Question

1) given the 8-bit binary number of 10111110 (two)

a)what is the hex representation of this number/

b) what is the decimal value if this number which is an 8-bit number as a two complement signed number. all steps included

c)what is the decimal value of this number as an 8-bit unsigned number, please write all steps

2) given the decimal number (-0.875) or (-7/8) convert it to single-precision floating-point. please show the easiest way to get to this,.

Answer 1

a)
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 10111110 to hexadecimal
1011 => B
1110 => E
So, in hexadecimal 10111110 is 0xBE
Answer: 0xBE

b)
since left most bit is 1, this number is negative number.
so, follow these steps below to convert this into a decimal value.
I. first flip all the bits. Flip all 0's to 1 and all 1's to 0.
   10111110 is flipped to 01000001
II. Add 1 to above result
01000001 + 1 = 01000010
III. Now convert this result to decimal value
=> 1000010
=> 1x2^6+0x2^5+0x2^4+0x2^3+0x2^2+1x2^1+0x2^0
=> 1x64+0x32+0x16+0x8+0x4+1x2+0x1
=> 64+0+0+0+0+2+0
=> 66
Answer: -66

c)
=> 10111110
=> 1x2^7+0x2^6+1x2^5+1x2^4+1x2^3+1x2^2+1x2^1+0x2^0
=> 1x128+0x64+1x32+1x16+1x8+1x4+1x2+0x1
=> 128+0+32+16+8+4+2+0
=> 190
Answer: 190

d)
Converting 0.875 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 0.875 to binary
      > Multiply 0.875 with 2.   Since 1.75 is >= 1. then add 1 to result
      > Multiply 0.75 with 2.    Since 1.5 is >= 1. then add 1 to result
      > Multiply 0.5 with 2.     Since 1.0 is >= 1. then add 1 to result
      > This is equal to 1, so, stop calculating
   0.875 of decimal is .111 in binary
   so, 0.875 in binary is .111
-0.875 in simple binary => .111
so, -0.875 in normal binary is .111 => 1.11 * 2^-1

single precision:
--------------------
sign bit is 1(-ve)
exp bits are (127-1=126) => 01111110
   Divide 126 successively by 2 until the quotient is 0
      > 126/2 = 63, remainder is 0
      > 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 1111110
   So, 126 of decimal is 1111110 in binary
frac bits are 11000000000000000000000

so, -0.875 in single-precision format is 1 01111110 11000000000000000000000
in hexadecimal it is 0xBF600000