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Recursion Announcements Recursive Functions Recursive Functions 4 Recursive Functions Definition : A function is called recursive if the body of that function calls itself, either directly or indirectly 4 Recursive Functions Definition

  1. Recursion

  2. Announcements

  3. Recursive Functions

  4. Recursive Functions � 4

  5. Recursive Functions Definition : A function is called recursive if the body of that function calls itself, either directly or indirectly � 4

  6. Recursive Functions Definition : A function is called recursive if the body of that function calls itself, either directly or indirectly Implication : Executing the body of a recursive function may require applying that function � 4

  7. Recursive Functions Definition : A function is called recursive if the body of that function calls itself, either directly or indirectly Implication : Executing the body of a recursive function may require applying that function � 4

  8. Recursive Functions Definition : A function is called recursive if the body of that function calls itself, either directly or indirectly Implication : Executing the body of a recursive function may require applying that function Drawing Hands, by M. C. Escher (lithograph, 1948) � 4

  9. Digit Sums 2+0+1+9 = 12 � 5

  10. Digit Sums 2+0+1+9 = 12 • If a number a is divisible by 9, then sum_digits(a) is also divisible by 9 � 5

  11. Digit Sums 2+0+1+9 = 12 • If a number a is divisible by 9, then sum_digits(a) is also divisible by 9 • Useful for typo detection! � 5

  12. Digit Sums 2+0+1+9 = 12 • If a number a is divisible by 9, then sum_digits(a) is also divisible by 9 • Useful for typo detection! The Bank of 61A 1234 5678 9098 7658 OSKI THE BEAR � 5

  13. Digit Sums 2+0+1+9 = 12 • If a number a is divisible by 9, then sum_digits(a) is also divisible by 9 • Useful for typo detection! The Bank of 61A A checksum digit is a function of all the other 1234 5678 9098 7658 digits; It can be computed to detect typos OSKI THE BEAR � 5

  14. Digit Sums 2+0+1+9 = 12 • If a number a is divisible by 9, then sum_digits(a) is also divisible by 9 • Useful for typo detection! The Bank of 61A A checksum digit is a function of all the other 1234 5678 9098 7658 digits; It can be computed to detect typos OSKI THE BEAR • Credit cards actually use the Luhn algorithm, which we'll implement after sum_digits � 5

  15. The Problem Within the Problem The sum of the digits of 6 is 6. Likewise for any one-digit (non-negative) number (i.e., < 10). The sum of the digits of 2019 is 201 9 Sum of these digits + This digit That is, we can break the problem of summing the digits of 2019 into a smaller instance of the same problem, plus some extra stuff. We call this recursion � 6

  16. Sum Digits Without a While Statement � 7

  17. Sum Digits Without a While Statement def split(n): """Split positive n into all but its last digit and its last digit.""" return n // 10, n % 10 � 7

  18. Sum Digits Without a While Statement def split(n): """Split positive n into all but its last digit and its last digit.""" return n // 10, n % 10 def sum_digits(n): """Return the sum of the digits of positive integer n.""" � 7

  19. Sum Digits Without a While Statement def split(n): """Split positive n into all but its last digit and its last digit.""" return n // 10, n % 10 def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n � 7

  20. Sum Digits Without a While Statement def split(n): """Split positive n into all but its last digit and its last digit.""" return n // 10, n % 10 def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) � 7

  21. Sum Digits Without a While Statement def split(n): """Split positive n into all but its last digit and its last digit.""" return n // 10, n % 10 def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last � 7

  22. The Anatomy of a Recursive Function def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last � 8

  23. The Anatomy of a Recursive Function • The def statement header is similar to other functions def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last � 8

  24. The Anatomy of a Recursive Function • The def statement header is similar to other functions def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last � 8

  25. The Anatomy of a Recursive Function • The def statement header is similar to other functions • Conditional statements check for base cases def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last � 8

  26. The Anatomy of a Recursive Function • The def statement header is similar to other functions • Conditional statements check for base cases def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last � 8

  27. The Anatomy of a Recursive Function • The def statement header is similar to other functions • Conditional statements check for base cases • Base cases are evaluated without recursive calls def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last � 8

  28. The Anatomy of a Recursive Function • The def statement header is similar to other functions • Conditional statements check for base cases • Base cases are evaluated without recursive calls def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last � 8

  29. The Anatomy of a Recursive Function • The def statement header is similar to other functions • Conditional statements check for base cases • Base cases are evaluated without recursive calls • Recursive cases are evaluated with recursive calls def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last � 8

  30. The Anatomy of a Recursive Function • The def statement header is similar to other functions • Conditional statements check for base cases • Base cases are evaluated without recursive calls • Recursive cases are evaluated with recursive calls def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last � 8

  31. The Anatomy of a Recursive Function • The def statement header is similar to other functions • Conditional statements check for base cases • Base cases are evaluated without recursive calls • Recursive cases are evaluated with recursive calls def sum_digits(n): """Return the sum of the digits of positive integer n.""" if n < 10: return n else: all_but_last, last = split(n) return sum_digits(all_but_last) + last (Demo) � 8

  32. Recursion in Environment Diagrams

  33. Recursion in Environment Diagrams � 10 http://pythontutor.com/composingprograms.html#code=def%20fact%28n%29%3A%0A%20%20%20%20if%20n%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%201%0A%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20return%20n%20*%20fact%28n%20-%201%29%0A%20%20%20%20%20%20%20%20%0Afact%283%29&cumulative=true&curInstr=0&mode=display&origin=composingprograms.js&py=3&rawInputLstJSON=%5B%5D

  34. Recursion in Environment Diagrams (Demo) � 10 http://pythontutor.com/composingprograms.html#code=def%20fact%28n%29%3A%0A%20%20%20%20if%20n%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%201%0A%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20return%20n%20*%20fact%28n%20-%201%29%0A%20%20%20%20%20%20%20%20%0Afact%283%29&cumulative=true&curInstr=0&mode=display&origin=composingprograms.js&py=3&rawInputLstJSON=%5B%5D

  35. Recursion in Environment Diagrams (Demo) � 10 http://pythontutor.com/composingprograms.html#code=def%20fact%28n%29%3A%0A%20%20%20%20if%20n%20%3D%3D%200%3A%0A%20%20%20%20%20%20%20%20return%201%0A%20%20%20%20else%3A%0A%20%20%20%20%20%20%20%20return%20n%20*%20fact%28n%20-%201%29%0A%20%20%20%20%20%20%20%20%0Afact%283%29&cumulative=true&curInstr=0&mode=display&origin=composingprograms.js&py=3&rawInputLstJSON=%5B%5D

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