Question

convert 22/7 to IEEE-754 single precision and double precision both.
Need a lot of explanation. (Atleast 1000 words)

Answer 1

1)
3.142857142857143
Converting 3.142857142857143 to binary
   Convert decimal part first, then the fractional part
   > First convert 3 to binary
   Divide 3 successively by 2 until the quotient is 0
       > 3/2 = 1, remainder is 1
       > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 11
   So, 3 of decimal is 11 in binary
   > Now, Convert 0.14285714 to binary
       > Multiply 0.14285714 with 2.    Since 0.28571429 is < 1. then add 0 to result
       > Multiply 0.28571429 with 2.    Since 0.57142857 is < 1. then add 0 to result
       > Multiply 0.57142857 with 2.    Since 1.14285714 is >= 1. then add 1 to result
       > Multiply 0.14285714 with 2.    Since 0.28571429 is < 1. then add 0 to result
       > Multiply 0.28571429 with 2.    Since 0.57142857 is < 1. then add 0 to result
       > Multiply 0.57142857 with 2.    Since 1.14285714 is >= 1. then add 1 to result
       > Multiply 0.14285714 with 2.    Since 0.28571429 is < 1. then add 0 to result
       > Multiply 0.28571429 with 2.    Since 0.57142857 is < 1. then add 0 to result
       > Multiply 0.57142857 with 2.    Since 1.14285714 is >= 1. then add 1 to result
       > Multiply 0.14285714 with 2.    Since 0.28571429 is < 1. then add 0 to result
       > Multiply 0.28571429 with 2.    Since 0.57142857 is < 1. then add 0 to result
       > Multiply 0.57142857 with 2.    Since 1.14285714 is >= 1. then add 1 to result
       > Multiply 0.14285714 with 2.    Since 0.28571429 is < 1. then add 0 to result
       > Multiply 0.28571429 with 2.    Since 0.57142857 is < 1. then add 0 to result
       > Multiply 0.57142857 with 2.    Since 1.14285714 is >= 1. then add 1 to result
       > Multiply 0.14285714 with 2.    Since 0.28571429 is < 1. then add 0 to result
       > Multiply 0.28571429 with 2.    Since 0.57142857 is < 1. then add 0 to result
       > Multiply 0.57142857 with 2.    Since 1.14285714 is >= 1. then add 1 to result
       > Multiply 0.14285714 with 2.    Since 0.28571429 is < 1. then add 0 to result
       > Multiply 0.28571429 with 2.    Since 0.57142857 is < 1. then add 0 to result
       > Multiply 0.57142857 with 2.    Since 1.14285714 is >= 1. then add 1 to result
       > Multiply 0.14285714 with 2.    Since 0.28571429 is < 1. then add 0 to result
       > Multiply 0.28571429 with 2.    Since 0.57142857 is < 1. then add 0 to result
       > Multiply 0.57142857 with 2.    Since 1.14285714 is >= 1. then add 1 to result
       > Multiply 0.14285714 with 2.    Since 0.28571428 is < 1. then add 0 to result
       > Multiply 0.28571428 with 2.    Since 0.57142857 is < 1. then add 0 to result
       > Multiply 0.57142857 with 2.    Since 1.14285713 is >= 1. then add 1 to result
       > Multiply 0.14285713 with 2.    Since 0.28571427 is < 1. then add 0 to result
       > Multiply 0.28571427 with 2.    Since 0.57142854 is < 1. then add 0 to result
       > Multiply 0.57142854 with 2.    Since 1.14285707 is >= 1. then add 1 to result
       > Multiply 0.14285707 with 2.    Since 0.28571415 is < 1. then add 0 to result
       > Multiply 0.28571415 with 2.    Since 0.57142830 is < 1. then add 0 to result
       > Multiply 0.57142830 with 2.    Since 1.14285660 is >= 1. then add 1 to result
       > Multiply 0.14285660 with 2.    Since 0.28571320 is < 1. then add 0 to result
       > Multiply 0.28571320 with 2.    Since 0.57142639 is < 1. then add 0 to result
       > Multiply 0.57142639 with 2.    Since 1.14285278 is >= 1. then add 1 to result
       > Multiply 0.14285278 with 2.    Since 0.28570557 is < 1. then add 0 to result
       > Multiply 0.28570557 with 2.    Since 0.57141113 is < 1. then add 0 to result
       > Multiply 0.57141113 with 2.    Since 1.14282227 is >= 1. then add 1 to result
       > Multiply 0.14282227 with 2.    Since 0.28564453 is < 1. then add 0 to result
       > Multiply 0.28564453 with 2.    Since 0.57128906 is < 1. then add 0 to result
       > Multiply 0.57128906 with 2.    Since 1.14257812 is >= 1. then add 1 to result
       > Multiply 0.14257812 with 2.    Since 0.28515625 is < 1. then add 0 to result
       > Multiply 0.28515625 with 2.    Since 0.57031250 is < 1. then add 0 to result
       > Multiply 0.57031250 with 2.    Since 1.14062500 is >= 1. then add 1 to result
       > Multiply 0.14062500 with 2.    Since 0.28125000 is < 1. then add 0 to result
       > Multiply 0.28125000 with 2.    Since 0.56250000 is < 1. then add 0 to result
       > Multiply 0.56250000 with 2.    Since 1.12500000 is >= 1. then add 1 to result
       > Multiply 0.12500000 with 2.    Since 0.25000000 is < 1. then add 0 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
   0.1428571428571428 of decimal is .001001001001001001001001001001001001001001001001001 in binary
   so, 3.142857142857143 in binary is 11.001001001001001001001001001001001001001001001001001
3.142857142857143 in simple binary => 11.001001001001001001001001001001001001001001001001001
so, 3.142857142857143 in normal binary is 11.001001001001001001001001001001001001001001001001001 => 1.1001001001001001001001 * 2^1

