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def multiplier(factor): def multiplyByFactor(number, factor=factor): return number*factor return multiplyByFactor This works because default values are frozen when a function is defined.
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Recursion
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You ve learned a lot about making functions and calling them. You also know that functions can call other functions. What might come as a surprise is that functions can call themselves. If you haven t encountered this sort of thing before, you may wonder what this word recursion is. It simply means referring to (or, in our case, calling ) yourself. A humorous definition goes like this: re cur sion \ri- k&r-zh&n\ n: see recursion. Recursive definitions (including recursive function definitions) include references to the term they are defining. Depending on the amount of experience you have with it, recursion can be either mind-boggling or quite straightforward. For a deeper understanding of it, you should probably buy yourself a good textbook on computer science, but playing around with the Python interpreter can certainly help. In general, you don t want recursive definitions like the humorous one of the word recursion because you won t get anywhere. You look up recursion, which again tells you to look up recursion, and so on. A similar function definition would be def recursion(): return recursion() It is obvious that this doesn t do anything it s just as silly as the mock dictionary definition. But what happens if you run it You re welcome to try: The program simply crashes (raises an exception) after a while. Theoretically, it should simply run forever. However, each time a function is called, it uses up a little bit of memory, and after enough function calls have been made (before the previous calls have returned), there is no more room, and the program ends with the error message maximum recursion depth exceeded. The sort of recursion you have in this function is called infinite recursion (just as a loop beginning with while True and containing no break or return statements is an infinite loop) because it never ends (in theory). What you want is a recursive function that does something useful. A useful recursive function usually consists of the following parts: A base case (for the smallest possible problem) when the function returns a value directly A recursive case, which contains one or more recursive calls on smaller parts of the problem The point here is that by breaking the problem up into smaller pieces, the recursion can t go on forever because you always end up with the smallest possible problem, which is covered by the base case.
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So you have a function calling itself. But how is that even possible It s really not as strange as it might seem. As I said before, each time a function is called, a new namespace is created for that specific call; that means that when a function calls itself, you are actually talking about two different functions (or, rather, the same function with two different namespaces). You might think of it as one creature of a certain species talking to another one of the same species.
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Two Classics: Factorial and Power
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In this section, we examine two classic recursive functions. First, let s say you want to compute the factorial of a number n. The factorial of n is defined as n (n 1) (n 2) . . . 1. It s used in many mathematical applications (for example, in calculating how many different ways there are of putting n people in a line). How do you calculate it You could always use a loop: def factorial(n): result = n for i in range(1,n): result *= i return result This works and is a straightforward implementation. Basically, what it does is this: first, it sets the result to n; then, the result is multiplied by each number from 1 to n 1 in turn; finally, it returns the result. But you can do this differently if you like. The key is the mathematical definition of the factorial, which can be stated as follows: The factorial of 1 is 1. The factorial of a number n greater than 1 is the product of n and the factorial of n 1. As you can see, this definition is exactly equivalent to the one given at the beginning of this section. Now, consider how you implement this definition as a function. It is actually pretty straightforward, once you understand the definition itself: def factorial(n): if n == 1: return 1 else: return n * factorial(n-1) This is a direct implementation of the definition. Just remember that the function call factorial(n) is a different entity from the call factorial(n-1). Let s consider another example. Assume you want to calculate powers, just like the built-in function pow, or the operator **. You can define the (integer) power of a number in several different ways, but let s start with a simple one: power(x,n) (x to the power of n) is the number x multiplied by itself n-1 times (so that x is used as a factor n times). So power(2,3) is 2 multiplied with itself twice, or 2 2 2 = 8. This is easy to implement:
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