Question

1. Please convert the following decimal numbers to 8-bit binary sign magnitude and to 8-bit 2’s complement. Please show your work if there are multiple steps. Sign Magnitude 2’s complement a. +67 b. +40 c. -28 d. -40

Answer 1

a)
67
-----
Since this is a positive number. we can directly convert this into binary
Divide 67 successively by 2 until the quotient is 0
   > 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 1000011
So, 67 of decimal is 1000011 in binary
So, 67 in normal binary format is 01000011
=====================================
||    1's complement: 01000011    ||
||    2's complement: 01000011    ||
||    sign-magnitude: 01000011    ||
=====================================

b)
40
-----
Since this is a positive number. we can directly convert this into binary
Divide 40 successively by 2 until the quotient is 0
   > 40/2 = 20, remainder is 0
   > 20/2 = 10, remainder is 0
   > 10/2 = 5, remainder 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 101000
So, 40 of decimal is 101000 in binary
So, 40 in normal binary format is 00101000
=====================================
||    1's complement: 00101000    ||
||    2's complement: 00101000    ||
||    sign-magnitude: 00101000    ||
=====================================

c)
-28
------
This is negative. so, follow these steps to convert this to various binary formats.
Divide 28 successively by 2 until the quotient is 0
   > 28/2 = 14, remainder is 0
   > 14/2 = 7, remainder 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 11100
So, 28 of decimal is 11100 in binary
So, 28 in normal binary format is 00011100
sign-magnitude:
-----------------
set 1 as left most bit, since this number is negative.
so, 00011100 becomes 10011100
=====================================
||    sign-magnitude: 10011100    ||
=====================================

1's complement:
-----------------
flip all the bits. Flip all 0's to 1 and all 1's to 0.
   00011100 is flipped to 11100011
=====================================
||    1's complement: 11100011    ||
=====================================

2's complement:
-----------------
Add 1 to above result
11100011 + 1 = 11100100
=====================================
||    2's complement: 11100100    ||
=====================================

d)
-40
------
This is negative. so, follow these steps to convert this to various binary formats.
Divide 40 successively by 2 until the quotient is 0
   > 40/2 = 20, remainder is 0
   > 20/2 = 10, remainder is 0
   > 10/2 = 5, remainder 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 101000
So, 40 of decimal is 101000 in binary
So, 40 in normal binary format is 00101000
sign-magnitude:
-----------------
set 1 as left most bit, since this number is negative.
so, 00101000 becomes 10101000
=====================================
||    sign-magnitude: 10101000    ||
=====================================

1's complement:
-----------------
flip all the bits. Flip all 0's to 1 and all 1's to 0.
   00101000 is flipped to 11010111
=====================================
||    1's complement: 11010111    ||
=====================================

2's complement:
-----------------
Add 1 to above result
11010111 + 1 = 11011000
=====================================
||    2's complement: 11011000    ||
=====================================