Question

What is the output of the following program lines (when embedded in a correct program that declares number to be of type \(i n t\) )? number \(=(1 / 3) * 3\) cout \(<<"(1 / 3) * 3\) is equal to \("<<\) number

Short answer

```cpp #include using namespace std; int main() { int number = (1 / 3) * 3; cout << "(1 / 3) * 3 is equal to " << number; return 0; } ``` Output: (1 / 3) * 3 is equal to 0

Step-by-step solution

Analyze the mathematical expression

We have the expression \((1 / 3) * 3\). Since both 1 and 3 are integers, the division operation will be an integer division. The result of the integer division will also be an integer, either rounded or truncated.

Calculate the division part of the expression

In this case, \((1 / 3)\) is an integer division since both 1 and 3 are integers. The result of an integer division is the largest integer that is less than or equal to the exact division result. Therefore, \(1 / 3\) as integers is equal to \(0\) because 1 is not enough to form a complete group of 3.

Calculate the multiplication part of the expression

Now that we have calculated the division part of the expression, we need to multiply the result by 3: \(0 * 3\). The result of this multiplication is simply \(0\).

Assign the expression result to the variable

The result of our expression is \(0\), so now we assign it to the integer variable number: number \(= 0\).

Display the output

Finally, the output is printed using cout as follows: cout \(<<"(1 / 3) * 3\) is equal to \("<<\) number. In this case, the output will be: \((1 / 3) * 3\) is equal to \(0\).

Key concepts

C++ Arithmetic Operations

Arithmetic operations in C++ are the backbone of many algorithms and calculations within the language. C++ provides a set of binary arithmetic operators that perform common mathematical operations, such as addition (+), subtraction (–), multiplication (*), and division (/). When performing arithmetic on integer data types, the outcome of these operations are also integers, leading to the concept of integer division.

Integer division is a crucial point to understand, as it behaves differently than floating-point division. In an integer division, the decimal part of the result is discarded and only the whole number is retained. For example, the expression \(1\/3\) in C++ yields \(0\) because both operands are integers and integer division truncates the result down to the nearest whole number, which is \(0\) in this case. This truncation is not equivalent to rounding; the division simply discards any remainder.

Understanding this behavior is essential, especially when expressions involve multiple arithmetic operations. In such cases, the order of operations follows mathematical conventions, often referred to as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). However, as seen in the exercise, if an integer division is performed before multiplication, the fact that the division result is truncated can influence the overall result of the expression significantly. To get a non-integer result from division, one or both operands must be converted to a floating-point type before the operation is carried out.

Integer Data Types

Within C++, integer data types are fundamental for representing whole numbers without any fractional or decimal component. The language provides several integer types, such as int, short, long, and long long, each with different sizes and thus ranges of values they can store.

For example, an int generally represents a 32-bit integer value, which can range from about -2 billion to +2 billion. Understanding the size and limits of these data types is critical to prevent overflow errors, where calculations exceed the maximum value the data type can hold, resulting in incorrect and unexpected behavior.

When working with integers, operations that involve division, such as those in the exercise, will always yield an integer result. It's vital to choose the correct integer type for the task at hand, considering the range of values you expect to work with. For instance, when variables might store larger values, opting for a long long type over an int could be a better choice to avoid overflow.

C++ Output with cout

Displaying output in C++ is typically done using cout, which is defined in the header file. The cout object represents the standard output stream, and it is used with the insertion operator (<<) to output literals, variables, and expressions to the console.

For instance, in the given exercise, cout is used to display text followed by a calculated number. When forming output with cout, you can chain multiple insertion operators to concatenate literals and variable values into a single line of output. This allows for readable and customizable output that can include both text and the results of expressions or the contents of variables.

cout is highly customizable and can be combined with manipulators like std::fixed or std::setprecision to control the formatting of the output, such as the number of decimal places to display for floating-point numbers. It is a powerful yet flexible way to produce user-friendly outputs that provide insights into the program's behavior or display results, as demonstrated in the exercise's solution.