Chapter 5  Conditionals and recursion 第 5 章 条件与递归

本页译自 Think Python 2e（Allen B. Downey）· Chapter 5 Conditionals and recursion。代码块保留英文原文不翻译；正文段段对照，中文块可用右下角按钮隐藏。

5.1 Modulus operator 5.1 取模运算符

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:

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 %0000 is zero, then % is divisible by %.

Also, you can extract the right-most digit or digits from a number. For example, %00000 yields the right-most digit of % (in base 10). Similarly %000000 yields the last two digits.

5.2 Boolean expressions 5.2 布尔表达式

A boolean expression is an expression that is either true or false. The following examples use the operator %0, which compares two operands and produces %000 if they are equal and %0000 otherwise:

%000 and %0000 are special values that belong to the type %000; they are not strings:

The %0 operator is one of the relational operators; the others are:

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 (%0). Remember that % is an assignment operator and %0 is a relational operator. There is no such thing as %0000 or %0000.

5.3 Logical operators 5.3 逻辑运算符

There are three logical operators: %00, %0, and %00. The semantics (meaning) of these operators is similar to their meaning in English. For example, %00000051











is true only if % is greater than 0 and less than 10.

%00000058









is true if either of the conditions is true, that is, if the number is divisible by 2 or 3.

Finally, the %00 operator negates a boolean expression, so %00000061



is true if %0000006 is false, that is, if % is less than or equal to %.

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."

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).

5.4 Conditional execution 5.4 条件执行

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 %0 statement:

The boolean expression after %0 is called the condition. If it is true, then the indented statement gets executed. If not, nothing happens.

%0 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 %000 statement, which does nothing.

5.5 Alternative execution 5.5 二选一执行

A second form of the %0 statement is alternative execution, in which there are two possibilities and the condition determines which one gets executed. The syntax looks like this:

If the remainder when % is divided by 2 is 0, then we know that % 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.

5.6 Chained conditionals 5.6 链式条件

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:

%000 is an abbreviation of "else if." Again, exactly one branch will be executed. There is no limit on the number of %000 statements. If there is an %000 clause, it has to be at the end, but there doesn’t have to be one.

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.

5.7 Nested conditionals 5.7 嵌套条件

One conditional can also be nested within another. We could have written the trichotomy example like this:

The outer conditional contains two branches. The first branch contains a simple statement. The second branch contains another %0 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:

The %0000 statement is executed only if we make it past both conditionals, so we can get the same effect with the %00 operator:

5.8 Recursion 5.8 递归

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:

If % is 0 or negative, it outputs the word, "Blastoff!" Otherwise, it outputs % and then calls a function named %00000108—itself—passing %00 as an argument.

What happens if we call this function like this?

The execution of %00000115 begins with %00, and since % is greater than 0, it outputs the value 3, and then calls itself...

The execution of %00000121 begins with %00, and since % is greater than 0, it outputs the value 2, and then calls itself...

这次 %00000124 的执行从 %00 开始，由于 % 大于 0，它输出值 2，然后调用自身……

The execution of %00000127 begins with %00, and since % is greater than 0, it outputs the value 1, and then calls itself...

这次 %00000130 的执行从 %00 开始，由于 % 大于 0，它输出值 1，然后调用自身……

The execution of %00000133 begins with %00, and since % is not greater than 0, it outputs the word, "Blastoff!" and then returns.

这次 %00000136 的执行从 %00 开始，由于 % 不大于 0，它输出「Blastoff!」这个词，然后返回。

The %00000139 that got %00 returns.

拿到 %00 的那次 %00000142 返回。

The %00000143 that got %00 returns.

拿到 %00 的那次 %00000146 返回。

The %00000147 that got %00 returns. And then you’re back in %0000014. So, the total output looks like this:

A function that calls itself is recursive; the process is called recursion.

As another example, we can write a function that prints a string % % times.

If %00000159 the %00000 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 %00000163: if % is greater than 0, it displays % and then calls itself to display % n−1 additional times. So the number of lines of output is %00000167
, which adds up to %.

For simple examples like this, it is probably easier to use a %00 loop. But we will see examples later that are hard to write with a %00 loop and easy to write with recursion, so it is good to start early.

5.9 Stack diagrams for recursive functions 5.9 递归函数的栈图

In Section 3.10, 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 %00000179 called with %0000.

Figure 5.1: Stack diagram.

As usual, the top of the stack is the frame for %0000018. It is empty because we did not create any variables in %0000018 or pass any arguments to it.

The four %00000187 frames have different values for the parameter %. The bottom of the stack, where %00, is called the base case. It does not make a recursive call, so there are no more frames.

Exercise 1 Draw a stack diagram for %000001 called with %00000194
and %00.

Exercise 2 Write a function called %000 that takes a function object and a number, %, as arguments, and that calls the given function % times.

5.10 Infinite recursion 5.10 无限递归

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:

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:

This traceback is a little bigger than the one we saw in the previous chapter. When the error occurs, there are 1000 %000002 frames on the stack!

5.11 Keyboard input 5.11 键盘输入

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 %00000209 that gets input from the keyboard. In Python 3, it is called %0000. 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 %00000211 returns what the user typed as a string.

Before getting input from the user, it is a good idea to print a prompt telling the user what to input. %00000216 can take a prompt as an argument:

The sequence %0 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 %00:

But if the user types something other than a string of digits, you get an error:

We will see how to handle this kind of error later.

5.12 Debugging 5.12 调试

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.

• 错误是什么类型，以及
• 它发生在哪里。

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.

In this example, the problem is that the second line is indented by one space. But the error message points to %, 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 SNR_db = 10 log₁₀ (P_signal / P_noise). In Python, you might write something like this:

But when you run it in Python 2, you get an error message.

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 %0000, 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.

