Question

convert 20.085 to IEEE-754 single precision and double precision both.
Need a lot of explanation. (Atleast 1000 words)

Answer 1

1)
20.085
Converting 20.085 to binary
   Convert decimal part first, then the fractional part
   > First convert 20 to binary
   Divide 20 successively by 2 until the quotient is 0
       > 20/2 = 10, remainder is 0
       > 10/2 = 5, remainder is 0
       > 5/2 = 2, remainder is 1
       > 2/2 = 1, remainder is 0
       > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 10100
   So, 20 of decimal is 10100 in binary
   > Now, Convert 0.08500000 to binary
       > Multiply 0.08500000 with 2.    Since 0.17000000 is < 1. then add 0 to result
       > Multiply 0.17000000 with 2.    Since 0.34000000 is < 1. then add 0 to result
       > Multiply 0.34000000 with 2.    Since 0.68000000 is < 1. then add 0 to result
       > Multiply 0.68000000 with 2.    Since 1.36000000 is >= 1. then add 1 to result
       > Multiply 0.36000000 with 2.    Since 0.72000000 is < 1. then add 0 to result
       > Multiply 0.72000000 with 2.    Since 1.44000000 is >= 1. then add 1 to result
       > Multiply 0.44000000 with 2.    Since 0.88000000 is < 1. then add 0 to result
       > Multiply 0.88000000 with 2.    Since 1.76000000 is >= 1. then add 1 to result
       > Multiply 0.76000000 with 2.    Since 1.52000000 is >= 1. then add 1 to result
       > Multiply 0.52000000 with 2.    Since 1.04000000 is >= 1. then add 1 to result
       > Multiply 0.04000000 with 2.    Since 0.08000000 is < 1. then add 0 to result
       > Multiply 0.08000000 with 2.    Since 0.16000000 is < 1. then add 0 to result
       > Multiply 0.16000000 with 2.    Since 0.32000000 is < 1. then add 0 to result
       > Multiply 0.32000000 with 2.    Since 0.64000000 is < 1. then add 0 to result
       > Multiply 0.64000000 with 2.    Since 1.28000000 is >= 1. then add 1 to result
       > Multiply 0.28000000 with 2.    Since 0.56000000 is < 1. then add 0 to result
       > Multiply 0.56000000 with 2.    Since 1.12000000 is >= 1. then add 1 to result
       > Multiply 0.12000000 with 2.    Since 0.24000000 is < 1. then add 0 to result
       > Multiply 0.24000000 with 2.    Since 0.48000000 is < 1. then add 0 to result
       > Multiply 0.48000000 with 2.    Since 0.96000000 is < 1. then add 0 to result
       > Multiply 0.96000000 with 2.    Since 1.92000000 is >= 1. then add 1 to result
       > Multiply 0.92000000 with 2.    Since 1.84000000 is >= 1. then add 1 to result
       > Multiply 0.84000000 with 2.    Since 1.68000001 is >= 1. then add 1 to result
       > Multiply 0.68000001 with 2.    Since 1.36000001 is >= 1. then add 1 to result
       > Multiply 0.36000001 with 2.    Since 0.72000003 is < 1. then add 0 to result
       > Multiply 0.72000003 with 2.    Since 1.44000006 is >= 1. then add 1 to result
       > Multiply 0.44000006 with 2.    Since 0.88000011 is < 1. then add 0 to result
       > Multiply 0.88000011 with 2.    Since 1.76000023 is >= 1. then add 1 to result
       > Multiply 0.76000023 with 2.    Since 1.52000046 is >= 1. then add 1 to result
       > Multiply 0.52000046 with 2.    Since 1.04000092 is >= 1. then add 1 to result
       > Multiply 0.04000092 with 2.    Since 0.08000183 is < 1. then add 0 to result
       > Multiply 0.08000183 with 2.    Since 0.16000366 is < 1. then add 0 to result
       > Multiply 0.16000366 with 2.    Since 0.32000732 is < 1. then add 0 to result
       > Multiply 0.32000732 with 2.    Since 0.64001465 is < 1. then add 0 to result
       > Multiply 0.64001465 with 2.    Since 1.28002930 is >= 1. then add 1 to result
       > Multiply 0.28002930 with 2.    Since 0.56005859 is < 1. then add 0 to result
       > Multiply 0.56005859 with 2.    Since 1.12011719 is >= 1. then add 1 to result
       > Multiply 0.12011719 with 2.    Since 0.24023438 is < 1. then add 0 to result
       > Multiply 0.24023438 with 2.    Since 0.48046875 is < 1. then add 0 to result
       > Multiply 0.48046875 with 2.    Since 0.96093750 is < 1. then add 0 to result
       > Multiply 0.96093750 with 2.    Since 1.92187500 is >= 1. then add 1 to result
       > Multiply 0.92187500 with 2.    Since 1.84375000 is >= 1. then add 1 to result
       > Multiply 0.84375000 with 2.    Since 1.68750000 is >= 1. then add 1 to result
       > Multiply 0.68750000 with 2.    Since 1.37500000 is >= 1. then add 1 to result
       > Multiply 0.37500000 with 2.    Since 0.75000000 is < 1. then add 0 to result
       > Multiply 0.75000000 with 2.    Since 1.50000000 is >= 1. then add 1 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
   0.08500000000000085 of decimal is .00010101110000101000111101011100001010001111011 in binary
   so, 20.085 in binary is 10100.00010101110000101000111101011100001010001111011
20.085 in simple binary => 10100.00010101110000101000111101011100001010001111011
so, 20.085 in normal binary is 10100.00010101110000101000111101011100001010001111011 => 1.010000010101110000101 * 2^4

