The modulus operator
works on integers and yields the remainder when the first operand is
divided by the second. In Python, the modulus operator is a percent sign
(%
). The syntax is the
same as for other operators:
>>> quotient = 7 / 3 >>> print quotient 2 >>> remainder = 7 % 3 >>> print remainder 1
So 7 divided by 3 is 2 with 1 left over.
The modulus operator turns out to be surprisingly useful. For
example, you can check whether one number is divisible by another—if
x % y
is zero, then x
is divisible by y
.
Also, you can extract the right-most digit or digits from a
number. For example, x % 10
yields
the right-most digit of x
(in base
10). Similarly x % 100
yields the
last two digits.
A Boolean expression is
an expression that is either true or false. The following examples use
the operator ==
, which compares two
operands and produces True
if they
are equal and False
otherwise:
>>> 5 == 5 True >>> 5 == 6 False
True
and False
are special values that belong to the
type bool
; they are not
strings:
>>> type(True) <type 'bool'> >>> type(False) <type 'bool'>
The ==
operator is one of the
relational operators; the others
are:
x
!=
y
# x is not equal to y
x
>
y
# x is greater than y
x
<
y
# x is less than y
x
>=
y
# x is greater than or equal to y
x
<=
y
# x is less than or equal to y
Although these operations are probably familiar to you, the Python
symbols are different from the mathematical symbols. A common error is
to use a single equal sign (=
)
instead of a double equal sign (==
).
Remember that =
is an assignment
operator and ==
is a relational
operator. There is no such thing as =<
or =>
.
There are three logical
operators: and
, or
, and not
. The semantics (meaning) of these
operators is similar to their meaning in English. For example, x > 0 and x < 10
is true only if
x
is greater than 0
and less than 10.
n%2 == 0 or n%3 == 0
is true if
either of the conditions is true, that is, if the
number is divisible by 2 or 3.
Finally, the not
operator
negates a boolean expression, so not (x >
y)
is true if x > y
is
false, that is, if x
is less than or
equal to y
.
Strictly speaking, the operands of the logical operators should be boolean expressions, but Python is not very strict. Any nonzero number is interpreted as “true.”
>>> 17 and True True
This flexibility can be useful, but there are some subtleties to it that might be confusing. You might want to avoid it (unless you know what you are doing).
In order to write useful programs, we almost always need the
ability to check conditions and change the behavior of the program
accordingly. Conditional statements give us this
ability. The simplest form is the if
statement:
if
x
>
0
:
'x is positive'
The boolean expression after if
is called the condition. If it is true,
then the indented statement gets executed. If not, nothing
happens.
if
statements have the same
structure as function definitions: a header followed by an indented
body. Statements like this are called compound
statements.
There is no limit on the number of statements that can appear in
the body, but there has to be at least one. Occasionally, it is useful
to have a body with no statements (usually as a place keeper for code
you haven’t written yet). In that case, you can use the pass
statement, which does nothing.
if
x
<
0
:
pass
# need to handle negative values!
A second form of the if
statement is alternative execution, in
which there are two possibilities and the condition determines which one
gets executed. The syntax looks like this:
if
x
%
2
==
0
:
'x is even'
else
:
'x is odd'
If the remainder when x
is
divided by 2 is 0, then we know that x
is even, and the program displays a message
to that effect. If the condition is false, the second set of statements
is executed. Since the condition must be true or false, exactly one of
the alternatives will be executed. The alternatives are called branches, because they are branches in the flow
of execution.
Sometimes there are more than two possibilities and we need more than two branches. One way to express a computation like that is a chained conditional:
if
x
<
y
:
'x is less than y'
elif
x
>
y
:
'x is greater than y'
else
:
'x and y are equal'
elif
is an abbreviation of
“else if.” Again, exactly one branch will be executed. There is no limit
on the number of elif
statements. If
there is an else
clause, it has to be
at the end, but there doesn’t have to be one.
if
choice
==
'a'
:
draw_a
()
elif
choice
==
'b'
:
draw_b
()
elif
choice
==
'c'
:
draw_c
()
Each condition is checked in order. If the first is false, the next is checked, and so on. If one of them is true, the corresponding branch executes, and the statement ends. Even if more than one condition is true, only the first true branch executes.
One conditional can also be nested within another. We could have written the trichotomy example like this:
if
x
==
y
:
'x and y are equal'
else
:
if
x
<
y
:
'x is less than y'
else
:
'x is greater than y'
The outer conditional contains two branches. The first branch
contains a simple statement. The second branch contains another if
statement, which has two branches of its
own. Those two branches are both simple statements, although they could
have been conditional statements as well.
Although the indentation of the statements makes the structure apparent, nested conditionals become difficult to read very quickly. In general, it is a good idea to avoid them when you can.
Logical operators often provide a way to simplify nested conditional statements. For example, we can rewrite the following code using a single conditional:
if
0
<
x
:
if
x
<
10
:
'x is a positive single-digit number.'