5.13 Glossary 5.13 术语表

modulus operator:

An operator, denoted with a percent sign (%), that works on integers and yields the remainder when one number is divided by another.

boolean expression:

An expression whose value is either %000 or %0000.

relational operator:

One of the operators that compares its operands: %0, %0, %000, %000, %0000, and %0000.

logical operator:

One of the operators that combines boolean expressions: %00, %0, and %00.

conditional statement:

A statement that controls the flow of execution depending on some condition.

condition:

The boolean expression in a conditional statement that determines which branch is executed.

compound statement:

A statement that consists of a header and a body. The header ends with a colon (:). The body is indented relative to the header.

branch:

One of the alternative sequences of statements in a conditional statement.

chained conditional:

A conditional statement with a series of alternative branches.

nested conditional:

A conditional statement that appears in one of the branches of another conditional statement.

recursion:

The process of calling the function that is currently executing.

base case:

A conditional branch in a recursive function that does not make a recursive call.

infinite recursion:

A recursion that doesn’t have a base case, or never reaches it. Eventually, an infinite recursion causes a runtime error.

modulus operator 取模运算符：

一种用百分号（%）表示的运算符，作用于整数，返回一个数除以另一个数时的余数。

boolean expression 布尔表达式：

值要么为 %000、要么为 %0000 的表达式。

relational operator 关系运算符：

用来比较操作数的运算符之一：%0、%0、%000、%000、%0000 和 %0000。

logical operator 逻辑运算符：

用来组合布尔表达式的运算符之一：%00、%0 和 %00。

conditional statement 条件语句：

依据某个条件来控制执行流的语句。

condition 条件：

条件语句中决定执行哪个分支的布尔表达式。

compound statement 复合语句：

由头部和主体构成的语句。头部以冒号（:）结尾。主体相对头部缩进。

branch 分支：

条件语句中若干备选语句序列之一。

chained conditional 链式条件：

带有一系列备选分支的条件语句。

nested conditional 嵌套条件：

出现在另一个条件语句的某个分支中的条件语句。

recursion 递归：

调用当前正在执行的函数这一过程。

base case 基础情形：

递归函数中不再进行递归调用的条件分支。

infinite recursion 无限递归：

没有基础情形、或永远到达不了基础情形的递归。最终，无限递归会引发运行时错误。

5.14 Exercises 5.14 习题

Exercise 3 Fermat’s Last Theorem says that there are no positive integers a, b, and c such that

aⁿ + bⁿ = cⁿ

for any values of n greater than 2.

1. Write a function named %00000256
   
   that takes four parameters—%, %, % and %—and that checks to see if Fermat’s theorem holds. If n is greater than 2 and it turns out to be true that aⁿ + bⁿ = cⁿ, the program should print, "Holy smokes, Fermat was wrong!" Otherwise the program should print, "No, that doesn’t work."
2. Write a function that prompts the user to input values for %, %, % and %, converts them to integers, and uses %00000265
   
   to check whether they violate Fermat’s theorem.

1. 写一个名为 %00000266
   
   的函数，它接受四个参数——%、%、% 和 %——并检查费马定理是否成立。如果 n 大于 2 且确实有 aⁿ + bⁿ = cⁿ，程序应当打印「Holy smokes，费马错了！」；否则程序应当打印「No，那不成立。」
2. 写一个函数，提示用户输入 %、%、% 和 % 的值，把它们转换成整数，并用 %00000275
   
   检查它们是否违背了费马定理。

Exercise 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.)

如果三段长度中有任意一段大于另外两段之和，就无法构成三角形；否则可以。（如果两段长度之和等于第三段，它们构成所谓「退化」三角形。）

1. Write a function named %00000276
   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.
2. Write a function that prompts the user to input three stick lengths, converts them to integers, and uses %00000277
   to check whether sticks with the given lengths can form a triangle.

1. 写一个名为 %00000278
   的函数，它接受三个整数作为参数，并根据给定长度的木棍能否构成三角形来打印「Yes」或「No」。
2. 写一个函数，提示用户输入三根木棍的长度，把它们转换成整数，并用 %00000279
   检查给定长度的木棍能否构成三角形。

The following exercises use TurtleWorld from Chapter 4:

Exercise 5 Read the following function and see if you can figure out what it does. Then run it (see the examples in Chapter 4).

Figure 5.2: A Koch curve.

Exercise 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

1. Draw a Koch curve with length x/3.
2. Turn left 60 degrees.
3. Draw a Koch curve with length x/3.
4. Turn right 120 degrees.
5. Draw a Koch curve with length x/3.
6. Turn left 60 degrees.
7. Draw a Koch curve with length x/3.

1. 画一条长度为 x/3 的科赫曲线。
2. 左转 60 度。
3. 画一条长度为 x/3 的科赫曲线。
4. 右转 120 度。
5. 画一条长度为 x/3 的科赫曲线。
6. 左转 60 度。
7. 画一条长度为 x/3 的科赫曲线。

The exception is if x is less than 3: in that case, you can just draw a straight line with length x.

1. Write a function called %000 that takes a turtle and a length as parameters, and that uses the turtle to draw a Koch curve with the given length.
2. Write a function called %00000282 that draws three Koch curves to make the outline of a snowflake. Solution: %00000283
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
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3. The Koch curve can be generalized in several ways. See %00000284
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   for examples and implement your favorite.

1. 写一个名为 %000 的函数，它接受一只乌龟和一个长度作为参数，并用这只乌龟画出给定长度的科赫曲线。
2. 写一个名为 %00000286 的函数，画出三条科赫曲线，构成雪花的轮廓。 参考解答：%00000287
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   。
3. 科赫曲线有几种推广方式。参见 %00000288
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   
   中的例子，并实现你最喜欢的一种。