4. Conditionals and recursion

4.1. The modulus operator

The modulus operator works on integers (and integer expressions) and yields the remainder when the first operand is divided by the second. In C++, the modulus operator is a percent sign, %. The syntax is exactly the same as for other operators:

int quotient = 7 / 3;
int remainder = 7 % 3;

The first operator, integer division, yields 2. The second operator yields 1, since 7 divided by 3 is 2 with 1 left over.

The modulus operater turns out to be surprisingly useful. For example, you can check whether one number is divisible by another: if x % y is zero, then x is divisible by y.

Also, you can use the modulus operator to extract the rightmost digit or digits from a number. For example, x % 10 yields the rightmost digit of x (in base 10). Similarly x % 100 yields the last two digits.

4.2. Conditional execution

In order to write useful programs, we almost always need the ability to check certain conditions and change the behavior of the program accordingly. Conditional statements give us this ability. The simplest form is the if statement:

if (x > 0) {
    cout << "x is positive" << endl;
}

The expression in parentheses is called the condition. If it is true, then the statements in brackets get executed. If the condition is not true, nothing happens.

The condition can contain any of the comparison operators:

x == y     // x equals y
x != y     // x is not equal to y
x > y      // x is greater than y
x < y      // x is less than y
x >= y     // x is greater than or equal to y
x <= y     // x is less than or equal to y

Although these operations are probably familiar to you, the syntax C++ uses is a little different from mathematical symbols like =, ≠ and ≤. A common error is to use a single = instead of a double ==. Remember that = is the assignment operator. Also, there is no such thing as =< or =>.

The two sides of a comparision operator have to be the same type. You can only compare ints to ints, and doubles to doubles. Comparing strings is more complicated, so we will put that off until we get to the Strings and things chapter.

4.3. Alternative execution

A second form of conditional execution is alternative execution, in which there are two possibilities, and the condition determines which one gets executed. The syntax looks like:

if (x % 2 == 0) {
    cout << "x is even" << endl;
} else {
    cout << "x is odd" << endl;
}

If the remainder when x is divided by 2 is zero, then we know that x is even, and the is code displays a message to that effect. If the condition is false, the second set of statements is executed. Since the condition must be true or false, exactly one of the alternatives will be executed.

This example provides us with another opportunity to talk about the usefulness of functions. If you think you might want to check the parity (eveness or oddness) of numbers often, you could “wrap” this code in a function, like this:

void print_parity(int x) {
    if (x % 2 == 0) {
        cout << "x is even" << endl;
    } else {
        cout << "x is odd" << endl;
    }
}

Now we have a function named print_parity that will display an appropriate message for any integer we care to provide. In our main function we would call this function like this:

print_parity(17);

4.4. Chained conditionals

Sometimes you want to check for a number of related conditions and choose one of several actions. One way of doing this is by chaining a series of ifs and elses:

if (x > 0) {
    cout << "x is positive" << endl;
} else if (x < 0) {
    cout << "x is negative" << endl;
} else {
    cout << "x is zero" << endl;
}

In this example, the law of trichotomy gaurantees that if x is neither greater than nor less than zero, we know that it is zero, so in the third branch of our conditional we only need an else.

These chains can be as long as you want, although they can be difficult to read if they get out of hand. One way to make them easier to read is to use standard indentation, as we have demonstrated. If you keep all the statements and curly-braces lined up, you are less likely to make syntax errors and you can find them more quickly if you do.

4.5. Nested conditionals

In addition to chaining, you can also nest one conditional within another. We could have written the previous example as:

if (x == 0) {
    cout << "x is zero" << endl;
} else {
    if (x > 0) {
        cout << "x is positive" << endl;
    } else {
        cout << "x is negative" << endl;
    }
}

There is now an outer conditional that contains two branches. The first branch contains a simple output statement, but the second branch contains another if statement, which has two branches of its own. Fortunately, those two branches are both output statements, although they could have been conditional statemtents as well.

Notice again that indentation helps make the structure apparent, but nevertheless, nested conditionals get difficult to read very quickly. In general, it is a good idea to avoid them when you can.

On the other hand, this kind of nested structure is common, and we will see it again often.

4.6. The return statement

The return statement allows you to terminate the execution of a function before you reach its end. One reason to use it is if you detect an error condition:

#include <math.h>

void print_logarithm(double x) {
    if (x <= 0.0) {
        cout << "Positive numbers only, please." << endl;
        return;
    }
    double result = log(x);
    cout << "The log of x is " << result << endl;
}

This defines a function named print_logarithm that has a double named x as a parameter. The first thing it does is check whether x is less than or equal to zero, in which case it displays an error message and then uses return to exit the function. The flow of execution immediately returns to the caller and the remaining lines of the function are not executed.

