# 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 `int`

s to `int`

s, and `double`

s to `double`

s. 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
`if`

s and `else`

s:

```
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 <cmath>
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
`cmath`

, 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:

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¶

- chaining¶
A way of joining several conditional statements in sequence.

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

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

- 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 (

`%`

).- 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.