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Arithmetic

Summary

An arithmetic expression is not computed from left to right as it is read, but according to a hidden structure, its tree, which is fixed by the precedence of the operators, their associativity and the parentheses; and at each node of that tree the type of the operands decides which operation is performed, so that 7 / 2 is 3 and 7 / 2.0 is 3.5. This lesson studies the five arithmetic operators of C++ with integers and with reals; integer division and the remainder, which since C++11 truncate toward zero, from which it follows that the remainder takes the sign of the dividend and differs from the remainder of Euclidean division when the dividend is negative; the way to obtain the Euclidean remainder; the tree of an expression; the type of the result when the operands are of different types, and the two ways in which an integer arithmetic expression loses its meaning: dividing by zero and overflowing in an intermediate computation.

Learning objectives

  1. Compute the quotient and the remainder of integer division in C++, also with negative operands, and relate them to the quotient and the remainder of Euclidean division.
  2. Draw the tree of an arithmetic expression according to precedence and associativity, and compute its value.
  3. Predict the type of the result of an operation between operands of different types.
  4. Use integer division and the remainder to break quantities down: the digits of a number, a duration in hours and minutes, a day of the week.

The arithmetic operators

C++ has five binary arithmetic operators: addition +, subtraction -, multiplication *, division / and remainder %; it also has the unary minus, which changes the sign of its only operand, as in -x, and a unary plus, +x, which leaves it as it is. The first four apply to integers and to reals; the remainder, only to integers. Now, the same notation does not designate the same operation in every case: if both operands of / are integers, the division is integer division and its result is an integer, the quotient without its fractional part; if either of them is real, the division is real. The lesson Characters, Booleans, conversions and constants showed the most visible consequence of this rule, that double r{7 / 2}; stores 3: the division was done between integers before the result was converted to a real.

ExampleThe five operators with two integers

The program applies the five operators to 17 and 5, and repeats the division with a real dividend.

C++ program
#include <print>​int main(){    const int a{17};    const int b{5};    std::println("{} + {} = {}", a, b, a + b);    std::println("{} - {} = {}", a, b, a - b);    std::println("{} * {} = {}", a, b, a * b);    std::println("{} / {} = {}", a, b, a / b);    std::println("{} % {} = {}", a, b, a % b);    const double x{17.0};    std::println("{} / {} = {}", x, b, x / b);}
Output
17 + 5 = 2217 - 5 = 1217 * 5 = 8517 / 5 = 317 % 5 = 217 / 5 = 3.4

17 divided by 5 gives 3 with 2 left over: they are the quotient and the remainder of sharing 17 objects out in groups of 5. The last line divides a real by an integer and obtains 3.4. Note that std::println writes the real 17.0 as “17”: the empty placeholder writes the shortest representation of the value, so what is written does not reveal the type; to see it, one asks for decimals with {:.1f}.

The % operator does not accept reals. Writing 7.5 % 2 is not a logic error that the program discovers when it runs, but a compilation error: the compiler says that operands of types double and int are invalid for the operator.

C++ program
#include <print>​int main(){    std::println("{}", 7.5 % 2);}
Compiler error messages
main.cpp: In function ‘int main()’:main.cpp:5:28: error: invalid operands of types ‘double’ and ‘int’ to binary ‘operator%’    5 |     std::println("{}", 7.5 % 2);      |                        ~~~ ^ ~

The remainder of a real division does exist, and it is computed by the function std::fmod from the header <cmath>, which will be studied in the lesson Bits and the math library, together with exponentiation: C++ has no exponentiation operator, and 2 ^ 3 is not 8, but something else that will be explained there.