single precision:
--------------------
sign bit is 0(+ve)
exponent bits are (127+1=128) => 10000000
   Divide 128 successively by 2 until the quotient 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 10000000
   So, 128 of decimal is 10000000 in binary
frac/significant bits are 10010010010010010010010

so, 3.142857142857143 in single-precision format is 0 10000000 10010010010010010010010
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 01000000010010010010010010010010 to hexadecimal
0100 => 4
0000 => 0
0100 => 4
1001 => 9
0010 => 2
0100 => 4
1001 => 9
0010 => 2
So, in hexadecimal 01000000010010010010010010010010 is 0x40492492

in hexadecimal it is 0x40492492

2)
3.142857142857143
Converting 3.142857142857143 to binary
   Convert decimal part first, then the fractional part
   > First convert 3 to binary
   Divide 3 successively by 2 until the quotient is 0
       > 3/2 = 1, remainder is 1
       > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 11
   So, 3 of decimal is 11 in binary
   > Now, Convert 0.14285714 to binary
       > Multiply 0.14285714 with 2.    Since 0.28571429 is < 1. then add 0 to result
       > Multiply 0.28571429 with 2.    Since 0.57142857 is < 1. then add 0 to result
       > Multiply 0.57142857 with 2.    Since 1.14285714 is >= 1. then add 1 to result
       > Multiply 0.14285714 with 2.    Since 0.28571429 is < 1. then add 0 to result
       > Multiply 0.28571429 with 2.    Since 0.57142857 is < 1. then add 0 to result
       > Multiply 0.57142857 with 2.    Since 1.14285714 is >= 1. then add 1 to result
       > Multiply 0.14285714 with 2.    Since 0.28571429 is < 1. then add 0 to result
       > Multiply 0.28571429 with 2.    Since 0.57142857 is < 1. then add 0 to result
       > Multiply 0.57142857 with 2.    Since 1.14285714 is >= 1. then add 1 to result
       > Multiply 0.14285714 with 2.    Since 0.28571429 is < 1. then add 0 to result
       > Multiply 0.28571429 with 2.    Since 0.57142857 is < 1. then add 0 to result
       > Multiply 0.57142857 with 2.    Since 1.14285714 is >= 1. then add 1 to result
       > Multiply 0.14285714 with 2.    Since 0.28571429 is < 1. then add 0 to result
       > Multiply 0.28571429 with 2.    Since 0.57142857 is < 1. then add 0 to result
       > Multiply 0.57142857 with 2.    Since 1.14285714 is >= 1. then add 1 to result
       > Multiply 0.14285714 with 2.    Since 0.28571429 is < 1. then add 0 to result
       > Multiply 0.28571429 with 2.    Since 0.57142857 is < 1. then add 0 to result
       > Multiply 0.57142857 with 2.    Since 1.14285714 is >= 1. then add 1 to result
       > Multiply 0.14285714 with 2.    Since 0.28571429 is < 1. then add 0 to result
       > Multiply 0.28571429 with 2.    Since 0.57142857 is < 1. then add 0 to result
       > Multiply 0.57142857 with 2.    Since 1.14285714 is >= 1. then add 1 to result
       > Multiply 0.14285714 with 2.    Since 0.28571429 is < 1. then add 0 to result
       > Multiply 0.28571429 with 2.    Since 0.57142857 is < 1. then add 0 to result
       > Multiply 0.57142857 with 2.    Since 1.14285714 is >= 1. then add 1 to result
       > Multiply 0.14285714 with 2.    Since 0.28571428 is < 1. then add 0 to result
       > Multiply 0.28571428 with 2.    Since 0.57142857 is < 1. then add 0 to result
       > Multiply 0.57142857 with 2.    Since 1.14285713 is >= 1. then add 1 to result
