Question

6-bit floating-point encoding:

1 sign bit, 3 exponent bits, 2 frac bits( mantissa/significand)

what is the exact 6-bit floating-point encoding for the following numbers:

17

0.5

-6

7.5

Please show the steps

Answer 1

1)
Converting 17.0 to binary
   Convert decimal part first, then the fractional part
   > First convert 17 to binary
   Divide 17 successively by 2 until the quotient is 0
      > 17/2 = 8, remainder is 1
      > 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 10001
   So, 17 of decimal is 10001 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, 17.0 in binary is 10001.0
17.0 in simple binary => 10001.0
so, 17.0 in normal binary is 10001.0 => 1. * 2^4

6-bit precision:
--------------------
sign bit is 0(+ve)
exp bits are (3+4=7) => 111
   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
frac bits are 00

so, 17.0 in 6-bit precision format is 0 111 00

2)
Converting 0.5 to binary
   Convert decimal part first, then the fractional part
   > First convert 0 to binary
   Divide 0 successively by 2 until the quotient is 0
   Read remainders from the bottom to top as
   So, 0 of decimal is  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, 0.5 in binary is .1
0.5 in simple binary => .1
so, 0.5 in normal binary is .1 => 1. * 2^-1

6-bit precision:
--------------------
sign bit is 0(+ve)
exp bits are (3-1=2) => 010
   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
frac bits are 00

so, 0.5 in 6-bit precision format is 0 010 00

3)
Converting 6.0 to binary
   Convert decimal part first, then the fractional part
   > First convert 6 to binary
   Divide 6 successively by 2 until the quotient is 0
      > 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 110
   So, 6 of decimal is 110 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, 6.0 in binary is 110.0
-6.0 in simple binary => 110.0
so, -6.0 in normal binary is 110.0 => 1.1 * 2^2

6-bit precision:
--------------------
sign bit is 1(-ve)
exp bits are (3+2=5) => 101
   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
frac bits are 10

so, -6.0 in 6-bit precision format is 1 101 10

4)
Converting 7.5 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 0.50000000 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, 7.5 in binary is 111.1
7.5 in simple binary => 111.1
so, 7.5 in normal binary is 111.1 => 1.11 * 2^2

6-bit precision:
--------------------
sign bit is 0(+ve)
exp bits are (3+2=5) => 101
   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
frac bits are 11

so, 7.5 in 6-bit precision format is 0 101 11