12.2: Simple Math Recursion
- Page ID
- 117600
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)By the end of this section you should be able to
- Identify a recursive case and a base case in a recursive algorithm.
- Demonstrate how to compute a recursive solution for the factorial function.
Calculating a factorial
The factorial of a positive integer is defined as the product of the integer and the positive integers less than the integer.
Ex: 5! = 5 * 4 * 3 * 2 * 1
Written as a general equation for a positive integer n:
n! = n * (n - 1) * (n - 2) * . . . * 1
The above formula for the factorial of n
results in a recursive formula: n! = n * (n - 1)!
Thus, the factorial of n
depends upon the value of the factorial at n - 1
. The factorial of n
can be found by repeating the factorial of n - 1
until (n - 1)! = 1!
(we know that 1! = 1
). This result can be used to build the overall solution as seen in the animation below.
Can the following algorithms be written recursively?
Defining a recursive function
Recursive algorithms are written in Python as functions. In a recursive function different actions are performed according to the input parameter value. A critical part of a recursive function is that the function must call itself.
A value for which the recursion applies is called the recursive case. In the recursive case, the function calls itself with a smaller portion of the input parameter. Ex: In the recursive function factorial()
, the initial parameter is an integer n
. In the function's recursive case, the argument passed to factorial()
is n - 1
, which is smaller than n
.
A value of n
for which the solution is known is called the base case. The base case stops the recursion. A recursive algorithm must include a base case; otherwise, the algorithm may result in an infinite computation.
To calculate a factorial, a recursive function, factorial()
is defined with an integer input parameter, n
. When n > 1
, the recursive case applies. The factorial()
calls itself with a smaller argument, n - 1
. When n == 1
, the solution is known because 1!
is 1
; therefore, n == 1
is a base case.
Note: 0!
is defined to be 1
; therefore, n == 0
is a second base case for factorial()
. When n < 1
, an error is returned.
For the questions below, the function rec_fact()
is another recursive function that calculates a factorial. What is the result of each definition of rec_fact()
if n = 17
is the initial input parameter?
0
Write a program that uses a recursive function to calculate the summation of numbers from 0 to a user specified positive integer n
.
Write a program that computes the sum of the digits of a positive integer using recursion.
Ex: The sum of the digits of 6721 is 16.
Hint: There are 10 base cases, which can be checked easily with the right condition.