Merge pull request #54 from oliviagillam/patch-13

Update lect16.md

_lectures/lect16.md

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 ---
-num: Lec 16
+num: Lec16
 lecture_date: 2020-02-26
-desc:
-ready: false
-pdfurl:
+desc: Recursion is Recursion
+ready: true
 ---
+
+
+# Thursday Lecture 2/27: Recursion!!!
+
+* Example: How could you find out how many people are in line if each person can only see the person directly in front of them?
+    * Each person asks the person in front of them "How many people are in front of you?" 
+    * Since each person does not know the total, they will ask the person in front of them. 
+    * Once the question reaches the first person in line, that person can confidently answer "There is no one in front of me."
+    * Then, the person before them (the second person in line) can take the answer they got (0 people), add 1 to it, and answer with confidence that "There is one person in front of me." 
+    * The 3rd person in line now adds 1 to it, and responds "There are two people in front of me." 
+    * This can go on and on until the last person in line knows how many people are in line.
+
+The simplest case is called the **Base case**: Theres nothing to do.
+
+**Example: Doing dishes**
+* Simplest case (base case): There are no dishes. You're done!
+* Next simplest case: There is one dish. You wash it and remove it from the sink. 
+   * Now, you are back to the base case -- there are no dishes. You're done!
+* Next simplest case: There are two dishes. 
+   * You wash one and remove it from the sink.
+   * Now, we are back to the previous simplest case when there is one dish. You wash it. 
+   * Now, you are back to the base case -- there are no dishes. You're done!
+* Next simplest case: There are 3 dishes. You wash one. Now there are 2 dishes. You wash one. Now there is one dish. You wash it. Now there are no dishes. You're done!
+* etc.
+
+**Each recursive case gets you closer to the base case.**
+
+
+```python3
+def doDishes(numDishes):
+    if numDishes == 0:
+        # return "You're done!"
+    else:
+        # wash one dish
+        # remove it from the sink
+        # return doDishes(numDishes - 1)
+```
+(* Assume `numDishes` is a non-negative integer)
+
+
+# The classic examples: Fibonacci and Factorials
+
+## Factorial
+
+```python3
+def factorial(n):
+    if (n == 0):
+        return 1
+    else:
+        return (n*factorial(n-1))
+```
+
+what if `n = 2.1`
+
+`2.1 - (1 * any integer)` will never equal 0. The base case will never be reached, so there will be an infinite recursion. 
+Python cannot handle an infinite recursion, so it will produce an error: `RecursionError: maximum recursion depth exceeded in comparison`. This is the error that you'll see if your function never reaches **the base case**.
+
+Let's create a new function that calls `factorial(n)`
+if and only if the user inputs a valid type:
+
+```python3
+def getFactorial(n):
+    '''
+    check that n is an integer>=0
+    return n
+    Otherwise, print an error
+    '''
+    if type(n) != int:
+        print("Error: incorrect type")
+    elif n < 0:
+        print("Error: incorrect value", n)
+    else:
+        return factorial(n)
+```
+* Question: What if the user inputs -2.1
+* Question: couldn't you do the same using only if statements and return statements?
+  * Answer: Yes, here is an example of that:
+```python3
+def getFactorial(n):
+    '''
+    check that n is an integer>=0
+    return n
+    Otherwise, print an error
+    '''
+    if type(n) != int:
+        print("Error: incorrect type")
+        return
+    if n < 0:
+        print("Error: incorrect value", n)
+        return
+    else:
+        return factorial(n)
+```
+
+* 1 1 2 3 5 8 13
+* 1 2 3 4 5 6 7
+```python3
+def fibonacci(n):
+    '''
+    returns the nth fibonacci term
+    '''
+    if n == 1:
+        return 1
+    if n == 2:
+        return 1
+    if n == 3:
+        return fibonacci(1) + fibonacci(2)
+    if n == 4:
+        return fibonacci(2) + fibonacci(3)
+    if n == 5:
+        return fibonacci(3) + fibonacci(4)
+
+        ...
+
+    # How do we generalize this
+```
+We can generalize this like so:
+
+```python3
+def fibonacci(n):
+    '''
+    returns the nth fibonacci term
+    '''
+    if n == 1:
+        return 1
+    if n == 2:
+        return 1
+    else:
+        return fibonacci(n - 2) + fibonacci(n - 1)
+```