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1 英语原意

常数：Constant

2 英文解释

constant

noun [ C ] MATHEMATICS, SCIENCE specialized

a particular number or amount that never changes

常数；常量，恒量

3 中文解释

常数，数学名词，指规定的数量与数字，如圆的周长和直径的比π﹑铁的膨胀系数为0.000012等。常数是具有一定含义的名称，用于代替数字或字符串，其值从不改变。

4 为什么叫常数？

常，读作cháng，形声字，始见于战国文字。本义是穿在下身的衣服，也泛指衣服。引申为恒久、经常之义，还用来指法典、伦常等。

常数里的“常”就是恒久不变的意思。

1 英语原意

几何：Geometry

2 英文解释

2.1 geometry noun [ U ]

the area of mathematics relating to the study of space and the relationships between points, lines, curves, and surfaces.

与空间以及点、线、曲线和曲面之间的关系有关的数学领域。

2.2 What is Geometry?

Geometry is all about shapes and their properties.

几何是关于形状及其属性的。

If you like playing with objects, or like drawing, then geometry is for you!

Geometry can be divided into:plane Geometry and Solid Geometry.

几何可分为：平面几何和立体几何。

Plane Geometry is about flat shapes like lines, circles and triangles ... shapes that can be drawn on a piece of paper.

平面几何是关于直线、圆形和三角形等平面形状......可以在一张纸上绘制的形状。

Solid Geometry is about three dimensional objects like cubes, prisms, cylinders and spheres.

立体几何是关于立方体、棱柱、圆柱体和球体等三维物体的。

Why? 为什么？

Why do we do Geometry? To discover patterns, find areas, volumes, lengths and angles, and better understand the world around us.

我们为什么要学几何？发现模式，找到面积，体积，长度和角度，更好地了解我们周围的世界。

3 中文解释

几何是研究空间结构及性质的一门学科。它是数学中最基本的研究内容之一，与分析、代数等等具有同样重要的地位，并且关系极为密切。

4 感想体会

几何一词最早起源于希腊语“γεωμετρία”，由“γέα”（土地）和“μετρεĭν”（测量）两个词合成而来，指土地的测量，即测地术。后来转化为拉丁语“geometria”。中文中的“几何”一词，最早是在明代利玛窦、徐光启合译《几何原本》时，由徐光启所创。当时并未给出所依根据，后世多认为一方面几何可能是拉丁化的希腊语GEO的音译，另一方面由于《几何原本》中也有利用几何方式来阐述数论的内容，也可能是magnitude（多少）的意译，所以一般认为几何是geometria的音、意并译。

1607年出版的《几何原本》中关于几何的译法在当时并未通行，同时代也存在着另一种译名——形学，如狄考文、邹立文、刘永锡编译的《形学备旨》，在当时也有一定的影响。

在1857年李善兰、伟烈亚力续译的《几何原本》后9卷出版后，几何之名虽然得到了一定的重视，但是直到20世纪初的时候才有了较明显的取代形学一词的趋势，如1910年《形学备旨》第11次印刷成都翻刊本徐树勋就将其改名为《续几何》。直至20世纪中期，已鲜有“形学”一词的使用出现。

“几何”最早的意思是“多少”的意思，比如诗词“对酒当歌，人生几何”，不知道后来为什么把Geometry翻译成几何了，其实Geometry更像是一种图形学。

1 英语原意

置信区间：Confidence Intervals

2 英文解释

2.1 单词解释

1) confidence noun (CERTAINTY)

the quality of being certain of your abilities or of having trust in people, plans, or the future

自信；信心，把握；信赖，信任

2) interval noun [C] (TIME/DISTANCE)

a period between two events or times

间隔;间距

2.2 What is Confidence Interval? 什么是置信区间？

A Confidence Interval is a range of values we are fairly sure our true value lies in.

置信区间是我们相当确定其真实值所在的范围。

Example: Average Height 

例子：平均身高

We measure the heights of 40 randomly chosen men, and get a mean height of 175cm,

我们测量了 40 名随机选择的男性的身高，得到的平均身高为 175 厘米，

We also know the standard deviation of men's heights is 20cm. 我们也知道男性身高的标准差是20厘米。

The 95% Confidence Interval (we show how to calculate it later) is: 95% 置信区间（稍后我们将介绍如何计算）为：

The "±" means "plus or minus", so 175cm ± 6.2cm means

±”表示“正负”，因此 175cm ± 6.2cm 可以表示为：

• 175cm − 6.2cm = 168.8cm to 
• 175cm + 6.2cm = 181.2cm

And our result says the true mean of ALL men (if we could measure all their heights) is likely to be between 168.8cm and 181.2cm

我们的结果是，所有男性的真实平均值（如果我们能测量他们所有的身高）可能在168.8厘米到181.2厘米之间

But it might not be!

但事实可能并非如此！

The "95%" says that 95% of experiments like we just did will include the true mean, but 5% won't.

“95%”表示，像我们刚才所做的那样，95%的值为真实平均值，但有5%不是。

So there is a 1-in-20 chance (5%) that our Confidence Interval does NOT include the true mean.

因此，我们的置信区间有 1/20 的几率（5%）不包括真实平均值。

2.3 Calculating the Confidence Interval 置信区间的计算

Step 1: start with

• the number of observations n
• the mean X
• and the standard deviation s

Note: we should use the standard deviation of the entire population, but in many cases we won't know it.

