Question

Convert 1.8125 to IEEE-754 representation. Show all your work.

Answer 1

1.8125
Converting 1.8125 to binary
   Convert decimal part first, then the fractional part
   > First convert 1 to binary
   Divide 1 successively by 2 until the quotient is 0
       > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 1
   So, 1 of decimal is 1 in binary
   > Now, Convert 0.81250000 to binary
       > Multiply 0.81250000 with 2.    Since 1.62500000 is >= 1. then add 1 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
   0.8125 of decimal is .1101 in binary
   so, 1.8125 in binary is 00000001.1101
1.8125 in simple binary => 1.1101
so, 1.8125 in normal binary is 1.1101 => 1.1101 * 2^0

single precision:
--------------------
sign bit is 0(+ve)
exponent bits are (127+0=127) => 01111111
   Divide 127 successively by 2 until the quotient is 0
       > 127/2 = 63, remainder is 1
       > 63/2 = 31, remainder is 1
       > 31/2 = 15, remainder is 1
       > 15/2 = 7, remainder is 1
       > 7/2 = 3, remainder is 1
       > 3/2 = 1, remainder is 1
       > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 1111111
   So, 127 of decimal is 1111111 in binary
frac/significant bits are 11010000000000000000000

so, 1.8125 in single-precision format is 0 01111111 11010000000000000000000
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 00111111111010000000000000000000 to hexadecimal
0011 => 3
1111 => F
1110 => E
1000 => 8
0000 => 0
0000 => 0
0000 => 0
0000 => 0
So, in hexadecimal 00111111111010000000000000000000 is 0x3FE80000

in hexadecimal it is 0x3FE80000