# Example - Factorial

Factorial n (usually written n!) is the product of all integers up to and including n (1 * 2 * 3 * ... * n). This problem is often (inappropriately) used as a programming example, especially to show how to write recursive functions.

### Recursive solution is a bad example

Writing a recursive factorial function is a mistake because
• The recursive solution's memory usage is O(N) instead of O(1).
• It's more complicated for students.
• It's inappropriate when the iterative solution is better.
Recursion is a great tool, but using it where it's not appropriate doesn't set a great example.

### Range and precision problems

Even the iterative solution has problems because the large numbers that factorial generates cannot be represented in the limited range of integers, or limited precision of floating-point numbers. Programs using these primitive types are very dangerous because they will produce garbage results without comment.

## Solution with `java.math.BigInteger`

The proper solution is to use `java.Math.BigInteger`. Here is a program with `int` and `BigInteger` solutions.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 ``` ```// numbers/Factorial.java - Computes factorial // Fred Swartz - 2003-11-02 import java.math.BigInteger; public class Factorial { public static void main(String[] args) { //-- BigInteger solution. BigInteger n = BigInteger.ONE; for (int i=1; i<=20; i++) { n = n.multiply(BigInteger.valueOf(i)); System.out.println(i + "! = " + n); } //-- int solution (BAD IDEA BECAUSE ONLY WORKS TO 12). int fact = 1; for (int i=1; i<=20; i++) { fact = fact * i; System.out.println(i + "! = " + fact); } } } ```

## Sample output

An `int` produces pure garbage after 12, and `long` only makes it to 20. The shame of integer arithmetic is that it doesn't stop when the values are bad -- it just produces garbage by throwing away bits in the result that don't fit in an `int` (32 bits) or long (64-bits). `BigInteger` values are limited only by the amount of memory.
`BigInteger` output`int` output
```1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
7! = 5040
8! = 40320
9! = 362880
10! = 3628800
11! = 39916800
12! = 479001600
13! = 6227020800
14! = 87178291200
15! = 1307674368000
16! = 20922789888000
17! = 355687428096000
18! = 6402373705728000
19! = 121645100408832000
20! = 2432902008176640000
```
```1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
7! = 5040
8! = 40320
9! = 362880
10! = 3628800
11! = 39916800
12! = 479001600