Question

4.1 Express -1.2345 x 10^-6 in IEEE 754 single-precision representation (please show work and explain)

Answer 1

-1.2345*10^-06
Converting 1.2345*10^-06 to binary
Convert 0.0000012345 to binary
      > Multiply 0.0000012345 with 2.    Since 0.0000012345 is < 1. then add 0 to result
      > Multiply 0.00000247 with 2.  Since 0.00000494 is < 1. then add 0 to result
      > Multiply 0.00000494 with 2.  Since 0.00000988 is < 1. then add 0 to result
      > Multiply 0.00000988 with 2.  Since 0.00001975 is < 1. then add 0 to result
      > Multiply 0.00001975 with 2.  Since 0.00003950 is < 1. then add 0 to result
      > Multiply 0.00003950 with 2.  Since 0.00007901 is < 1. then add 0 to result
      > Multiply 0.00007901 with 2.  Since 0.00015802 is < 1. then add 0 to result
      > Multiply 0.00015802 with 2.  Since 0.00031603 is < 1. then add 0 to result
      > Multiply 0.00031603 with 2.  Since 0.00063206 is < 1. then add 0 to result
      > Multiply 0.00063206 with 2.  Since 0.00126413 is < 1. then add 0 to result
      > Multiply 0.00126413 with 2.  Since 0.00252826 is < 1. then add 0 to result
      > Multiply 0.00252826 with 2.  Since 0.00505651 is < 1. then add 0 to result
      > Multiply 0.00505651 with 2.  Since 0.01011302 is < 1. then add 0 to result
      > Multiply 0.01011302 with 2.  Since 0.02022605 is < 1. then add 0 to result
      > Multiply 0.02022605 with 2.  Since 0.04045210 is < 1. then add 0 to result
      > Multiply 0.04045210 with 2.  Since 0.08090419 is < 1. then add 0 to result
      > Multiply 0.08090419 with 2.  Since 0.16180838 is < 1. then add 0 to result
      > Multiply 0.16180838 with 2.  Since 0.32361677 is < 1. then add 0 to result
      > Multiply 0.32361677 with 2.  Since 0.64723354 is < 1. then add 0 to result
      > Multiply 0.64723354 with 2.  Since 1.29446707 is >= 1. then add 1 to result
      > Multiply 0.29446707 with 2.  Since 0.58893414 is < 1. then add 0 to result
      > Multiply 0.58893414 with 2.  Since 1.17786829 is >= 1. then add 1 to result
      > Multiply 0.17786829 with 2.  Since 0.35573658 is < 1. then add 0 to result
      > Multiply 0.35573658 with 2.  Since 0.71147315 is < 1. then add 0 to result
      > Multiply 0.71147315 with 2.  Since 1.42294630 is >= 1. then add 1 to result
      > Multiply 0.42294630 with 2.  Since 0.84589261 is < 1. then add 0 to result
      > Multiply 0.84589261 with 2.  Since 1.69178522 is >= 1. then add 1 to result
      > Multiply 0.69178522 with 2.  Since 1.38357043 is >= 1. then add 1 to result
      > Multiply 0.38357043 with 2.  Since 0.76714086 is < 1. then add 0 to result
      > Multiply 0.76714086 with 2.  Since 1.53428173 is >= 1. then add 1 to result
      > Multiply 0.53428173 with 2.  Since 1.06856346 is >= 1. then add 1 to result
      > Multiply 0.06856346 with 2.  Since 0.13712691 is < 1. then add 0 to result
      > Multiply 0.13712691 with 2.  Since 0.27425382 is < 1. then add 0 to result
      > Multiply 0.27425382 with 2.  Since 0.54850765 is < 1. then add 0 to result
      > Multiply 0.54850765 with 2.  Since 1.09701530 is >= 1. then add 1 to result
      > Multiply 0.09701530 with 2.  Since 0.19403059 is < 1. then add 0 to result
      > Multiply 0.19403059 with 2.  Since 0.38806118 is < 1. then add 0 to result
      > Multiply 0.38806118 with 2.  Since 0.77612237 is < 1. then add 0 to result
      > Multiply 0.77612237 with 2.  Since 1.55224474 is >= 1. then add 1 to result
      > Multiply 0.55224474 with 2.  Since 1.10448947 is >= 1. then add 1 to result
