# Arithmetic operators

< cpp‎ | language

C++
 Language Standard library headers Concepts Utilities library Strings library Containers library Algorithms library Iterators library Numerics library Input/output library Localizations library Regular expressions library (C++11) Atomic operations library (C++11) Thread support library (C++11) Filesystem library (C++17) Technical Specifications

Expressions
 General value categories (lvalue, rvalue, xvalue) order of evaluation (sequence points) constant expressions unevaluated expressions primary expressions lambda-expression(C++11) Literals integer literals floating-point literals boolean literals character literals including escape sequences string literals null pointer literal(C++11) user-defined literal(C++11) Operators Assignment operators: a=b, a+=b, a-=b, a*=b, a/=b, a%=b, a&=b, a|=b, a^=b, a<<=b, a>>=b Increment and decrement: ++a, --a, a++, a-- Arithmetic operators: +a, -a, a+b, a-b, a*b, a/b, a%b, ~a, a&b, a|b, a^b, a<>b Logical operators: a||b, a&&b, !a Comparison operators: a==b, a!=b, ab, a<=b, a>=b Member access operators: a[b], *a, &a, a->b, a.b, a->*b, a.*b Other operators: a(...), a,b, a?b:c Alternative representations of operators Precedence and associativity Fold expression(C++17) new-expression delete-expression throw-expression alignof sizeof sizeof...(C++11) typeid noexcept(C++11) Operator overloading Conversions Implicit conversions const_cast static_cast reinterpret_cast dynamic_cast Explicit conversions (T)a, T(a) User-defined conversion

Returns the result of specific arithmetic operation.

Operator name Syntax Over​load​able Prototype examples (for class T)
Inside class definition Outside class definition
unary plus +a Yes T T::operator+() const; T operator+(const T &a);
unary minus -a Yes T T::operator-() const; T operator-(const T &a);
addition a + b Yes T T::operator+(const T2 &b) const; T operator+(const T &a, const T2 &b);
subtraction a - b Yes T T::operator-(const T2 &b) const; T operator-(const T &a, const T2 &b);
multiplication a * b Yes T T::operator*(const T2 &b) const; T operator*(const T &a, const T2 &b);
division a / b Yes T T::operator/(const T2 &b) const; T operator/(const T &a, const T2 &b);
modulo a % b Yes T T::operator%(const T2 &b) const; T operator%(const T &a, const T2 &b);
bitwise NOT ~a Yes T T::operator~() const; T operator~(const T &a);
bitwise AND a & b Yes T T::operator&(const T2 &b) const; T operator&(const T &a, const T2 &b);
bitwise OR a | b Yes T T::operator|(const T2 &b) const; T operator|(const T &a, const T2 &b);
bitwise XOR a ^ b Yes T T::operator^(const T2 &b) const; T operator^(const T &a, const T2 &b);
bitwise left shift a << b Yes T T::operator<<(const T2 &b) const; T operator<<(const T &a, const T2 &b);
bitwise right shift a >> b Yes T T::operator>>(const T2 &b) const; T operator>>(const T &a, const T2 &b);
Notes
• All built-in operators return values, and most user-defined overloads also return values so that the user-defined operators can be used in the same manner as the built-ins. However, in a user-defined operator overload, any type can be used as return type (including void). In particular, stream insertion and stream extraction overloads of operator<< and operator>> return T&.
• T2 can be any type including T

## Contents

### Explanation

All arithmetic operators compute the result of specific arithmetic operation and returns its result. The arguments are not modified.

#### Conversions

If the operand passed to an arithmetic operator is integral or unscoped enumeration type, then before any other action (but after lvalue-to-rvalue conversion, if applicable), the operand undergoes integral promotion. If an operand has array or function type, array-to-pointer and function-to-pointer conversions are applied.

