Question

convert -881/13 to IEEE-754 single precision and double precision both.
Need a lot of explanation. (Atleast 1000 words)

Answer 1

Answer:

1)
-67.76923076923077
Converting 67.76923076923077 to binary
   Convert decimal part first, then the fractional part
   > First convert 67 to binary
   Divide 67 successively by 2 until the quotient is 0
       > 67/2 = 33, remainder is 1
       > 33/2 = 16, remainder is 1
       > 16/2 = 8, remainder is 0
       > 8/2 = 4, remainder is 0
       > 4/2 = 2, remainder is 0
       > 2/2 = 1, remainder is 0
       > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 1000011
   So, 67 of decimal is 1000011 in binary
   > Now, Convert 0.76923077 to binary
       > Multiply 0.76923077 with 2.    Since 1.53846154 is >= 1. then add 1 to result
       > Multiply 0.53846154 with 2.    Since 1.07692308 is >= 1. then add 1 to result
       > Multiply 0.07692308 with 2.    Since 0.15384615 is < 1. then add 0 to result
       > Multiply 0.15384615 with 2.    Since 0.30769231 is < 1. then add 0 to result
       > Multiply 0.30769231 with 2.    Since 0.61538462 is < 1. then add 0 to result
       > Multiply 0.61538462 with 2.    Since 1.23076923 is >= 1. then add 1 to result
       > Multiply 0.23076923 with 2.    Since 0.46153846 is < 1. then add 0 to result
       > Multiply 0.46153846 with 2.    Since 0.92307692 is < 1. then add 0 to result
       > Multiply 0.92307692 with 2.    Since 1.84615385 is >= 1. then add 1 to result
       > Multiply 0.84615385 with 2.    Since 1.69230769 is >= 1. then add 1 to result
       > Multiply 0.69230769 with 2.    Since 1.38461538 is >= 1. then add 1 to result
       > Multiply 0.38461538 with 2.    Since 0.76923077 is < 1. then add 0 to result
       > Multiply 0.76923077 with 2.    Since 1.53846154 is >= 1. then add 1 to result
       > Multiply 0.53846154 with 2.    Since 1.07692308 is >= 1. then add 1 to result
       > Multiply 0.07692308 with 2.    Since 0.15384615 is < 1. then add 0 to result
       > Multiply 0.15384615 with 2.    Since 0.30769231 is < 1. then add 0 to result
       > Multiply 0.30769231 with 2.    Since 0.61538462 is < 1. then add 0 to result
       > Multiply 0.61538462 with 2.    Since 1.23076923 is >= 1. then add 1 to result
       > Multiply 0.23076923 with 2.    Since 0.46153846 is < 1. then add 0 to result
       > Multiply 0.46153846 with 2.    Since 0.92307693 is < 1. then add 0 to result
       > Multiply 0.92307693 with 2.    Since 1.84615386 is >= 1. then add 1 to result
       > Multiply 0.84615386 with 2.    Since 1.69230771 is >= 1. then add 1 to result
       > Multiply 0.69230771 with 2.    Since 1.38461542 is >= 1. then add 1 to result
       > Multiply 0.38461542 with 2.    Since 0.76923084 is < 1. then add 0 to result
       > Multiply 0.76923084 with 2.    Since 1.53846169 is >= 1. then add 1 to result
       > Multiply 0.53846169 with 2.    Since 1.07692337 is >= 1. then add 1 to result
       > Multiply 0.07692337 with 2.    Since 0.15384674 is < 1. then add 0 to result
       > Multiply 0.15384674 with 2.    Since 0.30769348 is < 1. then add 0 to result
       > Multiply 0.30769348 with 2.    Since 0.61538696 is < 1. then add 0 to result
       > Multiply 0.61538696 with 2.    Since 1.23077393 is >= 1. then add 1 to result
       > Multiply 0.23077393 with 2.    Since 0.46154785 is < 1. then add 0 to result
       > Multiply 0.46154785 with 2.    Since 0.92309570 is < 1. then add 0 to result
       > Multiply 0.92309570 with 2.    Since 1.84619141 is >= 1. then add 1 to result
       > Multiply 0.84619141 with 2.    Since 1.69238281 is >= 1. then add 1 to result
       > Multiply 0.69238281 with 2.    Since 1.38476562 is >= 1. then add 1 to result
       > Multiply 0.38476562 with 2.    Since 0.76953125 is < 1. then add 0 to result
       > Multiply 0.76953125 with 2.    Since 1.53906250 is >= 1. then add 1 to result
       > Multiply 0.53906250 with 2.    Since 1.07812500 is >= 1. then add 1 to result
       > Multiply 0.07812500 with 2.    Since 0.15625000 is < 1. then add 0 to result
       > Multiply 0.15625000 with 2.    Since 0.31250000 is < 1. then add 0 to result
       > Multiply 0.31250000 with 2.    Since 0.62500000 is < 1. then add 0 to result
       > Multiply 0.62500000 with 2.    Since 1.25000000 is >= 1. then add 1 to result
       > Multiply 0.25000000 with 2.    Since 0.50000000 is < 1. then add 0 to result
       > Multiply 0.50000000 with 2.    Since 1.00000000 is >= 1. then add 1 to result
       > This is equal to 1, so, stop calculating
   0.7692307692307736 of decimal is .11000100111011000100111011000100111011000101 in binary
   so, 67.76923076923077 in binary is 1000011.11000100111011000100111011000100111011000101
-67.76923076923077 in simple binary => 1000011.11000100111011000100111011000100111011000101
so, -67.76923076923077 in normal binary is 1000011.11000100111011000100111011000100111011000101 => 1.00001111000100111011 * 2^6

