What is “two's complement”?

Question

I'm in a computer systems course and have been struggling, in part, with two's complement. I want to understand it, but everything I've read hasn't brought the picture together for me. I've read the Wikipedia article and various other articles, including my text book.

What is two's complement, how can we use it and how can it affect numbers during operations like casts (from signed to unsigned and vice versa), bit-wise operations and bit-shift operations?

Answer 1

everything I've read hasn't brought the picture together for me

Positive numbers start at 1 and work up. Negative numbers start at the top and work down.

That's all. Nothing left to understand. The rest is just math and hardware.

Beauty of this system (for bit limited numbers): addition works. +1 plus -1 = zero

It's a system for bit-limited numbers (8 bit, 16 bit, 32 bit, whatever) because there has to be a "top" for where negative numbers start, and because the addition sometimes generates a 'carry' into or from the next column, which is magically discarded when there is no next column to carry into or out of.

16 bit Hex: Positive number start at 1 and work up to 0x7FFF. Negative numbers start at 0xFFFF and work down to to 8000.

4 bit binary: positive numbers start at 0001 and work up to 0111. Negative numbers start at 1111 and work down to 1000

3 bit binary, negative numbers:

111 = -1
110 = -2
101 = -3
100 = -4

Addition examples:

11+01 = 100 = 00 (2 bit answer)
110 + 001 = 111 (3 bits: -2 + 1 = -1)
111 + 011 = 1010 = 010 (3 bits: -1+3 = 2)

You can calculate the two's complement of a number (complement + 1) if you want to, but you don't have to do that to use two's complement, any more than you have to for positive numbers: you can just learn to use the representation the same way as you do for positive numbers. 1 is positive one. 11111111 is negative one. FFFF is negative one. FFFE is negative 2. FFFD is negative 3.

You sometimes see explanations of two's complement as "avoiding the problem of zero representation" but that's actually just a special case of the reason hardware designers use two's complement: addition just works.

There are other methods of representing negative numbers, other than just "starts at the top and works down". I've also used systems where there are two sign bits, or two separate registers, one for + numbers and one for - numbers, or where the sign bit represents the state of an inverter in front of a single-sided counting-up ADC. When people first started designing computer hardware, it wasn't clear that "starting at the top and counting down" would be the best representation for negative numbers. But it makes addition easy, and that's a winner.

Answer 2

Many of the answers so far nicely explain why two's complement is used to represent negative numbers, but do not tell us what two's complement number is, particularly not why a '1' is added, and in fact often added in a wrong way.

The confusion comes from a poor understanding of the definition of a complement number. A complement is the missing part that would make something complete.

The radix complement of an n digit number x in radix b is, by definition, b^n-x.

In binary 4 is represented by 100, which has 3 digits (n=3) and a radix of 2 (b=2). So its radix complement is b^n-x = 2^3-4=8-4=4 (or 100 in binary).

However, in binary obtaining a radix's complement is not as easy as getting its diminished radix complement, which is defined as (b^n-1)-y, just 1 less than that of radix complement. To get a diminished radix complement, you simply flip all the digits.

100 -> 011 (diminished (one's) radix complement)

to obtain the radix (two's) complement, we simply add 1, as the definition defined.

011 +1 ->100 (two's complement).

Now with this new understanding, let's take a look of the example given by Vincent Ramdhanie (see above second response):

Converting 1111 to decimal:

The number starts with 1, so it's negative, so we find the complement of 1111, which is 0000.
Add 1 to 0000, and we obtain 0001.
Convert 0001 to decimal, which is 1.
Apply the sign = -1.
Tada!

Should be understood as:

The number starts with 1, so it's negative. So we know it is a two's complement of some value x. To find the x represented by its two's complement, we first need find its 1's complement.

two's complement of x: 1111
one's complement of x: 1111-1 ->1110;
x = 0001, (flip all digits)

Apply the sign -, and the answer =-x =-1.

