A numerical operator is a symbol that makes the program perform a specific mathematical or logical manipulation. The numerical operators in C++ can be grouped into five types: arithmetic, assignment, comparison, logical and bitwise operators.

Arithmetic Operators

There are the four basic arithmetic operators, as well as the modulus operator (%) which is used to obtain the division remainder.

int x = 3 + 2; // 5 // addition
    x = 3 - 2; // 1 // subtraction
    x = 3 * 2; // 6 // multiplication
    x = 3 / 2; // 1 // division
    x = 3 % 2; // 1 // modulus (division remainder)

Notice that the division sign gives an incorrect result. This is because it operates on two integer values and will therefore truncate the result and return an integer. To get the correct value, one of the numbers must be explicitly converted to a floating-point number.

float f = 3 / (float)2; // 1.5

Assignment Operators

The second group is the assignment operators. Most importantly, the assignment operator (=) itself, which assigns a value to a variable.

Combined Assignment Operators

A common use of the assignment and arithmetic operators is to operate on a variable and then to save the result back into that same variable. These operations can be shortened with the combined assignment operators.

x += 5; // x = x+5;
x -= 5; // x = x-5;
x *= 5; // x = x*5;
x /= 5; // x = x/5;
x %= 5; // x = x%5;

Increment and Decrement Operators

Another common operation is to increment or decrement a variable by one. This can be simplified with the increment (++) and decrement (--) operators.

x++; // x = x+1;
x--; // x = x-1;

Both of these can be used either before or after a variable.

x++; // post-increment
x--; // post-decrement
++x; // pre-increment
--x; // pre-decrement

The result on the variable is the same whichever is used. The difference is that the post-operator returns the original value before it changes the variable, while the pre-operator changes the variable first and then returns the value.

int x, y;
x = 5; y = x++; // y=5, x=6
x = 5; y = ++x; // y=6, x=6

Comparison Operators

The comparison operators compare two values and return either true or false. They are mainly used to specify conditions, which are expressions that evaluate to either true or false.

bool b = (2 == 3); // false // equal to
     b = (2 != 3); // true  // not equal to
     b = (2 > 3);  // false // greater than
     b = (2 < 3);  // true  // less than
     b = (2 >= 3); // false // greater than or equal to
     b = (2 <= 3); // true  // less than or equal to

Logical Operators

The logical operators are often used together with the comparison operators. Logical and (&&) evaluates to true if both the left and right sides are true, and logical or (||) is true if either the left or right side is true. For inverting a Boolean result there is the logical not (!) operator. Note that for both “logical and” and “logical or” the right-hand side will not be evaluated if the result is already determined by the left-hand side.

bool b = (true && false); // false // logical and
     b = (true || false); // true  // logical or
     b = !(true);         // false // logical not

Bitwise Operators

The bitwise operators can manipulate individual bits inside an integer. For example, the “bitwise or” operator (|) makes the resulting bit 1 if the bits are set on either side of the operator.

int x = 5 & 4;  // 101 & 100 = 100 (4)  // and
x = 5 | 4;      // 101 | 100 = 101 (5)  // or
x = 5 ^ 4;      // 101 ^ 100 = 001 (1)  // xor
x = 4 << 1;     // 100 << 1  =1000 (8)  // left shift
x = 4 >> 1;     // 100 >> 1  =  10 (2)  // right shift
x = ~4;         // ~00000100 = 11111011 (-5) // invert

The bitwise operators also have combined assignment operators.

int x=5; x &= 4; // 101 & 100 = 100 (4) // and
    x=5; x |= 4; // 101 | 100 = 101 (5) // or
    x=5; x ^= 4; // 101 ^ 100 = 001 (1) // xor
    x=5; x <<= 1;// 101 << 1  =1010 (10)// left shift
    x=5; x >>= 1;// 101 >> 1  =  10 (2) // right shift

Operator Precedence

In C++, expressions are normally evaluated from left to right. However, when an expression contains multiple operators, the precedence of those operators decides the order in which they are evaluated. The order of precedence can be seen in the following table, where the operator with the lowest precedence will be evaluated first. This same basic order also applies to many other languages, such as C, Java, and C#.

To give an example, logical and (&&) binds weaker than relational operators, which in turn bind weaker than arithmetic operators.

bool b = 2+3 > 1*4 && 5/5 == 1; // true

To make things clearer, parentheses can be used to specify which part of the expression will be evaluated first. As seen in the table, parentheses are among the operators with lowest precedence.

bool b = ((2+3) > (1*4)) && ((5/5) == 1); // true