single precision:
--------------------
sign bit is 0(+ve)
exponent bits are (127+4=131) => 10000011
   Divide 131 successively by 2 until the quotient is 0
       > 131/2 = 65, remainder is 1
       > 65/2 = 32, remainder is 1
       > 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 10000011
   So, 131 of decimal is 10000011 in binary
frac/significant bits are 01000001010111000010100

so, 20.085 in single-precision format is 0 10000011 01000001010111000010100
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 01000001101000001010111000010100 to hexadecimal
0100 => 4
0001 => 1
1010 => A
0000 => 0
1010 => A
1110 => E
0001 => 1
0100 => 4
So, in hexadecimal 01000001101000001010111000010100 is 0x41A0AE14

in hexadecimal it is 0x41A0AE14

2)
20.085
Converting 20.085 to binary
   Convert decimal part first, then the fractional part
   > First convert 20 to binary
   Divide 20 successively by 2 until the quotient is 0
       > 20/2 = 10, remainder is 0
       > 10/2 = 5, remainder is 0
       > 5/2 = 2, remainder is 1
       > 2/2 = 1, remainder is 0
       > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 10100
   So, 20 of decimal is 10100 in binary
   > Now, Convert 0.08500000 to binary
       > Multiply 0.08500000 with 2.    Since 0.17000000 is < 1. then add 0 to result
       > Multiply 0.17000000 with 2.    Since 0.34000000 is < 1. then add 0 to result
       > Multiply 0.34000000 with 2.    Since 0.68000000 is < 1. then add 0 to result
       > Multiply 0.68000000 with 2.    Since 1.36000000 is >= 1. then add 1 to result
       > Multiply 0.36000000 with 2.    Since 0.72000000 is < 1. then add 0 to result
       > Multiply 0.72000000 with 2.    Since 1.44000000 is >= 1. then add 1 to result
       > Multiply 0.44000000 with 2.    Since 0.88000000 is < 1. then add 0 to result
       > Multiply 0.88000000 with 2.    Since 1.76000000 is >= 1. then add 1 to result
       > Multiply 0.76000000 with 2.    Since 1.52000000 is >= 1. then add 1 to result
       > Multiply 0.52000000 with 2.    Since 1.04000000 is >= 1. then add 1 to result
       > Multiply 0.04000000 with 2.    Since 0.08000000 is < 1. then add 0 to result
       > Multiply 0.08000000 with 2.    Since 0.16000000 is < 1. then add 0 to result
       > Multiply 0.16000000 with 2.    Since 0.32000000 is < 1. then add 0 to result
       > Multiply 0.32000000 with 2.    Since 0.64000000 is < 1. then add 0 to result
       > Multiply 0.64000000 with 2.    Since 1.28000000 is >= 1. then add 1 to result
       > Multiply 0.28000000 with 2.    Since 0.56000000 is < 1. then add 0 to result
       > Multiply 0.56000000 with 2.    Since 1.12000000 is >= 1. then add 1 to result
       > Multiply 0.12000000 with 2.    Since 0.24000000 is < 1. then add 0 to result
       > Multiply 0.24000000 with 2.    Since 0.48000000 is < 1. then add 0 to result
       > Multiply 0.48000000 with 2.    Since 0.96000000 is < 1. then add 0 to result
       > Multiply 0.96000000 with 2.    Since 1.92000000 is >= 1. then add 1 to result
       > Multiply 0.92000000 with 2.    Since 1.84000000 is >= 1. then add 1 to result
       > Multiply 0.84000000 with 2.    Since 1.68000001 is >= 1. then add 1 to result
       > Multiply 0.68000001 with 2.    Since 1.36000001 is >= 1. then add 1 to result