Answer 3

Reference: Two's Complement (Thomas Finley)

I invert all the bits and add 1. Programmatically:

  // In C++11
  int _powers[] = {
      1,
      2,
      4,
      8,
      16,
      32,
      64,
      128
  };

  int value = 3;
  int n_bits = 4;
  int twos_complement = (value ^ ( _powers[n_bits]-1)) + 1;

Answer 4

Two's complement of a given number is the number got by adding 1 with the ones' complement of the number.

Suppose, we have a binary number: 10111001101

Its 1's complement is: 01000110010

And its two's complement will be: 01000110011

Answer 5

In simple terms, two's complement is a way to store negative numbers in computer memory. Whereas positive numbers are stored as a normal binary number.

Let's consider this example,

The computer uses the binary number system to represent any number.

x = 5;

This is represented as 0101.

x = -5;

When the computer encounters the - sign, it computes its two's complement and stores it.

That is, 5 = 0101 and its two's complement is 1011.

The important rules the computer uses to process numbers are,

1. If the first bit is 1 then it must be a negative number.
2. If all the bits except first bit are 0 then it is a positive number, because there is no -0 in number system (1000 is not -0 instead it is positive 8).
3. If all the bits are 0 then it is 0.
4. Else it is a positive number.

Answer 6

Two's complement is one of the ways of expressing a negative number and most of the controllers and processors store a negative number in two's complement form.

Answer 7

The question is 'What is “two's complement”?'

The simple answer for those wanting to understand it theoretically (and me seeking to complement the other more practical answers): 2's complement is the representation for negative integers in the dual system that does not require additional characters, such as + and -.

Answer 8

Two's complement is a clever way of storing integers so that common math problems are very simple to implement.

To understand, you have to think of the numbers in binary.

It basically says,

• for zero, use all 0's.
• for positive integers, start counting up, with a maximum of 2^{(number of bits - 1)}-1.
• for negative integers, do exactly the same thing, but switch the role of 0's and 1's and count down (so instead of starting with 0000, start with 1111 - that's the "complement" part).

Let's try it with a mini-byte of 4 bits (we'll call it a nibble - 1/2 a byte).

• 0000 - zero
• 0001 - one
• 0010 - two
• 0011 - three
• 0100 to 0111 - four to seven

That's as far as we can go in positives. 2³-1 = 7.

For negatives:

• 1111 - negative one
• 1110 - negative two
• 1101 - negative three
• 1100 to 1000 - negative four to negative eight

Note that you get one extra value for negatives (1000 = -8) that you don't for positives. This is because 0000 is used for zero. This can be considered as Number Line of computers.

Distinguishing between positive and negative numbers

Doing this, the first bit gets the role of the "sign" bit, as it can be used to distinguish between nonnegative and negative decimal values. If the most significant bit is 1, then the binary can be said to be negative, where as if the most significant bit (the leftmost) is 0, you can say the decimal value is nonnegative.

"Sign-magnitude" negative numbers just have the sign bit flipped of their positive counterparts, but this approach has to deal with interpreting 1000 (one 1 followed by all 0s) as "negative zero" which is confusing.

"Ones' complement" negative numbers are just the bit-complement of their positive counterparts, which also leads to a confusing "negative zero" with 1111 (all ones).

You will likely not have to deal with Ones' Complement or Sign-Magnitude integer representations unless you are working very close to the hardware.

Answer 9

I wonder if it could be explained any better than the Wikipedia article.

The basic problem that you are trying to solve with two's complement representation is the problem of storing negative integers.

First, consider an unsigned integer stored in 4 bits. You can have the following

0000 = 0
0001 = 1
0010 = 2
...
1111 = 15

These are unsigned because there is no indication of whether they are negative or positive.

Sign Magnitude and Excess Notation

To store negative numbers you can try a number of things. First, you can use sign magnitude notation which assigns the first bit as a sign bit to represent +/- and the remaining bits to represent the magnitude. So using 4 bits again and assuming that 1 means - and 0 means + then you have

0000 = +0
0001 = +1
0010 = +2
...
1000 = -0
1001 = -1
1111 = -7

So, you see the problem there? We have positive and negative 0. The bigger problem is adding and subtracting binary numbers. The circuits to add and subtract using sign magnitude will be very complex.

What is

0010
1001 +
----

?