single precision:
--------------------
sign bit is 1(-ve)
exponent bits are (127+6=133) => 10000101
   Divide 133 successively by 2 until the quotient is 0
       > 133/2 = 66, remainder is 1
       > 66/2 = 33, remainder is 0
       > 33/2 = 16, remainder is 1
       > 16/2 = 8, remainder is 0
       > 8/2 = 4, remainder is 0
       > 4/2 = 2, remainder is 0
       > 2/2 = 1, remainder is 0
       > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 10000101
   So, 133 of decimal is 10000101 in binary
frac/significant bits are 00001111000100111011000

so, -67.76923076923077 in single-precision format is 1 10000101 00001111000100111011000
Hexadecimal Binary
0 0000
1 0001
2 0010
3 0011
4 0100
5 0101
6 0110
7 0111
8 1000
9 1001
A 1010
B 1011
C 1100
D 1101
E 1110
F 1111
Use this table to convert from binary to hexadecimal
Converting 11000010100001111000100111011000 to hexadecimal
1100 => C
0010 => 2
1000 => 8
0111 => 7
1000 => 8
1001 => 9
1101 => D
1000 => 8
So, in hexadecimal 11000010100001111000100111011000 is 0xC28789D8

in hexadecimal it is 0xC28789D8

2)
-67.76923076923077
Converting 67.76923076923077 to binary
   Convert decimal part first, then the fractional part
   > First convert 67 to binary
   Divide 67 successively by 2 until the quotient is 0
       > 67/2 = 33, remainder is 1
       > 33/2 = 16, remainder is 1
       > 16/2 = 8, remainder is 0
       > 8/2 = 4, remainder is 0
       > 4/2 = 2, remainder is 0
       > 2/2 = 1, remainder is 0
       > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 1000011
   So, 67 of decimal is 1000011 in binary
   > Now, Convert 0.76923077 to binary
       > Multiply 0.76923077 with 2.    Since 1.53846154 is >= 1. then add 1 to result
       > Multiply 0.53846154 with 2.    Since 1.07692308 is >= 1. then add 1 to result
       > Multiply 0.07692308 with 2.    Since 0.15384615 is < 1. then add 0 to result
       > Multiply 0.15384615 with 2.    Since 0.30769231 is < 1. then add 0 to result
       > Multiply 0.30769231 with 2.    Since 0.61538462 is < 1. then add 0 to result
       > Multiply 0.61538462 with 2.    Since 1.23076923 is >= 1. then add 1 to result
       > Multiply 0.23076923 with 2.    Since 0.46153846 is < 1. then add 0 to result
       > Multiply 0.46153846 with 2.    Since 0.92307692 is < 1. then add 0 to result
       > Multiply 0.92307692 with 2.    Since 1.84615385 is >= 1. then add 1 to result
       > Multiply 0.84615385 with 2.    Since 1.69230769 is >= 1. then add 1 to result
       > Multiply 0.69230769 with 2.    Since 1.38461538 is >= 1. then add 1 to result
       > Multiply 0.38461538 with 2.    Since 0.76923077 is < 1. then add 0 to result
       > Multiply 0.76923077 with 2.    Since 1.53846154 is >= 1. then add 1 to result
       > Multiply 0.53846154 with 2.    Since 1.07692308 is >= 1. then add 1 to result
       > Multiply 0.07692308 with 2.    Since 0.15384615 is < 1. then add 0 to result
       > Multiply 0.15384615 with 2.    Since 0.30769231 is < 1. then add 0 to result
       > Multiply 0.30769231 with 2.    Since 0.61538462 is < 1. then add 0 to result
       > Multiply 0.61538462 with 2.    Since 1.23076923 is >= 1. then add 1 to result
       > Multiply 0.23076923 with 2.    Since 0.46153846 is < 1. then add 0 to result
       > Multiply 0.46153846 with 2.    Since 0.92307693 is < 1. then add 0 to result
       > Multiply 0.92307693 with 2.    Since 1.84615386 is >= 1. then add 1 to result
       > Multiply 0.84615386 with 2.    Since 1.69230771 is >= 1. then add 1 to result
       > Multiply 0.69230771 with 2.    Since 1.38461542 is >= 1. then add 1 to result
       > Multiply 0.38461542 with 2.    Since 0.76923084 is < 1. then add 0 to result