For the binary operators (except shifts), if the promoted operands have different types, additional set of implicit conversions is applied, known as usual arithmetic conversions with the goal to produce the common type (also accessible via the std::common_type type trait)

• If either operand has scoped enumeration type, no conversion is performed: the other operand and the return type must have the same type
• Otherwise, if either operand is long double, the other operand is converted to long double
• Otherwise, if either operand is double, the other operand is converted to double
• Otherwise, if either operand is float, the other operand is converted to float
• Otherwise, the operand has integer type (because bool, char, char16_t, char32_t, wchar_t, and unscoped enumeration were promoted at this point) and integral conversions are applied to produce the common type, as follows:
• If both operands are signed or both are unsigned, the operand with lesser conversion rank is converted to the operand with the greater integer conversion rank
• Otherwise, if the unsigned operand's conversion rank is greater or equal to the conversion rank of the signed operand, the signed operand is converted to the unsigned operand's type.
• Otherwise, if the signed operand's type can represent all values of the unsigned operand, the unsigned operand is converted to the signed operand's type
• Otherwise, both operands are converted to the unsigned counterpart of the signed operand's type.

The conversion rank above increases in order bool, signed char, short, int, long, long long. The rank of any unsigned type is equal to the rank of the corresponding signed type. The rank of char is equal to the rank of signed char and unsigned char. The ranks of char16_t, char32_t, and wchar_t are equal to the ranks of their underlying types.

#### Overflows

Unsigned integer arithmetic is always performed modulo 2n
where n is the number of bits in that particular integer. E.g. for unsigned int, adding one to UINT_MAX gives 0, and subtracting one from 0 gives UINT_MAX.

When signed integer arithmetic operation overflows (the result does not fit in the result type), the behavior is undefined: it may wrap around according to the rules of the representation (typically 2's complement), it may trap on some platforms or due to compiler options (e.g. -ftrapv in GCC and Clang), or may be completely optimized out by the compiler.

#### Floating-point environment

If #pragma STDC FENV_ACCESS is supported and set to ON, all floating-point arithmetic operators obey the current floating-point rounding direction and report floating-point arithmetic errors as specified in math_errhandling unless part of a static initializer (in which case floating-point exceptions are not raised and the rounding mode is to nearest)

#### Floating-point contraction

Unless #pragma STDC FP_CONTRACT is supported and set to OFF, all floating-point arithmetic may be performed as if the intermediate results have infinite range and precision, that is, optimizations that omit rounding errors and floating-point exceptions are allowed. For example, C++ allows the implementation of (x*y) + z with a single fused multiply-add CPU instruction or optimization of a = x*x*x*x; as tmp = x *x; a = tmp*tmp.

Unrelated to contracting, intermediate results of floating-point arithmetic may have range and precision that is different from the one indicated by its type, see FLT_EVAL_METHOD

Formally, the C++ standard makes no guarantee on the accuracy of floating-point operations.

#### Unary arithmetic operators

The unary arithmetic operator expressions have the form

 + expression (1) - expression (2)
1) unary plus (promotion).
For the built-in operator, expression must have arithmetic, unscoped enumeration, or pointer type. Integral promotion is performed on the operand if it has integral or unscoped enumeration type and determines the type of the result.
2) unary minus (negation).
For the built-in operator, expression must have arithmetic or unscoped enumeration type. Integral promotion is performed on the operand and determines the type of the result.

The built-in unary plus operator returns the value of its operand. The only situation where it is not a no-op is when the operand has integral type or unscoped enumeration type, which is changed by integral promotion, e.g, it converts char to int or if the operand is subject to lvalue-to-rvalue, array-to-pointer, or function-to-pointer conversion.

The builtin unary minus operator calculates the negative of its promoted operand. For unsigned a, the value of -a is 2b
-a
, where b is the number of bits after promotion.