Another system is excess notation. You can store negative numbers, you get rid of the two zeros problem but addition and subtraction remains difficult.

So along comes two's complement. Now you can store positive and negative integers and perform arithmetic with relative ease. There are a number of methods to convert a number into two's complement. Here's one.

Convert Decimal to Two's Complement

1. Convert the number to binary (ignore the sign for now) e.g. 5 is 0101 and -5 is 0101

2. If the number is a positive number then you are done. e.g. 5 is 0101 in binary using two's complement notation.

3. If the number is negative then

   3.1 find the complement (invert 0's and 1's) e.g. -5 is 0101 so finding the complement is 1010

   3.2 Add 1 to the complement 1010 + 1 = 1011. Therefore, -5 in two's complement is 1011.

So, what if you wanted to do 2 + (-3) in binary? 2 + (-3) is -1. What would you have to do if you were using sign magnitude to add these numbers? 0010 + 1101 = ?

Using two's complement consider how easy it would be.

 2  =  0010
 -3 =  1101 +
 -------------
 -1 =  1111

Converting Two's Complement to Decimal

Converting 1111 to decimal:

1. The number starts with 1, so it's negative, so we find the complement of 1111, which is 0000.

2. Add 1 to 0000, and we obtain 0001.

3. Convert 0001 to decimal, which is 1.

4. Apply the sign = -1.

Tada!

Answer 10

2's complement is essentially a way of coming up with the additive inverse of a binary number. Ask yourself this: Given a number in binary form (present at a fixed length memory location), what bit pattern, when added to the original number (at the fixed length memory location), would make the result all zeros ? (at the same fixed length memory location). If we could come up with this bit pattern then that bit pattern would be the -ve representation (additive inverse) of the original number; as by definition adding a number to its additive inverse always results in zero. Example: take 5 which is 101 present inside a single 8 bit byte. Now the task is to come up with a bit pattern which when added to the given bit pattern (00000101) would result in all zeros at the memory location which is used to hold this 5 i.e. all 8 bits of the byte should be zero. To do that, start from the right most bit of 101 and for each individual bit, again ask the same question: What bit should I add to the current bit to make the result zero ? continue doing that taking in account the usual carry over. After we are done with the 3 right most places (the digits that define the original number without regard to the leading zeros) the last carry goes in the bit pattern of the additive inverse. Furthermore, since we are holding in the original number in a single 8 bit byte, all other leading bits in the additive inverse should also be 1's so that (and this is important) when the computer adds "the number" (represented using the 8 bit pattern) and its additive inverse using "that" storage type (a byte) the result in that byte would be all zeros.

 1 1 1
 ----------
   1 0 1
 1 0 1 1 ---> additive inverse
  ---------
   0 0 0

Answer 11

Two's complement is mainly used for the following reasons:

1. To avoid multiple representations of 0
2. To avoid keeping track of carry bit (as in one's complement) in case of an overflow.
3. Carrying out simple operations like addition and subtraction becomes easy.

Answer 12

The word complement derives from completeness. In the decimal world the numerals 0 through 9 provide a complement (complete set) of numerals or numeric symbols to express all decimal numbers. In the binary world the numerals 0 and 1 provide a complement of numerals to express all binary numbers. In fact The symbols 0 and 1 must be used to represent everything (text, images, etc) as well as positive (0) and negative (1). In our world the blank space to the left of number is considered as zero:

                  35=035=000000035.

In a computer storage location there is no blank space. All bits (binary digits) must be either 0 or 1. To efficiently use memory numbers may be stored as 8 bit, 16 bit, 32 bit, 64 bit, 128 bit representations. When a number that is stored as an 8 bit number is transferred to a 16 bit location the sign and magnitude (absolute value) must remain the same. Both 1's complement and 2's complement representations facilitate this. As a noun: Both 1's complement and 2's complement are binary representations of signed quantities where the most significant bit (the one on the left) is the sign bit. 0 is for positive and 1 is for negative. 2s complement does not mean negative. It means a signed quantity. As in decimal the magnitude is represented as the positive quantity. The structure uses sign extension to preserve the quantity when promoting to a register [] with more bits:

       [0101]=[00101]=[00000000000101]=5 (base 10)
       [1011]=[11011]=[11111111111011]=-5(base 10)

As a verb: 2's complement means to negate. It does not mean make negative. It means if negative make positive; if positive make negative. The magnitude is the absolute value:

        if a >= 0 then |a| = a
        if a < 0 then |a| = -a = 2scomplement of a

This ability allows efficient binary subtraction using negate then add. a - b = a + (-b)

The official way to take the 1's complement is for each digit subtract its value from 1.

