Question

Please show step by step

What is the binary value in two's complement representation for -23₁₀

Answer 1

If you want this in 8-bit 2's complement:
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This is negative. so, follow these steps to convert this into a 2's complement binary
Step 1:
Divide 23 successively by 2 until the quotient 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 10111
Adding 3 zeros on left hand side of this number to make this of length 8
So, 23 in normal binary is 00010111
Step 2: flip all the bits. Flip all 0's to 1 and all 1's to 0.
   00010111 is flipped to 11101000
Step 3:. Add 1 to above result
11101000 + 1 = 11101001
so, -23 in 2's complement binary is 11101001


If you want this in 16-bit 2's complement:
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This is negative. so, follow these steps to convert this into a 2's complement binary
Step 1:
Divide 23 successively by 2 until the quotient 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 10111
Adding 11 zeros on left hand side of this number to make this of length 16
So, 23 in normal binary is 0000000000010111
Step 2: flip all the bits. Flip all 0's to 1 and all 1's to 0.
   0000000000010111 is flipped to 1111111111101000
Step 3:. Add 1 to above result
1111111111101000 + 1 = 1111111111101001
so, -23 in 2's complement binary is 1111111111101001