       > Multiply 0.36000001 with 2.    Since 0.72000003 is < 1. then add 0 to result
       > Multiply 0.72000003 with 2.    Since 1.44000006 is >= 1. then add 1 to result
       > Multiply 0.44000006 with 2.    Since 0.88000011 is < 1. then add 0 to result
       > Multiply 0.88000011 with 2.    Since 1.76000023 is >= 1. then add 1 to result
       > Multiply 0.76000023 with 2.    Since 1.52000046 is >= 1. then add 1 to result
       > Multiply 0.52000046 with 2.    Since 1.04000092 is >= 1. then add 1 to result
       > Multiply 0.04000092 with 2.    Since 0.08000183 is < 1. then add 0 to result
       > Multiply 0.08000183 with 2.    Since 0.16000366 is < 1. then add 0 to result
       > Multiply 0.16000366 with 2.    Since 0.32000732 is < 1. then add 0 to result
       > Multiply 0.32000732 with 2.    Since 0.64001465 is < 1. then add 0 to result
       > Multiply 0.64001465 with 2.    Since 1.28002930 is >= 1. then add 1 to result
       > Multiply 0.28002930 with 2.    Since 0.56005859 is < 1. then add 0 to result
       > Multiply 0.56005859 with 2.    Since 1.12011719 is >= 1. then add 1 to result
       > Multiply 0.12011719 with 2.    Since 0.24023438 is < 1. then add 0 to result
       > Multiply 0.24023438 with 2.    Since 0.48046875 is < 1. then add 0 to result
       > Multiply 0.48046875 with 2.    Since 0.96093750 is < 1. then add 0 to result
       > Multiply 0.96093750 with 2.    Since 1.92187500 is >= 1. then add 1 to result
       > Multiply 0.92187500 with 2.    Since 1.84375000 is >= 1. then add 1 to result
       > Multiply 0.84375000 with 2.    Since 1.68750000 is >= 1. then add 1 to result
       > Multiply 0.68750000 with 2.    Since 1.37500000 is >= 1. then add 1 to result
       > Multiply 0.37500000 with 2.    Since 0.75000000 is < 1. then add 0 to result
       > Multiply 0.75000000 with 2.    Since 1.50000000 is >= 1. then add 1 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
   0.08500000000000085 of decimal is .00010101110000101000111101011100001010001111011 in binary
   so, 20.085 in binary is 10100.00010101110000101000111101011100001010001111011
20.085 in simple binary => 10100.00010101110000101000111101011100001010001111011
so, 20.085 in normal binary is 10100.00010101110000101000111101011100001010001111011 => 1.010000010101110000101000111101011100001010001111011 * 2^4

64-bit format:
--------------------
sign bit is 0(+ve)
exponent bits are (1023+4=1027) => 10000000011
   Divide 1027 successively by 2 until the quotient is 0
       > 1027/2 = 513, remainder is 1
       > 513/2 = 256, remainder is 1
       > 256/2 = 128, remainder is 0
       > 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 10000000011
   So, 1027 of decimal is 10000000011 in binary
frac/significant bits are 0100000101011100001010001111010111000010100011110110

so, 20.085 in 64-bit format is 0 10000000011 0100000101011100001010001111010111000010100011110110
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 0100000000110100000101011100001010001111010111000010100011110110 to hexadecimal
0100 => 4
0000 => 0
0011 => 3
0100 => 4
0001 => 1
0101 => 5
1100 => C
0010 => 2
1000 => 8
1111 => F
0101 => 5
1100 => C
0010 => 2
1000 => 8
1111 => F
0110 => 6
So, in hexadecimal 0100000000110100000101011100001010001111010111000010100011110110 is 0x403415C28F5C28F6

in hexadecimal it is 0x403415C28F5C28F6