用Python探索数学：编程解密代数、统计与微积分

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"Doing Math with Python 由 Saha 原版 PDF" 这本书旨在结合编程、数学和科学这三个主题，作者Amit Saha希望通过编程来探索高中阶段的数学概念。书中涉及了一系列熟悉的话题，但使用的方法是计算机而非传统的笔和纸。以下是书中涵盖的一些关键知识点： 1. **测量单位操作**：学习如何用Python处理和转换不同的度量单位，这对于在科学计算中至关重要。 2. **抛体运动分析**：通过编程模拟物体的抛射运动，理解重力和速度的影响，以及预测物体的轨迹。 3. **统计分析**：包括计算平均数（mean）、中位数（median）和众数（mode），以及确定变量间的线性相关性，这些都是数据分析的基础。 4. **代数方程求解**：使用Python的库如sympy解决代数方程，这对于理解和应用代数概念非常有帮助。 5. **简单摆动的运动描述**：研究简谐振动，理解摆动周期和振幅的关系，这是物理学中的经典问题。 6. **概率模拟**：通过编程模拟骰子游戏，学习概率和统计分布，增强对随机事件理解。 7. **几何图形创建**：利用Python生成几何形状，例如绘制圆、正多边形等，有助于视觉化和理解几何概念。 8. **微积分基础**：探讨函数的极限（limits）、导数（derivatives）和积分（integrals），这些是微积分的核心概念，对于物理、工程等多个领域都极其重要。 9. **Python编程基础**：除了数学概念，读者还将学习如何使用Python进行数据处理和计算，提升编程技能。这本书不仅适合对数学感兴趣的高中生和大学生，也适用于任何想要通过编程来加深数学理解的成年人。它提供了一种新颖且实用的方法来学习和教授数学，使得抽象的概念变得更具交互性和可视化。技术审阅者Jeremy Kun的专业把关确保了书中的代码和数学概念的准确性。通过这本书，读者可以将编程与数学相结合，提升解决问题的能力，并在探索中获得乐趣。

WORKING WITH NUMBERS

Let’s take our ﬁrst steps toward using

Python to explore the world of math and

science. We’ll keep it simple now so you can

get a handle on using Python itself. We’ll start

by performing basic mathematical operations, and

then we’ll write simple programs for manipulating

and understanding numbers. Let’s get started!

Basic Mathematical Operations

The Python interactive shell is going to be our friend in this book. Start the

Python 3 IDLE shell and say “hello” (see Figure 1-1) by typing print('Hello

IDLE')

and then pressing enter. (For instructions on how to install Python

and start IDLE, see Appendix A.) IDLE obeys your command and prints

the words back to the screen. Congratulations—you just wrote a program!

When you see the >>> prompt again, IDLE is ready for more instructions.

4 Chapter 1

Labels: Attaching Names to Numbers

As we start designing more complex Python programs, we’ll assign names

to numbers—at times for convenience, but mostly out of necessity. Here’s a

simple example:

u >>> a = 3

>>> a + 1

v >>> a = 5

>>> a + 1

At u, we assign the name a to the number 3. When we ask Python to

evaluate the result of the expression a + 1, it sees that the number that

a refers to is 3, and then it adds 1 and displays the output (4). At v, we

change the value of a to 5, and this is reﬂected in the second addition

operation. Using the name a is convenient because you can simply change

the number that a points to and Python uses this new value when a is

referred to anywhere after that.

This kind of name is called a label. You may have been introduced to

the term variable to describe the same idea elsewhere. However, consider-

ing that variable is also a mathematical term (used to refer to something like

x in the equation x + 2 = 3), in this book I use the term variable only in the

context of mathematical equations and expressions.

Different Kinds of Numbers

You may have noticed that I’ve used two kinds of numbers to demonstrate

the mathematical operations—numbers without a decimal point, which you

already know as integers, and numbers with a decimal point, which program-

mers call ﬂoating point numbers. We humans have no trouble recognizing

and working with numbers whether they’re written as integers, ﬂoating

point decimals, fractions, or roman numerals. But in some of the programs

that we write in this book, it will only make sense to perform a task on a

particular type of number, so we’ll often have to write a bit of code to have

the programs check whether the numbers we input are of the right type.

Python considers integers and ﬂoating point numbers to be different

types. If you use the function type(), Python will tell you what kind of num-

ber you’ve just input. For example:

>>> type(3)

>>> type(3.5)

>>> type(3.0)

Working with Numbers 5

Here, you can see that Python classiﬁes the number 3 as an integer

(type 'int') but classiﬁes 3.0 as a ﬂoating point number (type 'float'). We

all know that 3 and 3.0 are mathematically equivalent, but in many situa-

tions, Python will treat these two numbers differently because they are two

different types.

Some of the programs we write in this chapter will work properly only

with an integer as an input. As we just saw, Python won’t recognize a num-

ber like 1.0 or 4.0 as an integer, so if we want to accept numbers like that

as valid input in these programs, we’ll have to convert them from ﬂoating

point numbers to integers. Luckily, there’s a function built in to Python that

does just that:

>>> int(3.8)

>>> int(3.0)

The function int() takes the input ﬂoating point number, gets rid of

anything that comes after the decimal point, and returns the resulting inte-

ger. The float() function works similarly to perform the reverse conversion:

>>> float(3)

3.0

float()

takes the integer that was input and adds a decimal point to

turn it into a ﬂoating point number.

Working with Fractions

Python can also handle fractions, but to do that, we’ll need to use Python’s

fractions module. You can think of a module as a program written by someone

else that you can use in your own programs. A module can include classes,

functions, and even label deﬁnitions. It can be part of Python’s standard

library or distributed from a third-party location. In the latter case, you

would have to install the module before you could use it.

The fractions module is part of the standard library, meaning that it’s

already installed. It deﬁnes a class Fraction, which is what we’ll use to enter

fractions into our programs. Before we can use it, we’ll need to import it,

which is a way of telling Python that we want to use the class from this mod-

ule. Let’s see a quick example—we’ll create a new label, f, which refers to

the fraction 3/4:

u >>> from fractions import Fraction

v >>> f = Fraction(3, 4)

w >>> f

Fraction(3, 4)

We ﬁrst import the Fraction class from the fractions module u.

Next, we create an object of this class by passing the numerator and

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