Question

Given: Decimal Numbers 123₁₀ , 994₁₀ , and -372₁₀

3.1 Express the above numbers in 12-bit Hexadecimal representation (show all work)

Answer 1

Using 2's complement representation to convert these decimal numbers to binary

1)
Since this is a positive number. we can directly convert this into binary
Divide 123 successively by 2 until the quotient is 0
   > 123/2 = 61, remainder is 1
   > 61/2 = 30, remainder is 1
   > 30/2 = 15, remainder is 0
   > 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 1111011
So, 123 of decimal is 1111011 in binary
Adding 5 zeros on left hand side of this number to make this of length 12
so, 123 in 2's complement binary is 000001111011

Now, let's convert {} to hexadecimal 000001111011
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 000001111011 to hexadecimal
0000 => 0
0111 => 7
1011 => B
So, in hexadecimal 000001111011 is 0x07B
Answer: 0x07B

2)
Since this is a positive number. we can directly convert this into binary
Divide 994 successively by 2 until the quotient is 0
   > 994/2 = 497, remainder is 0
   > 497/2 = 248, remainder is 1
   > 248/2 = 124, remainder is 0
   > 124/2 = 62, remainder is 0
   > 62/2 = 31, remainder is 0
   > 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 1111100010
So, 994 of decimal is 1111100010 in binary
Adding 2 zeros on left hand side of this number to make this of length 12
so, 994 in 2's complement binary is 001111100010

Now, let's convert {} to hexadecimal 001111100010
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 001111100010 to hexadecimal
0011 => 3
1110 => E
0010 => 2
So, in hexadecimal 001111100010 is 0x3E2
Answer: 0x3E2

3)
This is negative. so, follow these steps to convert this into a 2's complement binary
Step 1:
Divide 372 successively by 2 until the quotient is 0
   > 372/2 = 186, remainder is 0
   > 186/2 = 93, remainder is 0
   > 93/2 = 46, remainder is 1
   > 46/2 = 23, remainder is 0
   > 23/2 = 11, remainder is 1
   > 11/2 = 5, remainder is 1
   > 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 101110100
So, 372 of decimal is 101110100 in binary
Adding 3 zeros on left hand side of this number to make this of length 12
So, 372 in normal binary is 000101110100
Step 2: flip all the bits. Flip all 0's to 1 and all 1's to 0.
   000101110100 is flipped to 111010001011
Step 3:. Add 1 to above result
111010001011 + 1 = 111010001100
so, -372 in 2's complement binary is 111010001100

Now, let's convert {} to hexadecimal 111010001100
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 111010001100 to hexadecimal
1110 => E
1000 => 8
1100 => C
So, in hexadecimal 111010001100 is 0xE8C
Answer: 0xE8C