In overload resolution against user-defined operators, for every promoted arithmetic type A and for every type T, the following function signatures participate in overload resolution:

 A operator+(A) T* operator+(T*) A operator-(A)
#include <iostream>
int main()
{
char c = 0x6a;
int n1 = 1;
unsigned char n2 = 1;
unsigned int n3 = 1;
std::cout << "char: " << c << " int: " << +c << '\n'
<< "-1, where 1 is signed: " << -n1 << '\n'
<< "-1, where 1 is unsigned char: " << -n2 << '\n'
<< "-1, where 1 is unsigned int: " << -n3 << '\n';
char a[3];
std::cout << "size of array: " << sizeof a << '\n'
<< "size of pointer: " << sizeof +a << '\n';
}

Output:

char: j int: 106
-1, where 1 is signed: -1
-1, where 1 is unsigned char: -1
-1, where 1 is unsigned int: 4294967295
size of array: 3
size of pointer: 8

The binary additive arithmetic operator expressions have the form

 lhs + rhs (1) lhs - rhs (2)
For the built-in operator, lhs and rhs must be one of the following:
• both have arithmetic or unscoped enumeration type. In this case, the usual arithmetic conversions are performed on both operands and determine the type of the result..
• one is a pointer to complete object type, the other has integral or unscoped enumeration type. In this case, the result type has the type of the pointer.
2) subtraction
For the built-in operator, lhs and rhs must be one of the following:
• both have arithmetic or unscoped enumeration type. In this case, the usual arithmetic conversions are performed on both operands and determine the type of the result..
• lhs is a pointer to complete object type, rhs has integral or unscoped enumeration type. In this case, the result type has the type of the pointer.
• both are pointers to the same complete object types, ignoring cv-qualifiers. In this case, the result type is std::ptrdiff_t.

With operands of arithmetic or enumeration type, the result of binary plus is the sum of the operands (after usual arithmetic conversions), and the result of the binary minus operator is the result of subtracting the second operand from the first (after usual arithmetic conversions), except that, if the type supports IEEE floating-point arithmetic (see std::numeric_limits::is_iec559),

• if one operand is NaN, the result is NaN
• infinity minus infinity is NaN and FE_INVALID is raised
• infinity plus the negative infinity is NaN and FE_INVALID is raised

If any of the operands is a pointer, the following rules apply:

• A pointer to non-array object is treated as a pointer to the first element of an array with size 1.
• If the pointer P points to the ith element of an array, then the expressions P+n, n+P, and P-n are pointers of the same type that point to the i+nth, i+nth, and i-nth element of the same array, respectively. The result of pointer addition may also be a one-past-the-end pointer (that is, pointer P such that the expression P-1 points to the last element of the array). Any other situations (that is, attempts to generate a pointer that isn't pointing at an element of the same array or one past the end) invoke undefined behavior.
• If the pointer P points to the ith element of an array, and the pointer Q points at the jth element of the same array, the expression P-Q has the value i-j, if the value fits in std::ptrdiff_t. Both operands must point to the elements of the same array (or one past the end), otherwise the behavior is undefined. If the result does not fit in std::ptrdiff_t, the behavior is undefined.
• In any case, if the pointed-to type is different from the array element type, disregarding cv qualifications, at every level if the elements are themselves pointers, the behavior of pointer arithmetic is undefined. In particular, pointer arithmetic with pointer to base, which is pointing at an element of an array of derived objects is undefined.
• If the value 0 is added or subtracted from a pointer, the result is the pointer, unchanged. If two pointers point at the same object or are both one past the end of the same array, or both are null pointers, then the result of subtraction is equal to .

These pointer arithmetic operators allow pointers to satisfy the RandomAccessIterator concept.