       > Multiply 0.14285713 with 2.    Since 0.28571427 is < 1. then add 0 to result
       > Multiply 0.28571427 with 2.    Since 0.57142854 is < 1. then add 0 to result
       > Multiply 0.57142854 with 2.    Since 1.14285707 is >= 1. then add 1 to result
       > Multiply 0.14285707 with 2.    Since 0.28571415 is < 1. then add 0 to result
       > Multiply 0.28571415 with 2.    Since 0.57142830 is < 1. then add 0 to result
       > Multiply 0.57142830 with 2.    Since 1.14285660 is >= 1. then add 1 to result
       > Multiply 0.14285660 with 2.    Since 0.28571320 is < 1. then add 0 to result
       > Multiply 0.28571320 with 2.    Since 0.57142639 is < 1. then add 0 to result
       > Multiply 0.57142639 with 2.    Since 1.14285278 is >= 1. then add 1 to result
       > Multiply 0.14285278 with 2.    Since 0.28570557 is < 1. then add 0 to result
       > Multiply 0.28570557 with 2.    Since 0.57141113 is < 1. then add 0 to result
       > Multiply 0.57141113 with 2.    Since 1.14282227 is >= 1. then add 1 to result
       > Multiply 0.14282227 with 2.    Since 0.28564453 is < 1. then add 0 to result
       > Multiply 0.28564453 with 2.    Since 0.57128906 is < 1. then add 0 to result
       > Multiply 0.57128906 with 2.    Since 1.14257812 is >= 1. then add 1 to result
       > Multiply 0.14257812 with 2.    Since 0.28515625 is < 1. then add 0 to result
       > Multiply 0.28515625 with 2.    Since 0.57031250 is < 1. then add 0 to result
       > Multiply 0.57031250 with 2.    Since 1.14062500 is >= 1. then add 1 to result
       > Multiply 0.14062500 with 2.    Since 0.28125000 is < 1. then add 0 to result
       > Multiply 0.28125000 with 2.    Since 0.56250000 is < 1. then add 0 to result
       > Multiply 0.56250000 with 2.    Since 1.12500000 is >= 1. then add 1 to result
       > Multiply 0.12500000 with 2.    Since 0.25000000 is < 1. then add 0 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
   0.1428571428571428 of decimal is .001001001001001001001001001001001001001001001001001 in binary
   so, 3.142857142857143 in binary is 11.001001001001001001001001001001001001001001001001001
3.142857142857143 in simple binary => 11.001001001001001001001001001001001001001001001001001
so, 3.142857142857143 in normal binary is 11.001001001001001001001001001001001001001001001001001 => 1.1001001001001001001001001001001001001001001001001001 * 2^1

64-bit format:
--------------------
sign bit is 0(+ve)
exponent bits are (1023+1=1024) => 10000000000
   Divide 1024 successively by 2 until the quotient is 0
       > 1024/2 = 512, remainder is 0
       > 512/2 = 256, remainder is 0
       > 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 10000000000
   So, 1024 of decimal is 10000000000 in binary
frac/significant bits are 1001001001001001001001001001001001001001001001001001

so, 3.142857142857143 in 64-bit format is 0 10000000000 1001001001001001001001001001001001001001001001001001
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 0100000000001001001001001001001001001001001001001001001001001001 to hexadecimal
0100 => 4
0000 => 0
0000 => 0
1001 => 9
0010 => 2
0100 => 4
1001 => 9
0010 => 2
0100 => 4
1001 => 9
0010 => 2
0100 => 4
1001 => 9
0010 => 2
0100 => 4
1001 => 9
So, in hexadecimal 0100000000001001001001001001001001001001001001001001001001001001 is 0x4009249249249249

in hexadecimal it is 0x4009249249249249