Modulo operation with negative numbers

Question

In a C program i was trying the below operations Just to check the behavior     x   5    -3    y    -5     3    z    -5     -3     printf   d   d   d   x  y  z      gave me output as  2  -2   -2  in gcc   I was expecting a positive result every time  Can a modulus be negative  Can anybody explain this behavior

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Based on the C99 Specification  a     a   b    b   a   b  We can write a function to calculate  a   b     a -  a   b    b   int remainder int a  int b        return a -  a   b    b      For modulo operation  we can have the following function  assuming b  gt  0   int mod int a  int b        int r   a   b      return r  lt  0   r   b   r      My conclusion is that a   b in C is a remainder operation and NOT a modulo operation

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According to C99 standard  section 6 5 5 Multiplicative operators  the following is required    a   b    b   a   b   a   Conclusion  The sign of the result of a remainder operation  according to C99  is the same as the dividend s one   Let s see some examples  dividend   divisor    When only dividend is negative   -3   2    2     -3   2   -3   -3   2    2   -2   -3   2  must be -1   When only divisor is negative   3   -2    -2     3   -2   3   3   -2    -2   2   3   -2  must be 1   When both divisor and dividend are negative   -3   -2    -2     -3   -2   -3   -3   -2    -2   -2   -3   -2  must be -1        6 5 5  Multiplicative operators      Syntax         multiplicative-expression          cast-expression   multiplicative-expression   cast-expression   multiplicative-expression   cast-expression   multiplicative-expression   cast-expression            Constraints         Each of the operands shall have arithmetic type  The   operands of the   operator shall have integer type          Semantics         The usual arithmetic conversions are performed on the   operands    The result of the binary   operator is the product of   the operands    The  result  of  the   operator is the quotient from   the division of the first operand by the second  the   result of the   operator is the remainder  In both   operations  if the value of the second operand is zero    the behavior is undefined    When integers are divided  the result of the   operator   is the algebraic quotient with any fractional part   discarded  1   If the quotient a b is representable    the expression  a b  b   a b shall equal a           1   This is often called  truncation toward zero

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C99 requires that when a b is representable    a b    b   a b shall equal a  This makes sense  logically  Right   Let s see what this leads to     Example A  5  -3  is -1      -1     -3    5  -3    5  This can only happen if 5  -3  is 2     Example B   -5  3 is -1      -1    3    -5  3   -5  This can only happen if  -5  3 is -2

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The   operator in C is not the modulo operator but the remainder operator   Modulo and remainder operators differ with respect to negative values    With a remainder operator  the sign of the result is the same as the sign of the dividend while with a modulo operator the sign of the result is the same as the divisor   C defines the   operation for a   b as     a     a   b   b    a   b   with   the integer division with truncation towards 0  That s the truncation that is done towards 0  and not towards negative inifinity  that defines the   as a remainder operator rather than a modulo operator

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I believe it s more useful to think of mod as it s defined in abstract arithmetic  not as an operation  but as a whole different class of arithmetic  with different elements  and different operators  That means addition in mod 3 is not the same as the  normal  addition  that is  integer addition   So when you do   5   -3   You are trying to map the integer 5 to an element in the set of mod -3  These are the elements of mod -3     0  -2  -1     So   0   gt  0  1   gt  -2  2   gt  -1  3   gt  0  4   gt  -2  5   gt  -1   Say you have to stay up for some reason 30 hours  how many hours will you have left of that day  30 mod -24   But what C implements is not mod  it s a remainder  Anyway  the point is that it does make sense to return negatives

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Modulus operator gives the remainder  Modulus operator in c usually takes the sign of the numerator   x   5    -3    - here numerator is positive hence it results in 2 y    -5     3   - here numerator is negative hence it results -2 z    -5     -3  - here numerator is negative hence it results -2   Also modulus remainder  operator can only be used with integer type and cannot be used with floating point

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It seems the problem is that   is not floor operation  int mod int m  float n        return m - floor m n  n

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The other answers have explained in C99 or later  division of integers involving negative operands always truncate towards zero    Note that  in C89  whether the result round upward or downward is implementation-defined  Because  a b    b   a b equals a in all standards  the result of   involving negative operands is also implementation-defined in C89

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In Mathematics  where these conventions stem from  there is no assertion that modulo arithmetic should yield a positive result   Eg    1 mod 5   1  but it can also equal -4  That is  1 5 yields a remainder 1 from 0 or -4 from 5   Both factors of 5   Similarly  -1 mod 5   -1  but it can also equal 4  That is  -1 5 yields a remainder -1 from 0 or 4 from -5   Both factors of 5   For further reading look into equivalence classes in Mathematics

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Can a modulus be negative         can be negative as it is the remainder operator  the remainder after division  not after Euclidean division   Since C99 the result may be 0  negative or positive       a   b  7    3 -- gt   1    7   -3 -- gt   1   -7    3 -- gt  -1   -7   -3 -- gt  -1     The modulo OP wanted is a classic Euclidean modulo  not        I was expecting a positive result every time    To perform a Euclidean modulo that is well defined whenever a b is defined  a b are of any sign and the result is never negative   int modulo Euclidean int a  int b      int m   a   b    if  m  lt  0           m     b  lt  0    -b   b     avoid this form  it is UB when b    INT MIN     m    b  lt  0    m - b   m   b        return m     modulo Euclidean  7   3  -- gt   1   modulo Euclidean  7  -3  -- gt   1   modulo Euclidean -7   3  -- gt   2   modulo Euclidean -7  -3  -- gt   2

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The result of Modulo operation depends on the sign of numerator  and thus you re getting -2 for y and z  Here s the reference  http   www chemie fu-berlin de chemnet use info libc libc 14 html     Integer Division      This section describes functions for performing integer division    These functions are redundant in the GNU C library  since in GNU C the       operator always rounds towards zero  But in other C   implementations      may round differently with negative arguments    div and ldiv are useful because they specify how to round the   quotient  towards zero  The remainder has the same sign as the   numerator

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I don t think there isn t any need to check if the number is negative  A simple function to find the positive modulo would be this - Edit  Assuming N  gt  0 and N   N - 1  lt   INT MAX int modulo int x int N       return  x   N   N   N     This will work for both positive and negative values of x  Original P S  also as pointed out by  chux  If your x and N may reach something like INT MAX-1 and INT MAX respectively  just replace int with long long int  And If they are crossing limits of long long as well  i e  near LLONG MAX   then you shall handle positive and negative cases separately as described in other answers here

[c] Modulo operation with negative numbers

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