        1'scomp(0101) = 1010.

This is the same as flipping or inverting each bit individually. This results in a negative zero which is not well loved so adding one to te 1's complement gets rid of the problem. To negate or take the 2s complement first take the 1s complement then add 1.

        Example 1                             Example 2
         0101  --original number              1101
         1's comp  1010                       0010
         add 1     0001                       0001
         2's comp  1011  --negated number     0011

In the examples the negation works as well with sign extended numbers.

Adding:
1110 Carry 111110 Carry 0110 is the same as 000110 1111 111111 sum 0101 sum 000101

SUbtracting:

    1110  Carry                      00000   Carry
     0110          is the same as     00110
    -0111                            +11001
  ----------                        ----------
sum  0101                       sum   11111

Notice that when working with 2's complement, blank space to the left of the number is filled with zeros for positive numbers butis filled with ones for negative numbers. The carry is always added and must be either a 1 or 0.

Cheers

Answer 13

To bitwise complement a number is to flip all the bits in it. To two’s complement it, we flip all the bits and add one.

Using 2’s complement representation for signed integers, we apply the 2’s complement operation to convert a positive number to its negative equivalent and vice versa. So using nibbles for an example, 0001 (1) becomes 1111 (-1) and applying the op again, returns to 0001.

The behaviour of the operation at zero is advantageous in giving a single representation for zero without special handling of positive and negative zeroes. 0000 complements to 1111, which when 1 is added. overflows to 0000, giving us one zero, rather than a positive and a negative one.

A key advantage of this representation is that the standard addition circuits for unsigned integers produce correct results when applied to them. For example adding 1 and -1 in nibbles: 0001 + 1111, the bits overflow out of the register, leaving behind 0000.

For a gentle introduction, the wonderful Computerphile have produced a video on the subject.

Answer 14

I read a fantastic explanation on Reddit by jng, using the odometer as an analogy.

It is a useful convention. The same circuits and logic operations that add / subtract positive numbers in binary still work on both positive and negative numbers if using the convention, that's why it's so useful and omnipresent.

Imagine the odometer of a car, it rolls around at (say) 99999. If you increment 00000 you get 00001. If you decrement 00000, you get 99999 (due to the roll-around). If you add one back to 99999 it goes back to 00000. So it's useful to decide that 99999 represents -1. Likewise, it is very useful to decide that 99998 represents -2, and so on. You have to stop somewhere, and also by convention, the top half of the numbers are deemed to be negative (50000-99999), and the bottom half positive just stand for themselves (00000-49999). As a result, the top digit being 5-9 means the represented number is negative, and it being 0-4 means the represented is positive - exactly the same as the top bit representing sign in a two's complement binary number.

Understanding this was hard for me too. Once I got it and went back to re-read the books articles and explanations (there was no internet back then), it turned out a lot of those describing it didn't really understand it. I did write a book teaching assembly language after that (which did sell quite well for 10 years).

Answer 15

Like most explanations I've seen, the ones above are clear about how to work with 2's complement, but don't really explain what they are mathematically. I'll try to do that, for integers at least, and I'll cover some background that's probably familiar first.

Recall how it works for decimal:
  2345
is a way of writing
  2 × 10³ + 3 × 10² + 4 × 10¹ + 5 × 10⁰.

In the same way, binary is a way of writing numbers using just 0 and 1 following the same general idea, but replacing those 10s above with 2s. Then in binary,
  1111
is a way of writing
  1 × 2³ + 1 × 2² + 1 × 2¹ + 1 × 2⁰
and if you work it out, that turns out to equal 15 (base 10). That's because it is
  8+4+2+1 = 15.