      > Multiply 0.10448947 with 2.  Since 0.20897894 is < 1. then add 0 to result
      > Multiply 0.20897894 with 2.  Since 0.41795789 is < 1. then add 0 to result
      > Multiply 0.41795789 with 2.  Since 0.83591577 is < 1. then add 0 to result
      > Multiply 0.83591577 with 2.  Since 1.67183155 is >= 1. then add 1 to result
      > Multiply 0.67183155 with 2.  Since 1.34366310 is >= 1. then add 1 to result
      > Multiply 0.34366310 with 2.  Since 0.68732619 is < 1. then add 0 to result
      > Multiply 0.68732619 with 2.  Since 1.37465239 is >= 1. then add 1 to result
      > Multiply 0.37465239 with 2.  Since 0.74930477 is < 1. then add 0 to result
      > Multiply 0.74930477 with 2.  Since 1.49860954 is >= 1. then add 1 to result
      > Multiply 0.49860954 with 2.  Since 0.99721909 is < 1. then add 0 to result
      > Multiply 0.99721909 with 2.  Since 1.99443817 is >= 1. then add 1 to result
      > Multiply 0.99443817 with 2.  Since 1.98887634 is >= 1. then add 1 to result
      > Multiply 0.98887634 with 2.  Since 1.97775269 is >= 1. then add 1 to result
      > Multiply 0.97775269 with 2.  Since 1.95550537 is >= 1. then add 1 to result
      > Multiply 0.95550537 with 2.  Since 1.91101074 is >= 1. then add 1 to result
      > Multiply 0.91101074 with 2.  Since 1.82202148 is >= 1. then add 1 to result
      > Multiply 0.82202148 with 2.  Since 1.64404297 is >= 1. then add 1 to result
      > Multiply 0.64404297 with 2.  Since 1.28808594 is >= 1. then add 1 to result
      > Multiply 0.28808594 with 2.  Since 0.57617188 is < 1. then add 0 to result
      > Multiply 0.57617188 with 2.  Since 1.15234375 is >= 1. then add 1 to result
      > Multiply 0.15234375 with 2.  Since 0.30468750 is < 1. then add 0 to result
      > Multiply 0.30468750 with 2.  Since 0.60937500 is < 1. then add 0 to result
      > Multiply 0.60937500 with 2.  Since 1.21875000 is >= 1. then add 1 to result
      > Multiply 0.21875000 with 2.  Since 0.43750000 is < 1. then add 0 to result
      > Multiply 0.43750000 with 2.  Since 0.87500000 is < 1. then add 0 to result
      > Multiply 0.87500000 with 2.  Since 1.75000000 is >= 1. then add 1 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
   -1.2345*10^-06 of decimal is .00000000000000000001010010110110001000110001101010111111110100100111 in binary
   so, -1.2345*10^-06 in binary is .00000000000000000001010010110110001000110001101010111111110100100111
-1.2345*10^-06 in simple binary => .00000000000000000001010010110110001000110001101010111111110100100111
so, -1.2345*10^-06 in normal binary is .00000000000000000001010010110110001000110001101010111111110100100111 => 1.01001011011000100011 * 2^-20

single precision:
--------------------
sign bit is 1(-ve)
exponent bits are (127-20=107) => 01101011
   Divide 107 successively by 2 until the quotient is 0
      > 107/2 = 53, remainder is 1
      > 53/2 = 26, remainder is 1
      > 26/2 = 13, remainder is 0
      > 13/2 = 6, remainder is 1
      > 6/2 = 3, remainder is 0
      > 3/2 = 1, remainder is 1
      > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 1101011
   So, 107 of decimal is 1101011 in binary
frac/significant bits are 01001011011000100011000

so, -1.2345*10^-06 in single-precision format is 1 01101011 01001011011000100011000
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 10110101101001011011000100011000 to hexadecimal
1011 => B
0101 => 5
1010 => A
0101 => 5
1011 => B
0001 => 1
0001 => 1
1000 => 8
So, in hexadecimal 10110101101001011011000100011000 is 0xB5A5B118

in hexadecimal it is 0xB5A5B118