In overload resolution against user-defined operators, for every pair of promoted arithmetic types L and R and for every object type T, the following function signatures participate in overload resolution:

 LR operator+(L, R) LR operator-(L, R) T* operator+(T*, std::ptrdiff_t) T* operator+(std::ptrdiff_t, T*) T* operator-(T*, std::ptrdiff_t) std::ptrdiff_t operator-(T*, T*)

where LR is the result of usual arithmetic conversions on L and R

#include <iostream>
int main()
{
char c = 2;
unsigned int un = 2;
int  n = -10;
std::cout <<  " 2 + (-10), where 2 is a char    = " << c + n << '\n'
<<  " 2 + (-10), where 2 is unsigned  = " << un + n << '\n'
<<  " -10 - 2.12  = " << n - 2.12 << '\n';

char a[4] = {'a', 'b', 'c', 'd'};
char* p = &a[1];
std::cout << "Pointer addition examples: " << *p << *(p + 2)
<< *(2 + p) << *(p - 1) << '\n';
char* p2 = &a[4];
std::cout << "Pointer difference: " << p2 - p << '\n';
}

Output:

2 + (-10), where 2 is a char    = -8
2 + (-10), where 2 is unsigned  = 4294967288
-10 - 2.12  = -12.12
Pointer difference: 3

#### Multiplicative operators

The binary multiplicative arithmetic operator expressions have the form

 lhs * rhs (1) lhs / rhs (2) lhs % rhs (3)
1) multiplication
For the built-in operator, lhs and rhs must both have arithmetic or unscoped enumeration type.
2) division
For the built-in operator, lhs and rhs must both have arithmetic or unscoped enumeration type.
3) remainder
For the built-in operator, lhs and rhs must both have integral or unscoped enumeration type

For all three operators, the usual arithmetic conversions are performed on both operands and determine the type of the result.

The binary operator * performs multiplication of its operands (after usual arithmetic conversions), except that, for floating-point multiplication,

• multiplication of a NaN by any number gives NaN
• multiplication if infinity by zero gives NaN and FE_INVALID is raised

The binary operator / divides the first operand by the second (after usual arithmetic conversions).

For integral operands, it yields the algebraic quotient.

 The quotient is rounded in implementation-defined direction. (until C++11) The quotient is truncated towards zero (fractional part is discarded). (since C++11)

If the second operand is zero, the behavior is undefined, except that if floating-point division is taking place and the type supports IEEE floating-point arithmetic (see std::numeric_limits::is_iec559), then:

• if one operand is NaN, the result is NaN
• dividing a non-zero number by ±0.0 gives the correctly-signed infinity and FE_DIVBYZERO is raised
• dividing 0.0 by 0.0 gives NaN and FE_INVALID is raised

The binary operator % yields the remainder of the integer division of the first operand by the second (after usual arithmetic conversions; note that the operand types must be integral types). If the quotient a/b is representable in the result type, (a/b)*b + a%b == a. If the second operand is zero, the behavior is undefined. If the quotient a/b is not representable in the result type, the behavior of both a/b and a%b is undefined (that means INT_MIN%-1 is undefined on 2's complement systems)

Note: Until C++11, if one or both operands to binary operator % were negative, the sign of the remainder was implementation-defined, as it depends on the rounding direction of integer division. The function std::div provided well-defined behavior in that case.

Note: for floating-point remainder, see std::remainder and std::fmod.

In overload resolution against user-defined operators, for every pair of promoted arithmetic types LA and RA and for every pair of promoted integral types LI and RI the following function signatures participate in overload resolution:

 LRA operator*(LA, RA) LRA operator/(LA, RA) LRI operator%(LI, RI)

where LRx is the result of usual arithmetic conversions on Lx and Rx

#include <iostream>
int main()
{
char c = 2;
unsigned int un = 2;
int  n = -10;
std::cout <<  "2 * (-10), where 2 is a char    = " << c * n << '\n'
<<  "2 * (-10), where 2 is unsigned  = " << un * n << '\n'
<<  "-10 / 2.12  = " << n / 2.12 << '\n'
<<  "-10 / 21  = " << n / 21 << '\n'
<<  "-10 % 21  = " << n % 21 << '\n';
}

Output:

