## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

No credit card required

“91974˙CH02˙ﬁnal” 2012/12/14 13:58 page 80 #36
80 CHAPTER 2
π
thon
2.7 Summary
This chapter has introduced the following additional building blocks of programming and
Python:
Math modules
Boolean expressions
Selection statements
One very important pattern, the accumulator pattern, was also introduced. This pattern
uses an accumulator variable that is continuously updated as part of an iteration. It is
important to initialize the accumulator variable to an appropriate starting value. Finally,
we used the Monte Carlo simulation technique, which is based on random numbers, to
approximate the value of pi.
Key Terms
accumulator pattern initialization relational expression
accumulator variable Leibniz formula relational operator
approximation logical operator running sum
Archimedes approach Monte Carlo simulation selection statement
Boolean expression nested selection short-circuit evaluation
compound Boolean expression pi tail nesting
Python Keywords
and help math random
elif if else not return
False if or sin
for import print True
“91974˙CH02˙ﬁnal” 2012/12/14 13:58 page 81 #37
Programming Exercises 81
Bibliography
[AAR04] Runde Angel, Christine Abbot, and Dennis Runde. A Survey of Mathematics
with Applications. Prentice Hall, 2004.
[BB87] J. M. Borwein and P. B. Borwein. Ramanujan and pi. Scientiﬁc American, 1987.
[Wei02] Eric W. Weisstein. CRC Concise Encyclopedia of Mathematics. Chapman & Hall/
CRC, 2002.
Programming Exercises
2.1 Write a function to compute the circumference of a circle with a radius of r.User
as the parameter to the function. Use π from the math module.
2.2 Write a function to compute the area of a circle with a radius of r.User as the
parameter to the function. Use π from the math module.
2.3 Write a function to compute the volume of a sphere with a radius of r.User as the
parameter to the function. Use π from the math module.
2.4 Write a function that will return an approximate value for π by evaluating the equa-
tion π = 16(arctan
1
5
) 4(arctan
1
239
), where arctan is the arc tangent function from
the math library.
2.5 Write a function that will return an approximate value for π by evaluating the equa-
tion π =
ln(640320
3
+744)
163
, where ln is the log function from the math library.
2.6 Write a function isEven that takes a number as a parameter and returns True if the
number is even and False otherwise.
2.7 Write a function that will return an approximate value for π by evaluating the equa-
tion π =
12(1
1
3×3
+
1
5×3
2
1
7×3
3
+ .... Use a parameter called n to denote the
number of terms to include in the summation.
2.8 We have used the square root function from the math library a number of times in
this chapter. Write your own square root approximation function using the equation
X
k+1
=
1
2
× (X
k
+
n
X
k
) where X
0
= 1. This equation says that the
n can be found
by repeatedly computing the next X
i
term. The larger the number of terms used, the
better the answer. Allow your function to have two input parameters, the number
that you want the square root of and the number of terms to compute.
“91974˙CH02˙ﬁnal” 2012/12/14 13:58 page 82 #38
82 CHAPTER 2
π
thon
2.9
Buﬀon’s needle is a very old (eighteenth-century) simulation for approximating pi.
Research this simulation and implement it using Python and turtle graphics. How
good is the approximation?

## With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

No credit card required