Question

Using IEEE 754 single precision floating point, write the hexadecimal

representation for each of the following:

a. Zero

b. -2.0 (base 10)

c. 256. 0078125 (base 10)

d. Negative infinity

Answer 1

a.  Zero
0 is stored as all 0's
Answer: 00000000

b.  -2.0
-2.0
Converting 2.0 to binary
   Convert decimal part first, then the fractional part
   > First convert 2 to binary
   Divide 2 successively by 2 until the quotient is 0
      > 2/2 = 1, remainder is 0
      > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 10
   So, 2 of decimal is 10 in binary
   > Now, Convert 0.00000000 to binary
      > Multiply 0.00000000 with 2.  Since 0.00000000 is < 1. then add 0 to result
      > This is equal to 1, so, stop calculating
   0.0 of decimal is .0 in binary
   so, 2.0 in binary is 00000010.0
-2.0 in simple binary => 10.0
so, -2.0 in normal binary is 10.0 => 1. * 2^1

single precision:
--------------------
sign bit is 1(-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 00000000000000000000000

so, -2.0 in single-precision format is 1 10000000 00000000000000000000000
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 11000000000000000000000000000000 to hexadecimal
1100 => C
0000 => 0
0000 => 0
0000 => 0
0000 => 0
0000 => 0
0000 => 0
0000 => 0
So, in hexadecimal 11000000000000000000000000000000 is 0xC0000000

in hexadecimal it is 0xC0000000
Answer: C0000000

c.
256.0078125
Converting 256.0078125 to binary
   Convert decimal part first, then the fractional part
   > First convert 256 to binary
   Divide 256 successively by 2 until the quotient 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 100000000
   So, 256 of decimal is 100000000 in binary
   > Now, Convert 0.00781250 to binary
      > Multiply 0.00781250 with 2.  Since 0.01562500 is < 1. then add 0 to result
      > Multiply 0.01562500 with 2.  Since 0.03125000 is < 1. then add 0 to result
      > Multiply 0.03125000 with 2.  Since 0.06250000 is < 1. then add 0 to result
      > Multiply 0.06250000 with 2.  Since 0.12500000 is < 1. then add 0 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
   0.0078125 of decimal is .0000001 in binary
   so, 256.0078125 in binary is 10000000.0000001
256.0078125 in simple binary => 100000000.0000001
so, 256.0078125 in normal binary is 100000000.0000001 => 1.000000000000001 * 2^8

single precision:
--------------------
sign bit is 0(+ve)
exponent bits are (127+8=135) => 10000111
   Divide 135 successively by 2 until the quotient is 0
      > 135/2 = 67, remainder is 1
      > 67/2 = 33, remainder is 1
      > 33/2 = 16, remainder is 1
      > 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 10000111
   So, 135 of decimal is 10000111 in binary
frac/significant bits are 00000000000000100000000

so, 256.0078125 in single-precision format is 0 10000111 00000000000000100000000
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 01000011100000000000000100000000 to hexadecimal
0100 => 4
0011 => 3
1000 => 8
0000 => 0
0000 => 0
0001 => 1
0000 => 0
0000 => 0
So, in hexadecimal 01000011100000000000000100000000 is 0x43800100

in hexadecimal it is 0x43800100
Answer: 43800100

d)
Negative infinity
sign bit = 1
exponent bits = 11111111
fractional bits = 00000000000000000000000

let's convert 1 11111111 00000000000000000000000 to hexadecimal
Converting 11111111100000000000000000000000 to hexadecimal
1111 => F
1111 => F
1000 => 8
0000 => 0
0000 => 0
0000 => 0
0000 => 0
0000 => 0
So, in hexadecimal 11111111100000000000000000000000 is 0xFF800000
Answer: FF800000