2 * (-10), where 2 is a char    = -20
2 * (-10), where 2 is unsigned  = 4294967276
-10 / 2.12  = -4.71698
-10 / 21  = 0
-10 % 21  = -10

#### Bitwise logic operators

The bitwise arithmetic operator expressions have the form

 ~ rhs (1) lhs & rhs (2) lhs | rhs (3) lhs ^ rhs (4)
1) bitwise NOT
2) bitwise AND
3) bitwise OR
4) bitwise XOR
For the built-in operators, lhs and rhs must both have integral or unscoped enumeration type. Usual arithmetic conversions are performed on both operands and determine the type of the result.

The result of operator~ is the bitwise NOT (one's complement) value of the argument (after promotion). The result of operator& is the bitwise AND value of the operands (after usual arithmetic conversions). The result of operator| is the bitwise OR value of the operands (after usual arithmetic conversions). The result of operator^ is the bitwise XOR value of the operands (after usual arithmetic conversions)

In overload resolution against user-defined operators, for every pair of promoted integral types L and R the following function signatures participate in overload resolution:

 R operator~(R) LR operator&(L, R) LR operator^(L, R) LR operator|(L, R)

where LR is the result of usual arithmetic conversions on L and R

#include <iostream>
int main()
{
std::cout << std::hex << std::showbase;
uint32_t a = 0x12345678;
std::cout << "Value: " << a << " mask: " << mask << '\n'
<< "Setting bits: " << (a | mask) << '\n'
<< "Clearing bits: " << (a & ~mask) << '\n'
<< "Selecting bits: " << (a & mask) << '\n';
}

Output:

Value: 0x12345678 mask: 0xf0
Setting bits: 0x123456f8
Clearing bits: 0x12345608
Selecting bits: 0x70

#### Bitwise shift operators

The bitwise shift operator expressions have the form

 lhs << rhs (1) lhs >> rhs (2)
1) left shift of lhs by rhs bits
2) right shift of lhs by rhs bits
For the built-in operators, lhs and rhs must both have integral or unscoped enumeration type. Integral promotions are performed on both operands.

The return type is the type of the left operand after integral promotions.

For unsigned and positive a, the value of a << b is the value of a * 2b
, reduced modulo maximum value of the return type plus 1 (that is, bitwise left shift is performed and the bits that get shifted out of the destination type are discarded).

For signed and positive a,

 the value of a << b is a * 2b if it is representable the return type, otherwise the behavior is undefined. (until C++14) the value of a << b is a * 2b if it is representable in the unsigned version of the return type (which is then converted to signed: this makes it legal to create INT_MIN as 1<<31), otherwise the behavior is undefined. (since C++14)

For negative a, the behavior of a << b is undefined.

For unsigned a and for signed a with nonnegative values, the value of a >> b is the integer part of a/2b
. For negative a, the value of a >> b is implementation-defined (in most implementations, this performs arithmetic right shift, so that the result remains negative).

In any case, if the value of the right operand is negative or is greater or equal to the number of bits in the promoted left operand, the behavior is undefined.

In overload resolution against user-defined operators, for every pair of promoted integral types L and R, the following function signatures participate in overload resolution:

 L operator<<(L, R) L operator>>(L, R)
#include <iostream>
enum {ONE=1, TWO=2};
int main()
{
std::cout << std::hex << std::showbase;
char c = 0x10;
unsigned long long ull = 0x123;
std::cout << "0x123 << 1 = " << (ull << 1) << '\n'
<< "0x123 << 63 = " << (ull << 63) << '\n' // overflow in unsigned
<< "0x10 << 10 = " << (c << 10) << '\n';   // char is promoted to int
long long ll = -1000;
std::cout << std::dec << "-1000 >> 1 = " << (ll >> ONE) << '\n';
}

Output:

0x123 << 1 = 0x246
0x123 << 63 = 0x8000000000000000
0x10 << 10 = 0x4000
-1000 >> 1 = -500

### Standard library

Arithmetic operators are overloaded for many standard library types.