       > Multiply 0.76923084 with 2.    Since 1.53846169 is >= 1. then add 1 to result
       > Multiply 0.53846169 with 2.    Since 1.07692337 is >= 1. then add 1 to result
       > Multiply 0.07692337 with 2.    Since 0.15384674 is < 1. then add 0 to result
       > Multiply 0.15384674 with 2.    Since 0.30769348 is < 1. then add 0 to result
       > Multiply 0.30769348 with 2.    Since 0.61538696 is < 1. then add 0 to result
       > Multiply 0.61538696 with 2.    Since 1.23077393 is >= 1. then add 1 to result
       > Multiply 0.23077393 with 2.    Since 0.46154785 is < 1. then add 0 to result
       > Multiply 0.46154785 with 2.    Since 0.92309570 is < 1. then add 0 to result
       > Multiply 0.92309570 with 2.    Since 1.84619141 is >= 1. then add 1 to result
       > Multiply 0.84619141 with 2.    Since 1.69238281 is >= 1. then add 1 to result
       > Multiply 0.69238281 with 2.    Since 1.38476562 is >= 1. then add 1 to result
       > Multiply 0.38476562 with 2.    Since 0.76953125 is < 1. then add 0 to result
       > Multiply 0.76953125 with 2.    Since 1.53906250 is >= 1. then add 1 to result
       > Multiply 0.53906250 with 2.    Since 1.07812500 is >= 1. then add 1 to result
       > Multiply 0.07812500 with 2.    Since 0.15625000 is < 1. then add 0 to result
       > Multiply 0.15625000 with 2.    Since 0.31250000 is < 1. then add 0 to result
       > Multiply 0.31250000 with 2.    Since 0.62500000 is < 1. then add 0 to result
       > Multiply 0.62500000 with 2.    Since 1.25000000 is >= 1. then add 1 to result
       > Multiply 0.25000000 with 2.    Since 0.50000000 is < 1. then add 0 to result
       > Multiply 0.50000000 with 2.    Since 1.00000000 is >= 1. then add 1 to result
       > This is equal to 1, so, stop calculating
   0.7692307692307736 of decimal is .11000100111011000100111011000100111011000101 in binary
   so, 67.76923076923077 in binary is 1000011.11000100111011000100111011000100111011000101
-67.76923076923077 in simple binary => 1000011.11000100111011000100111011000100111011000101
so, -67.76923076923077 in normal binary is 1000011.11000100111011000100111011000100111011000101 => 1.00001111000100111011000100111011000100111011000101 * 2^6

64-bit format:
--------------------
sign bit is 1(-ve)
exponent bits are (1023+6=1029) => 10000000101
   Divide 1029 successively by 2 until the quotient is 0
       > 1029/2 = 514, remainder is 1
       > 514/2 = 257, remainder is 0
       > 257/2 = 128, remainder is 1
       > 128/2 = 64, remainder is 0
       > 64/2 = 32, remainder is 0
       > 32/2 = 16, remainder is 0
       > 16/2 = 8, remainder is 0
       > 8/2 = 4, remainder is 0
       > 4/2 = 2, remainder is 0
       > 2/2 = 1, remainder is 0
       > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 10000000101
   So, 1029 of decimal is 10000000101 in binary
frac/significant bits are 0000111100010011101100010011101100010011101100010100

so, -67.76923076923077 in 64-bit format is 1 10000000101 0000111100010011101100010011101100010011101100010100
Hexadecimal Binary
0 0000
1 0001
2 0010
3 0011
4 0100
5 0101
6 0110
7 0111
8 1000
9 1001
A 1010
B 1011
C 1100
D 1101
E 1110
F 1111
Use this table to convert from binary to hexadecimal
Converting 1100000001010000111100010011101100010011101100010011101100010100 to hexadecimal
1100 => C
0000 => 0
0101 => 5
0000 => 0
1111 => F
0001 => 1
0011 => 3
1011 => B
0001 => 1
0011 => 3
1011 => B
0001 => 1
0011 => 3
1011 => B
0001 => 1
0100 => 4
So, in hexadecimal 1100000001010000111100010011101100010011101100010011101100010100 is 0xC050F13B13B13B14

in hexadecimal it is 0xC050F13B13B13B14

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