11+0x2.9103830456733704e-11+0x1.4551915228366852e-11+0x7.275957614183426e-12+0x3.637978807091713e-12+0x1.8189894035458565e-12+0x9.094947017729282e-13+0x4.547473508864641e-13+0x2.2737367544323206e-13+0x1.1368683772161603e-13+0x5.684341886080802e-14+0x2.842170943040401e-14+0x1.4210854715202004e-14+0x7.105427357601002e-15+0x3.552713678800501e-15+0x1.7763568394002505e-15+0x8.881784197001252e-16+0x4.440892098500626e-16+1x2.220446049250313e-16
   => 1+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+0.0+2.220446049250313e-16
   => 1.0000000000000002
so, 1.0000000000000002 * 2^52 in decimal is 4503599627370497.0
so, 0100001100110000000000000000000000000000000000000000000000000001 in 64-bit format is 4503599627370497.0
Answer: 4503599627370497.0

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