#### Unary arithmetic operators

 operator+operator- implements unary + and unary - (public member function of std::chrono::duration)  operator+operator- applies unary operators to complex numbers (function template)  operator+operator-operator~operator! applies a unary arithmetic operator to each element of the valarray (public member function of std::valarray) 

 operator+operator- modifies the time point by the given duration (function template)  operator+ concatenates two strings or a string and a char (function template)  operator+operator- advances or decrements the iterator (public member function of std::reverse_iterator) operator+operator- advances or decrements the iterator (public member function of std::move_iterator) operator+operator-operator*operator/ performs complex number arithmetics on two complex values or a complex and a scalar (function template)  operator+operator-operator*operator/operator%operator&operator|operator^operator<>operator&&operator|| applies binary operators to each element of two valarrays, or a valarray and a value (function template) 

#### Multiplicative operators

 operator+operator-operator*operator/operator% implements arithmetic operations with durations as arguments (function template)  operator+operator-operator*operator/ performs complex number arithmetics on two complex values or a complex and a scalar (function template)  operator+operator-operator*operator/operator%operator&operator|operator^operator<>operator&&operator|| applies binary operators to each element of two valarrays, or a valarray and a value (function template) 

#### Bitwise logic operators

 operator&=operator|=operator^=operator~ performs binary AND, OR, XOR and NOT (public member function of std::bitset)  operator&operator|operator^ performs binary logic operations on bitsets (function template)  operator~ applies a unary arithmetic operator to each element of the valarray (public member function of std::valarray) operator^operator&operator| applies binary operators to each element of two valarrays, or a valarray and a value (function template)

#### Bitwise shift operators

 operator<> applies binary operators to each element of two valarrays, or a valarray and a value (function template) operator<> performs binary shift left and shift right (public member function of std::bitset)

#### Stream insertion/extraction operators

Throughout the standard library, bitwise shift operators are commonly overloaded with I/O stream ( or one of the classes derived from it) as both the left operand and return type. Such operators are known as stream insertion and stream extraction operators:

 operator>> extracts formatted data (public member function of std::basic_istream)  operator>>(std::basic_istream) extracts characters and character arrays (function template)  operator<< inserts formatted data (public member function of std::basic_ostream)  operator<<(std::basic_ostream) inserts character data (function template)  operator<> serializes and deserializes a complex number (function template)  operator<> performs stream input and output of bitsets (function template)  operator<> performs stream input and output on strings (function template)  operator<> performs stream input and output on pseudo-random number engine (function template)  operator<> performs stream input and output on pseudo-random number distribution (function template) 

Common operators
assignment increment
decrement
arithmetic logical comparison member
access
other

a = b
a += b
a -= b
a *= b
a /= b
a %= b
a &= b
a |= b
a ^= b
a <<= b
a >>= b

++a
--a
a++
a--

+a
-a
a + b
a - b
a * b
a / b
a % b
~a
a & b
a | b
a ^ b
a << b
a >> b

!a
a && b
a || b

a == b
a != b
a < b
a > b
a <= b
a >= b

a[b]
*a
&a
a->b
a.b
a->*b
a.*b

a(...)
a, b
? :

Special operators

static_cast converts one type to another related type
dynamic_cast converts within inheritance hierarchies
const_cast adds or removes cv qualifiers
reinterpret_cast converts type to unrelated type
C-style cast converts one type to another by a mix of static_cast, const_cast, and reinterpret_cast
new creates objects with dynamic storage duration
delete destructs objects previously created by the new expression and releases obtained memory area
sizeof queries the size of a type
sizeof... queries the size of a parameter pack (since C++11)
typeid queries the type information of a type
noexcept checks if an expression can throw an exception (since C++11)
alignof queries alignment requirements of a type (since C++11)