What is 2 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   Hence  I wanted to start this community wiki post to define what Two s Complement is  how to use it and how it can affect numbers during operations like casts  from signed to unsigned and vice versa   bit-wise operations and bit-shift operations   What I m hoping for is a clear and concise definition that is easily understood by a programmer

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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

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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

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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  so instead of starting with 0000  start with 1111 - that s the  quot complement quot  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  quot sign quot  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   quot Sign-magnitude quot  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  quot negative zero quot  which is confusing   quot Ones  complement quot  negative numbers are just the bit-complement of their positive counterparts  which also leads to a confusing  quot negative zero quot  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

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Many of the answers so far nicely explain why two s complement is used to represent negative number  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 represent 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       start of Vincent  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   end of Vincent     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

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Two s complement is mainly used for the following reasons     To avoid multiple representations of 0 To avoid keeping track of carry bit  as in one s complement  in case of an overflow  Carrying out simple operations like addition and subtraction becomes easy

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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

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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

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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

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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  gt   0 then  a    a         if a  lt  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

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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

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You can also use an online calculator to calculate the two s complement binary representation of a decimal number  http   www convertforfree com twos-complement-calculator

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Two s complement is one of the way of expressing a negative number and most of the controllers and processors store a negative number in 2 s complement form

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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

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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

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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   nbsp  nbsp 2345 is a way of writing  nbsp  nbsp 2  times  103   3  times  102   4  times  101   5  times  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  nbsp  nbsp 1111is a way of writing nbsp  nbsp 1  times  23   1  times  22   1  times  21   1  times  20and if you work it out  that turns out to equal 15  base 10    That s because it is nbsp  nbsp 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 nbsp  nbsp 00001111or nbsp  nbsp 0  times  27   0  times  26   0  times  25   0  times  24   1  times  23   1  times  22   1  times  21   1  times  20  To form the 2 s complement negative  we first complement all the  binary  digits to form  nbsp  nbsp 11110000and add 1 to form  nbsp  nbsp 11110001but 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 nbsp  nbsp -1  times  27   1  times  26   1  times  25   1  times  24   0  times  23   0  times  22   0  times  21   1  times  20Notice 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 nbsp  nbsp -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

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2 s complement of a given number is the no  got by adding 1 with the 1 s complement of the no  suppose  we have a binary no   10111001101  It s 1 s complement is   01000110010 And it s 2 s complement will be   01000110011

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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  Convert the number to binary  ignore the sign for now  e g  5 is 0101 and -5 is 0101  If the number is a positive number then you are done  e g  5 is 0101 in binary using two s complement notation   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   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

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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

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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  quot the number quot   represented using the 8 bit pattern  and its additive inverse using  quot that quot  storage type  a byte  the result in that byte would be all zeros   1 1 1  ----------    1 0 1  1 0 1 1 --- gt  additive inverse   ---------    0 0 0

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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

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In simple term 2 s Complement is a way to store negative number in Computer Memory  Whereas Positive Numbers are stored as Normal Binary Number   Let s consider this example   Computer uses Binary Number System to represent any number   x   5    This is represented as 0101   x   -5    When the computer encouters - sign  it computes it s 2 s complement and stores it  i e 5   0101 and it s 2 s complement is 1011   Important rules computer uses to process numbers are    If the first bit is 1 then it must be negative number  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  If all the bits are 0 then it is 0  Else it is a positive number

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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

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REFERENCE  https   www cs cornell edu  tomf notes cps104 twoscomp html  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

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