Question

1. Convert 5.5 to hexadecimal notation using IEEE 754 single precision. Please show your
work and answer must be in hexadecimal notation.

2. (4 points) Convert -7.875 to hexadecimal notation using IEEE 754 single precision. Please show your
work and answer must be in hexadecimal notation.

Answer 1

1)
5.5
Converting 5.5 to binary
   Convert decimal part first, then the fractional part
   > First convert 5 to binary
   Divide 5 successively by 2 until the quotient 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 101
   So, 5 of decimal is 101 in binary
   > Now, Convert 0.50000000 to binary
      > Multiply 0.50000000 with 2.  Since 1.00000000 is >= 1. then add 1 to result
      > This is equal to 1, so, stop calculating
   0.5 of decimal is .1 in binary
   so, 5.5 in binary is 00000101.1
5.5 in simple binary => 101.1
so, 5.5 in normal binary is 101.1 => 1.011 * 2^2

single precision:
--------------------
sign bit is 0(+ve)
exponent bits are (127+2=129) => 10000001
   Divide 129 successively by 2 until the quotient is 0
      > 129/2 = 64, remainder is 1
      > 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 10000001
   So, 129 of decimal is 10000001 in binary
frac/significant bits are 01100000000000000000000

so, 5.5 in single-precision format is 0 10000001 01100000000000000000000
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 01000000101100000000000000000000 to hexadecimal
0100 => 4
0000 => 0
1011 => B
0000 => 0
0000 => 0
0000 => 0
0000 => 0
0000 => 0
So, in hexadecimal 01000000101100000000000000000000 is 0x40B00000

in hexadecimal it is 0x40B00000

2)
-7.875
Converting 7.875 to binary
   Convert decimal part first, then the fractional part
   > First convert 7 to binary
   Divide 7 successively by 2 until the quotient is 0
      > 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 111
   So, 7 of decimal is 111 in binary
   > Now, Convert 0.87500000 to binary
      > 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
   0.875 of decimal is .111 in binary
   so, 7.875 in binary is 00000111.111
-7.875 in simple binary => 111.111
so, -7.875 in normal binary is 111.111 => 1.11111 * 2^2

single precision:
--------------------
sign bit is 1(-ve)
exponent bits are (127+2=129) => 10000001
   Divide 129 successively by 2 until the quotient is 0
      > 129/2 = 64, remainder is 1
      > 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 10000001
   So, 129 of decimal is 10000001 in binary
frac/significant bits are 11111000000000000000000

so, -7.875 in single-precision format is 1 10000001 11111000000000000000000
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 11000000111111000000000000000000 to hexadecimal
1100 => C
0000 => 0
1111 => F
1100 => C
0000 => 0
0000 => 0
0000 => 0
0000 => 0
So, in hexadecimal 11000000111111000000000000000000 is 0xC0FC0000

in hexadecimal it is 0xC0FC0000