Question

Add the following IEEE 754 single-precision floating-point numbers  2.25 + 13.0625. Write your final answer out in the IEEE 754 format.

Answer 1

2.25
Converting 2.25 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 0.25000000 to binary
      > 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
   0.25 of decimal is .01 in binary
   so, 2.25 in binary is 00000010.01
2.25 in simple binary => 10.01
so, 2.25 in normal binary is 10.01 => 1.001 * 2^1

13.0625
Converting 13.0625 to binary
   Convert decimal part first, then the fractional part
   > First convert 13 to binary
   Divide 13 successively by 2 until the quotient 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 1101
   So, 13 of decimal is 1101 in binary
   > Now, Convert 0.06250000 to binary
      > 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
   0.0625 of decimal is .0001 in binary
   so, 13.0625 in binary is 00001101.0001
13.0625 in simple binary => 1101.0001
so, 13.0625 in normal binary is 1101.0001 => 1.1010001 * 2^3

so, 2.25 + 13.0625
= 1.001 * 2^1 + 1.1010001 * 2^3
= 0.01001 * 2^3 + 1.1010001 * 2^3
= 1.1110101 * 2^3

single precision:
--------------------
sign bit is 0(+ve)
exponent bits are (127+3=130) => 10000010
   Divide 130 successively by 2 until the quotient is 0
      > 130/2 = 65, remainder is 0
      > 65/2 = 32, remainder is 1
      > 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 10000010
   So, 130 of decimal is 10000010 in binary
frac/significant bits are 11101010000000000000000

so, 2.25 + 13.0625 in single-precision format is 0 10000010 11101010000000000000000

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 01000001011101010000000000000000 to hexadecimal
0100 => 4
0001 => 1
0111 => 7
0101 => 5
0000 => 0
0000 => 0
0000 => 0
0000 => 0
So, in hexadecimal 01000001011101010000000000000000 is 0x41750000

in hexadecimal it is 0x41750000