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# Comparing Iterative and Recursive Factorial Functions: Comparing iterative and recursive factorial functions

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- what I want to do in this video is make it clear
- the distinction between
- an I-rative, or I should say iterative,
- I always pronounce it wrong
- iterative function definition
- and a recursive function definition
- We'll do it really by... Just kinda understanding where the interation is happening over here
- and where the recurssion is happening here on the right.
- So when we start off we see that product is set to be equal to 1
- and then we enter our for loop and the for loop really is the meat of this interative function definition.
- And understanding what's happening in the for loop let's... Let me make a little table here.
- So I'm gonna make a table for the value of our variable i and I'll also figure out what the value of
- product times i plus 1,
- cuz every interation to this for loop we are going to evaluate this bussiness right over here
- and then I'm going to make a calumn for the new value of our product, the new value of our product.
- Let me underline these things and then we have the new value of our product.
- So we learned in many videos ago that in python we say for i in range.
- This range part right over here...
- This range part right over here returns a list and it returns a list of the number of elements
- that we have passed number... We passed into it right over here.
- So if we assume and I should have said it from the get code.
- Let's assume that we are calling just to make something specific,
- let's say this is the result of a call of factorial of 3.
- So the argument that we passed this factorial is 3 so the variable number will refer to 3.
- So when you call range of number it will literally return a list: 0, 1, 2.
- So 3 elements starting at 0, the last element is 3 minus 1. It is 2.
- And so each loop trough this for loop i is going to be assigned to each sucessive element
- in this list.
- So on the first time trough this for loop i is going to be assigned to 0.
- So our i is going to refer to 0.
- And then product times i plus 1, well in this first loop product appeared before we even entered the
- loop, product was defined to be 1.
- So product is going to be 1 and it is 1 times -- I don't want to do it in that colour, I'll just do it
- in mag... I'll do it in magneta -- 1 times i which is 0. 1 times 0 plus 1.
- Plus 1 and this... and then our new value of product is essentially this evaluated,
- we have it right over here.
- Product is equal to all of this bussiness.
- So our new value is 1 times 0 plus 1 and that's just 1 times 1 and that's just going to be 1.
- That's all we had inside the for loop clause,
- cuz that was the only stuff that was intended within this for loop
- and then so we go back up and we are going to interate trough the next interation of our loop.
- I guess you could say and now i is going to be assigned to 1.
- So now i is going to be 1.
- This expression over here, we are going to take our old product.
- So product is still 1, so product is 1 and it is going to be times i which is now 1 plus 1.
- And what's this going to be equal to?
- Well if you evaluate all of this you get 1 times 2 so now the new value for product is 2.
- After our second interation trough the loop our second pass trough the loop.
- Now it will go back to the beginning of the for loop and i will be assigned to the next element in the
- list. It wil be now assigned to 2.
- So i is now 2. This thing over here, we are going to take...
- This is going to be product. Product is now 2.
- So it's going to be 2 times i, well i is now 2 plus 1 and so what is this, it's 2 times 3 or 6.
- Or 6 and then it will go and we'll say okay, can we assign i to any more elements in this?
- No we have run out of elements so we brake out of the for loop and we just return the product.
- Or the variable product, what it's referring to and that's what I should really say.
- We should return the value that the product is referring to and that value is 6.
- So when you call factorial 3 it will return 6.
- So if you were to say factorial, if you say factorial of 3 plus factorial of 3
- and you were to evaluate this expression this expression would evaluate to 6
- and this expression over here would evaluate to 6 because that is what the function would return
- and then if you add those up they would evaluate to 12.
- So this is why we call it interative.
- We kept interating trough this same set of instructions and now let's compare the recurssive definition.
- And this one is a little more fun in a lot of ways.
- Once again we are going to call factorial of 3.
- Factorial of 3.
- So 3 is our argument. That's the value that number will refer to and it will take on
- and says if number is less than or equal to 1.
- Well 3 is not less than or equal to 1.
- So we are not gonna do this part over here.
- We are going to do else clause.
- So we are going to return number. We are going to return number times factorial of all of this.
- So this is going to evaluate to number which is 3. That's the argument we passed.
- Times factorial of number minus 1.
- Well number minus 1 is going to evaluate to 2. 3 minus 1 is 2.
- So times factorial of 2.
- Well that's just another function called a factorial so we go back.
- Okay, factorial but now the argument is 2. so the number is 2.
- We go here. If number is less than or equal to 1 we do this,
- but the number isn't less than or equal to 1. It's 2. So now we do else.
- So what we now want to return is the number times the factorial number minus 1.
- Well in this situation...
- In this situation the number is now 2 and we are gonna want to multiply that times the factorial...
- times the factorial of 2 minus 1.
- Well 2 minus 1 is just 1.
- Times the factorial of 1.
- Well we just made another function call so the interpretator kinda has to remember that we made this
- whole series of function calls and has to keep digging deeper and deeper and deeper.
- So now we have called factorial of 1.
- Factorial of 1. What is the argument? Number is referring to 1.
- If number is less than or equal to 1.
- Number is less than or equal to 1.
- Now this is what we call a base case. We are kinda going down to it.
- So the number is less than or equal to 1. Return 1.
- So in this situation, when we call factorial 1 it literally returns 1.
- And so we now know that factorial of 2 evaluates to 2 times 1.
- So this evaluates to 2 and now we know factorial of 3 evaluates to 3 times 2 which will evaluate to 6.
- So very different ways of thinking about it, but getting the exact same result.
- Once again if you take factorial of 3 plus factorial 3, it doesn't matter which way we implement it.
- We'll get 6 plus 6 or 12.

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