This is all well and good for positive numbers. It even works for negative numbers if you're willing to just stick a minus sign in front of them, as humans do with decimal numbers. That can even be done in computers, sort of, but I haven't seen such a computer since the early 1970's. I'll leave the reasons for a different discussion.

For computers it turns out to be more efficient to use a complement representation for negative numbers. And here's something that is often overlooked. Complement notations involve some kind of reversal of the digits of the number, even the implied zeroes that come before a normal positive number. That's awkward, because the question arises: all of them? That could be an infinite number of digits to be considered.

Fortunately, computers don't represent infinities. Numbers are constrained to a particular length (or width, if you prefer). So let's return to positive binary numbers, but with a particular size. I'll use 8 digits ("bits") for these examples. So our binary number would really be
  00001111
or
  0 × 2⁷ + 0 × 2⁶ + 0 × 2⁵ + 0 × 2⁴ + 1 × 2³ + 1 × 2² + 1 × 2¹ + 1 × 2⁰

To form the 2's complement negative, we first complement all the (binary) digits to form
  11110000
and add 1 to form
  11110001
but how are we to understand that to mean -15?

The answer is that we change the meaning of the high-order bit (the leftmost one). This bit will be a 1 for all negative numbers. The change will be to change the sign of its contribution to the value of the number it appears in. So now our 11110001 is understood to represent
  -1 × 2⁷ + 1 × 2⁶ + 1 × 2⁵ + 1 × 2⁴ + 0 × 2³ + 0 × 2² + 0 × 2¹ + 1 × 2⁰
Notice that "-" in front of that expression? It means that the sign bit carries the weight -2⁷, that is -128 (base 10). All the other positions retain the same weight they had in unsigned binary numbers.

Working out our -15, it is
  -128 + 64 + 32 + 16 + 1
Try it on your calculator. it's -15.

Of the three main ways that I've seen negative numbers represented in computers, 2's complement wins hands down for convenience in general use. It has an oddity, though. Since it's binary, there have to be an even number of possible bit combinations. Each positive number can be paired with its negative, but there's only one zero. Negating a zero gets you zero. So there's one more combination, the number with 1 in the sign bit and 0 everywhere else. The corresponding positive number would not fit in the number of bits being used.

What's even more odd about this number is that if you try to form its positive by complementing and adding one, you get the same negative number back. It seems natural that zero would do this, but this is unexpected and not at all the behavior we're used to because computers aside, we generally think of an unlimited supply of digits, not this fixed-length arithmetic.

This is like the tip of an iceberg of oddities. There's more lying in wait below the surface, but that's enough for this discussion. You could probably find more if you research "overflow" for fixed-point arithmetic. If you really want to get into it, you might also research "modular arithmetic".

Answer 16

The simplest answer:

1111 + 1 = (1)0000. So 1111 must be -1. Then -1 + 1 = 0.

It's perfect to understand these all for me.

Answer 17

Looking at the two's complement system from a math point of view it really makes sense. In ten's complement, the idea is to essentially 'isolate' the difference.

Example: 63 - 24 = x

We add the complement of 24 which is really just (100 - 24). So really, all we are doing is adding 100 on both sides of the equation.

Now the equation is: 100 + 63 - 24 = x + 100, that is why we remove the 100 (or 10 or 1000 or whatever).

Due to the inconvenient situation of having to subtract one number from a long chain of zeroes, we use a 'diminished radix complement' system, in the decimal system, nine's complement.

When we are presented with a number subtracted from a big chain of nines, we just need to reverse the numbers.

Example: 99999 - 03275 = 96724

That is the reason, after nine's complement, we add 1. As you probably know from childhood math, 9 becomes 10 by 'stealing' 1. So basically it's just ten's complement that takes 1 from the difference.

In Binary, two's complement is equatable to ten's complement, while one's complement to nine's complement. The primary difference is that instead of trying to isolate the difference with powers of ten (adding 10, 100, etc. into the equation) we are trying to isolate the difference with powers of two.

It is for this reason that we invert the bits. Just like how our minuend is a chain of nines in decimal, our minuend is a chain of ones in binary.

