Two's Complement on representing negative numbers

Lee Buhrmann 0
Viewed 60

I am currently trying to gain a more intuitive understanding of two's complement and its uses; however, I cannot seem to perform subtraction using two's complement correctly. I understand that when a negative number is stored in a signed int variable the procedure is to perform two's complement on the number having the MSB be 1 to represent the negative sign.

So in a 4 bit system 1010 represents -6.

Now I am following this guide on how to subtract two binary #s

For example I have the number 0101 (5 in decimal) and 1010 (-6 in decimal). If I wanted to do the equation 5 - (-6) it would look like 0101 - 1010 in binary. Next I would take the 1010 and perform twos' complement on it to get 0110. Now I take 0101 + 0110 and get 1011. I don't have a carry so I perform two's complement on the result giving me 0101, but this says the answer is -5 when it should be 11.

twos-complement
3 Answers

4 bits can represent 16 different values. With two's complement, they are -8 to 7.

1000  -8
1001  -7
1010  -6
1011  -5
1100  -4
1101  -3
1110  -2
1111  -1
0000   0
0001   1
0010   2
0011   3
0100   4
0101   5
0110   6
0111   7

You can't possibly get an answer of 11. It's too large to fit in 4 bits as a two's complement number. As such, the calculation you are attempting results in an overflow. (See below for how to detect overflows.)

This means 4 bits is not enough to compute 5 - -6. You need at least 5 bits.

  0(...0)0101    5
- 1(...1)1010   -6
------------------
  0(...0)0101    5
+ 0(...0)0110    6
------------------
  0(...0)1011   11

Detecting Overflow in Two's Complement

With two's complement numbers, an addition overflows when the carry into the sign bit is different than the carry out of it.

  0101    5
+ 0110    6
-----------
  1011   -5   Carry in: 1  Carry out: 0  OVERFLOW!

  1011   -5
+ 1010   -6
-----------
  0101    5   Carry in: 0  Carry out: 1  OVERFLOW!

  0001    1
+ 0010    2
-----------
  0011    3   Carry in: 0  Carry out: 0  No overflow

  1111   -1
+ 1110   -2
-----------
  1101   -3   Carry in: 1  Carry out: 1  No overflow
ikegami0

For example I have the number 0101 (5 in decimal) and 1010 (-6 in decimal). If I wanted to do the equation 5 - (-6) it would look like 0101 - 1010 in binary.

Ok.

Next I would take the 1010 and perform twos' complement on it to get 0110.

Ok. And 0110 binary is 6 decimal is -(-6) decimal.

Now I take 0101 + 0110 and get 1011.

Ok.

I don't have a carry so I perform two's complement on the result giving me 0101,

I take you to mean that as part of the process of interpreting the result, not computing it. The result itself is 1011 (4-bit, two's complement binary).

but this says the answer is -5 when it should be 11.

The answer in the operational system you have chosen, with 4-bit two's complement, is -5, exactly as you have computed. With three data bits and one sign bit, the maximum result your data type can represent is 7 (0111 binary). This underscores the importance of choosing data types appropriate for the computations you want to perform.

John Bollinger0

Note that 0b1011 is 11 in binary, if interpreted as unsigned or with more than 3 data bits, so both 11 and -5 are the result (which you use depends on how its interpreted).

In terms of modular arithmetic, 4 bits is a base of 24, and:

11 ≡ -5 (mod 16)

Which answer you use depends on whether the nibble is interpreted as signed or unsigned.

outis0
Related