We used a floating-point value on the right side of the condition because there is a floating-point variable on the left.

Remember that any time you want to use a function from the math library, you have to include the header file math.h, which we have included here as a reminder.

4.7. Recursion

We mentioned in the last chapter that it is legal for one function to call another, and we have seen several examples of that. We neglected to mention that it is also legal for a function to call itself. It may not be obvious why that is a good thing, but it turns out to be one of the most magical and interesting things a program can do.

For example, look at the following function:

void countdown(int n) {
    if (n == 0) {
        cout << "Blastoff!" << endl;
    } else {
        cout << n << endl;
        countdown(n-1);
    }
}

The name of the function is countdown and it has a single integer parameter. If the parameter is zero, it outputs the word “Blastoff!”. Otherwise, it outputs the parameter and then calls a function named countdown - itself - passing n-1 as an argument.

What happens if we call this function like this:

int main()
{
    countdown(3);
    return 0;
}

The execution of countdown begins with n=3, and since n is not zero, it outputs the value 3, then calls itself…

The execution of countdown begins with n=2, and since n is not zero, it outputs the value 2, then calls itself…

The execution of countdown begins with n=1, and since n is not zero, it outputs the value 1, then calls itself…

The execution of countdown begins with n=0, and since n is zero, it outputs the word “Blastoff!” and then returns.

The countdown that got n=1 returns.

The countdown that got n=2 returns.

The countdown that got n=3 returns.

And then we’re back in main (what a trip!), so the total output looks like:

3
2
1
Blastoff!

As a second example, let’s look again at the functions new_line and three_lines.

void new_line() {
    cout << end;
}

void three_lines() {
    new_line(); new_line(); new_line();
}

Although these work, they would not be much help if we wanted to output 2 newlines, or 42. A better alternative would be:

void n_lines(int n) {
    if (n > 0) {
        cout << endl;
        n_lines(n-1);
    }
}

This function is similar to countdown; as long as n is greater than zero, it outputs one newline, and then calls itself to output n-1 additional newlines. Thus the total number of newlines is 1 + (n - 1), which comes out to n, just like we wanted.

The process of a function calling itself is called recursion, and such functions are said to be recursive.

4.8. Infinite recursion

In the examples in the previous section, notice that each time the functions get called recursively, the argument gets smaller by one, so eventually it gets to zero. When the argument is zero, the function returns immediately, without making any recursive calls. This case - when the function completes with making a recursive call - called the base case.

If a recursion never reaches a base case, it will go on making recursive calls forever and the program will never terminate. This is known as infinite recursion, and is generally not considered a good idea.

In most programming environments, a program with infinite recursion will not really run forever. Eventually, something will break and the program will report an error. You will explore this first example we have seen of a run-time error in the exercises.

4.9. Stack diagrams for recursive functions

In the previous chapter we used a stack diagram to represent the state of a program during a function call. The same kind of diagram can make it easier to interpret a recursive function.

Remember that every time a function gets called it creates a new instance that contains the function’s local variables and parameters.

This figure shows a stack diagram for countdown, called with n = 3:

countdown stack diagram

There is one instance of main and four instances of countdown, each with a different value for the parameter n. The bottom of the stack, countdown with n = 0, is the base case. It does not make a recursive call, so there are no more instances of countdown.

The instance of main is empty because main does not have parameters or local variables.

4.10. Glossary

modulus operator

An operator that works on integers and yields the remainder when one number is divided by another. In C++ it is denoted with a percent sign (%).

conditional

A block of statements that may or may not be executed depending on some condition.

chaining

A way of joining several conditional statements in sequence.

nesting

Putting a conditional statement inside one or both branches of another conditional statement.

recursion

The process of calling the same function you are currently executing.

infinite recursion

A function that calls itself recursively without ever reaching the base case. Eventually an infinite recursion will cause a run-time error.

4.11. Exercises

  1. What happens if you call the countdown function introduced in the Recursion section with an argument that is a negative number?

  2. Draw a stack diagram for n_lines, invoked with the parameter n equal to 4.

  3. Write a function named compare that takes two int arguments, a and b, and prints out whether a is greater than b, a is less than b, or a is equal to b.