Example: 111111 - 101001 = 010110

Because chains of ones are 1 below a nice power of two, they 'steal' 1 from the difference like nine's do in decimal.

When we are using negative binary number's, we are really just saying:

0000 - 0101 = x

1111 - 0101 = 1010

1111 + 0000 - 0101 = x + 1111

In order to 'isolate' x, we need to add 1 because 1111 is one away from 10000 and we remove the leading 1 because we just added it to the original difference.

1111 + 1 + 0000 - 0101 = x + 1111 + 1

10000 + 0000 - 0101 = x + 10000

Just remove 10000 from both sides to get x, it's basic algebra.

Answer 18

Two complement is found out by adding one to 1'st complement of the given number. Lets say we have to find out twos complement of 10101 then find its ones complement, that is, 01010 add 1 to this result, that is, 01010+1=01011, which is the final answer.

Answer 19

2’s Complements: When we add an extra one with the 1’s complements of a number we will get the 2’s complements. For example: 100101 it’s 1’s complement is 011010 and 2’s complement is 011010+1 = 011011 (By adding one with 1's complement) For more information this article explain it graphically.

Answer 20

Lets get the answer 10 – 12 in binary form using 8 bits: What we will really do is 10 + (-12)

We need to get the compliment part of 12 to subtract it from 10. 12 in binary is 00001100. 10 in binary is 00001010.

To get the compliment part of 12 we just reverse all the bits then add 1. 12 in binary reversed is 11110011. This is also the Inverse code (one's complement). Now we need to add one, which is now 11110100.

So 11110100 is the compliment of 12! Easy when you think of it this way.

Now you can solve the above question of 10 - 12 in binary form.

00001010
11110100
-----------------
11111110  

Answer 21

2's complement is very useful for finding the value of a binary, however I thought of a much more concise way of solving such a problem(never seen anyone else publish it):

take a binary, for example: 1101 which is [assuming that space "1" is the sign] equal to -3.

using 2's complement we would do this...flip 1101 to 0010...add 0001 + 0010 ===> gives us 0011. 0011 in positive binary = 3. therefore 1101 = -3!

What I realized:

instead of all the flipping and adding, you can just do the basic method for solving for a positive binary(lets say 0101) is (2³ * 0) + (2² * 1) + (2¹ * 0) + (2⁰ * 1) = 5.

Do exactly the same concept with a negative!(with a small twist)

take 1101, for example:

for the first number instead of 2³ * 1 = 8 , do -(2³ * 1) = -8.

then continue as usual, doing -8 + (2² * 1) + (2¹ * 0) + (2⁰ * 1) = -3

Answer 22

It is a clever means of encoding negative integers in such a way that approximately half of the combination of bits of a data type are reserved for negative integers, and the addition of most of the negative integers with their corresponding positive integers results in a carry overflow that leaves the result to be binary zero.

So, in 2's complement if one is 0x0001 then -1 is 0x1111, because that will result in a combined sum of 0x0000 (with an overflow of 1).

Answer 23

Imagine that you have a finite number of bits/trits/digits/whatever. You define 0 as all digits being 0, and count upwards naturally:

00
01
02
..

Eventually you will overflow.

98
99
00

We have two digits and can represent all numbers from 0 to 100. All those numbers are positive! Suppose we want to represent negative numbers too?

What we really have is a cycle. The number before 2 is 1. The number before 1 is 0. The number before 0 is... 99.

So, for simplicity, let's say that any number over 50 is negative. "0" through "49" represent 0 through 49. "99" is -1, "98" is -2, ... "50" is -50.

This representation is ten's complement. Computers typically use two's complement, which is the same except using bits instead of digits.

The nice thing about ten's complement is that addition just works. You do not need to do anything special to add positive and negative numbers!

Answer 24

I liked lavinio's answer, but shifting bits adds some complexity. Often there's a choice of moving bits while respecting the sign bit or while not respecting the sign bit. This is the choice between treating the numbers as signed (-8 to 7 for a nibble, -128 to 127 for bytes) or full-range unsigned numbers (0 to 15 for nibbles, 0 to 255 for bytes).