The print
statement is executed
only if we make it past both conditionals, so we can get the same effect
with the and
operator:
if
0
<
x
and
x
<
10
:
'x is a positive single-digit number.'
It is legal for one function to call another; it is also legal for a function to call itself. It may not be obvious why that is a good thing, but it turns out to be one of the most magical things a program can do. For example, look at the following function:
def
countdown
(
n
):
if
n
<=
0
:
'Blastoff!'
else
:
n
countdown
(
n
-
1
)
If n
is 0 or negative, it
outputs the word, “Blastoff!” Otherwise, it outputs n
and then calls a function named countdown
—itself—passing n-1
as an argument.
What happens if we call this function like this?
>>> countdown(3)
The execution of countdown
begins with n=3
, and since n
is greater than 0, it outputs the value 3,
and then calls itself...
The execution of
countdown
begins withn=2
, and sincen
is greater than 0, it outputs the value 2, and then calls itself...The execution of
countdown
begins withn=1
, and sincen
is greater than 0, it outputs the value 1, and then calls itself...The execution of
countdown
begins withn=0
, and sincen
is not greater than 0, it outputs the word, “Blastoff!” and then returns.The
countdown
that gotn=1
returns.The
countdown
that gotn=2
returns.
The countdown
that got n=3
returns.
And then you’re back in __main__
. So, the total output looks like
this:
3 2 1 Blastoff!
A function that calls itself is recursive; the process is called recursion.
As another example, we can write a function that prints a string
n
times.
def
print_n
(
s
,
n
):
if
n
<=
0
:
return
s
print_n
(
s
,
n
-
1
)
If n <= 0
the return
statement exits the function. The flow
of execution immediately returns to the caller, and the remaining lines
of the function are not executed.
The rest of the function is similar to countdown
: if n
is greater than 0, it displays s
and then calls itself to display s
n-1 additional times.
So the number of lines of output is 1 + (n -
1)
, which adds up to n
.
For simple examples like this, it is probably easier to use a
for
loop. But we will see examples
later that are hard to write with a for
loop and easy to write with recursion, so
it is good to start early.
In Stack Diagrams, we used a stack diagram to represent the state of a program during a function call. The same kind of diagram can help interpret a recursive function.
Every time a function gets called, Python creates a new function frame, which contains the function’s local variables and parameters. For a recursive function, there might be more than one frame on the stack at the same time.
Figure 5-1 shows a stack diagram for countdown
called with n = 3
.
As usual, the top of the stack is the frame for __main__
. It is empty because we
did not create any variables in __main__
or pass any arguments to it.
The four countdown
frames have
different values for the parameter n
.
The bottom of the stack, where n=0
,
is called the base case. It does not
make a recursive call, so there are no more frames.
If a recursion never reaches a base case, it goes on making recursive calls forever, and the program never terminates. This is known as infinite recursion, and it is generally not a good idea. Here is a minimal program with an infinite recursion:
def
recurse
():
recurse
()
In most programming environments, a program with infinite recursion does not really run forever. Python reports an error message when the maximum recursion depth is reached:
File "<stdin>", line 2, in recurse File "<stdin>", line 2, in recurse File "<stdin>", line 2, in recurse . . . File "<stdin>", line 2, in recurse RuntimeError: Maximum recursion depth exceeded
This traceback is a little bigger than the one we saw in the
previous chapter. When the error occurs, there are 1000 recurse
frames on the stack!
The programs we have written so far are a bit rude in the sense that they accept no input from the user. They just do the same thing every time.
Python 2 provides a built-in function called raw_input
that gets input from
the keyboard. In Python 3, it is called input
. When this function is called, the
program stops and waits for the user to type something. When the user
presses Return or Enter, the program resumes and raw_input
returns what the user
typed as a string.
>>> input = raw_input() What are you waiting for? >>> print input What are you waiting for?
Before getting input from the user, it is a good idea to print a
prompt telling the user what to input. raw_input
can take a prompt as an
argument:
>>> name = raw_input('What...is your name? ') What...is your name? Arthur, King of the Britons! >>> print name Arthur, King of the Britons!
The sequence
at
the end of the prompt represents a newline, which is a special character that causes
a line break. That’s why the user’s input appears below the
prompt.
If you expect the user to type an integer, you can try to convert
the return value to int
:
>>> prompt = 'What...is the airspeed velocity of an unladen swallow? ' >>> speed = raw_input(prompt) What...is the airspeed velocity of an unladen swallow? 17 >>> int(speed) 17
But if the user types something other than a string of digits, you get an error:
>>> speed = raw_input(prompt) What...is the airspeed velocity of an unladen swallow? What do you mean, an African or a European swallow? >>> int(speed) ValueError: invalid literal for int()
The traceback Python displays when an error occurs contains a lot of information, but it can be overwhelming, especially when there are many frames on the stack. The most useful parts are usually:
What kind of error it was, and
Where it occurred.
Syntax errors are usually easy to find, but there are a few gotchas. Whitespace errors can be tricky because spaces and tabs are invisible and we are used to ignoring them.
>>> x = 5 >>> y = 6 File "<stdin>", line 1 y = 6 ^ SyntaxError: invalid syntax
In this example, the problem is that the second line is indented
by one space. But the error message points to y
, which is misleading. In general, error
messages indicate where the problem was discovered, but the actual error
might be earlier in the code, sometimes on a previous line.
The same is true of runtime errors.
Suppose you are trying to compute a signal-to-noise ratio in decibels. The formula is . In Python, you might write something like this:
import
math
signal_power
=
9
noise_power
=
10
ratio
=
signal_power
/
noise_power
decibels
=
10
*
math
.
log10
(
ratio
)
decibels
But when you run it in Python 2, you get an error message.
Traceback (most recent call last): File "snr.py", line 5, in ? decibels = 10 * math.log10(ratio) OverflowError: math range error
The error message indicates line 5, but there is nothing wrong
with that line. To find the real error, it might be useful to print the
value of ratio
, which turns out to be
0. The problem is in line 4, because dividing two integers does floor
division. The solution is to represent signal power and noise power with
floating-point values.
In general, error messages tell you where the problem was discovered, but that is often not where it was caused.
In Python 3, this example does not cause an error; the division operator performs floating-point division even with integer operands.
An operator, denoted with a percent sign (%
), that works on integers and yields
the remainder when one number is divided by another.
One of the operators that compares its operands: ==
, !=
, >
, <
, >=
, and <=
.
One of the operators that combines boolean expressions:
and
, or
, and not
.
A statement that controls the flow of execution depending on some condition.
The boolean expression in a conditional statement that determines which branch is executed.
A statement that consists of a header and a body. The header ends with a colon (:). The body is indented relative to the header.
One of the alternative sequences of statements in a conditional statement.
A conditional statement with a series of alternative branches.
A conditional statement that appears in one of the branches of another conditional statement.
The process of calling the function that is currently executing.
A conditional branch in a recursive function that does not make a recursive call.
A recursion that doesn’t have a base case, or never reaches it. Eventually, an infinite recursion causes a runtime error.
Exercise 5-3.
Fermat’s Last Theorem says that there are no integers a, b, and c such that
for any values of n greater than 2.
Write a function named check_fermat
that takes four
parameters—a
, b
, c
and n
—and that checks to see if
Fermat’s theorem holds. If n is greater than
2 and it turns out to be true that
the program should print, “Holy smokes, Fermat was wrong!” Otherwise the program should print, “No, that doesn’t work.”
Write a function that prompts the user to input values for
a
, b
, c
and n
, converts them to
integers, and uses check_fermat
to check whether they violate
Fermat’s theorem.
Exercise 5-4.
If you are given three sticks, you may or may not be able to arrange them in a triangle. For example, if one of the sticks is 12 inches long and the other two are one inch long, it is clear that you will not be able to get the short sticks to meet in the middle. For any three lengths, there is a simple test to see if it is possible to form a triangle:
If any of the three lengths is greater than the sum of the other two, then you cannot form a triangle. Otherwise, you can. (If the sum of two lengths equals the third, they form what is called a “degenerate” triangle.)
Write a function named is_triangle
that takes three integers as
arguments, and that prints either “Yes” or “No,” depending on
whether you can or cannot form a triangle from sticks with the
given lengths.
Write a function that prompts the user to input three stick
lengths, converts them to integers, and uses is_triangle
to check
whether sticks with the given lengths can form a triangle.
The following exercises use TurtleWorld from Chapter 4:
Exercise 5-5.
Read the following function and see if you can figure out what it does. Then run it (see the examples in Chapter 4).
def
draw
(
t
,
length
,
n
):
if
n
==
0
:
return
angle
=
50
fd
(
t
,
length
*
n
)
lt
(
t
,
angle
)
draw
(
t
,
length
,
n
-
1
)
rt
(
t
,
2
*
angle
)
draw
(
t
,
length
,
n
-
1
)
lt
(
t
,
angle
)
bk
(
t
,
length
*
n
)
Exercise 5-6.
The Koch curve is a fractal that looks something like Figure 5-2. To draw a Koch curve with length x, all you have to do is:
Draw a Koch curve with length x/3.
Turn left 60 degrees.
Draw a Koch curve with length x/3.
Turn right 120 degrees.
Draw a Koch curve with length x/3.
Turn left 60 degrees.
Draw a Koch curve with length x/3.
The exception is if x is less than 3: in that case, you can just draw a straight line with length x.
Write a function called koch
that takes a turtle and a length as
parameters, and that uses the turtle to draw a Koch curve with the
given length.
Write a function called snowflake
that draws three Koch curves
to make the outline of a snowflake.
Solution: http://thinkpython.com/code/koch.py.
The Koch curve can be generalized in several ways. See http://en.wikipedia.org/wiki/Koch_snowflake for examples and implement your favorite.
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