We can use the standard deviation for the sample if we have enough observations (at least n=30, hopefully more).

Using our example:

• number of observations n = 40
• mean X = 175
• standard deviation s = 20

Step 2: decide what Confidence Interval we want: 95% or 99% are common choices. Then find the "Z" value for that Confidence Interval here:

or 95% the Z value is 1.960

Step 3: use that Z value in this formula for the Confidence Interval

Where:

• X is the mean
• Z is the chosen Z-value from the table above
• s is the standard deviation
• n is the number of observations

And we have:

Which is:

In other words: from 168.8cm to 181.2cm

The value after the ± is called the margin of error

The margin of error in our example is 6.20cm

3 中文解释

置信区间是指由样本统计量所构造的总体参数的估计区间。在统计学中，一个概率样本的置信区间（Confidence interval）是对这个样本的某个总体参数的区间估计。置信区间展现的是这个参数的真实值有一定概率落在测量结果的周围的程度，其给出的是被测量参数的测量值的可信程度，即前面所要求的“一个概率”。

置信区间是一种常用的区间估计方法，所谓置信区间就是分别以统计量的置信上限和置信下限为上下界构成的区间 [2]。对于一组给定的样本数据，其平均值为μ，标准偏差为σ，则其整体数据的平均值的100(1-α)%置信区间为(μ-Ζα/2σ , μ+Ζα/2σ) ，其中α为非置信水平在正态分布内的覆盖面积 ，Ζα/2即为对应的标准分数。

4 感想体会

置信区间 Confidence Intervals，Confidence 是信心的意思，Intervals是间隔多长的意思，Confidence Intervals就是样本实测值有多大信心落在一个制定的区间范围内。中文翻译成置信区间，让人一头雾水。

参考文献：https://www.mathsisfun.com/data/confidence-interval.html

1 英语原意

博弈论：Game theory

2 英文解释

Game Theory can help us find the ...

博弈论可以帮助我们找到…

• best decision in a competitive situation, or
  在竞争形势下的最佳决策，或
• fairest decision in a cooperative situation
  在合作的情况下最公平的决定

... where the outcome for each player depends on their decision and the decisions of other players.

...每个参与者的结果取决于他们的决定和其他参与者的决定。

It is useful in business, military, sports, finance, personal life, games and more.

它在商业、军事、体育、金融、个人生活、游戏等方面都很有用。

Let's have a look at an example to see how Game Theory can help us find the best decision.

让我们看一个例子，看看博弈论如何帮助我们找到最佳决策。

2.1 Prisoner's Dilemma 囚徒困境

Casey and Dana are arrested after a burglary. They are in separate rooms and cannot cooperate.

凯西和戴娜在一次入室盗窃后被捕。他们在不同的房间，无法合作。

Casey has been told:凯西被告知：

if you both stay quiet you both get 1 month in prison for trespass

如果你们都保持沉默，你们都将因非法侵入罪入狱一个月

if you accuse Dana: you go free, Dana gets 10 months

如果你指控Dana你就自由了，而Dana要被关10个月 

if Dana accuses you: you get 10 months, Dana goes free

如果Dana指控你，你将被关10个月，而Dana就自由了

if you both blame each other you both get 6 months

如果你们互相指责，你们都被判6个月监禁。

What do you advise Casey to do? 你建议凯西怎么做？

... think about it for a bit ...

So maybe both Casey and Dana should keep quiet, right? They get only 1 month each that way..

所以也许凯西和戴娜都应该保持沉默，对吗？这样的话，他们每人只会被关一个月。

But that outcome is called 但这个结果被称为 unstable.不稳定

Because either side can do better by making the "I go free, you get 10 months" decision.

因为任何一方都可以做得更好，只需要指控对方，对方被关10个月，而自己就可以自由了。

So what to do?那该怎么办？

Well, sadly, Casey is better off blaming Dana.

很遗憾，凯西最好还是指控戴娜.

1 Whole Numbers

Whole Numbers are simply the numbers 0, 1, 2, 3, 4, 5, ... (and so on)

No Fractions!

Examples: 0, 7, 212 and 1023 are all whole numbers

(But numbers like ½, 1.1 and −5 are not whole numbers.)

2 Counting Numbers

Counting Numbers are Whole Numbers, but without the zero. Because you can't "count" zero.

So they are 1, 2, 3, 4, 5, ... (and so on).

3 Natural Numbers 自然数

"Natural Numbers" can mean either "Counting Numbers" {1, 2, 3, ...}, or "Whole Numbers" {0, 1, 2, 3, ...}, depending on the subject.

4 Integers 整数

Integers are like whole numbers, but they also include negative numbers ... but still no fractions allowed!

So, integers can be negative {−1, −2,−3, −4, ... }, positive {1, 2, 3, 4, ... }, or zero {0}

We can put that all together like this:

Integers = { ..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ... }

Examples: −16, −3, 0, 1 and 198 are all integers.

(But numbers like ½, 1.1 and 3.5 are not integers)

These are all integers, and they continue left and right infinitely:

整数（integer），是正整数、零、负整数的集合。

5 疑问和不同意见

Some People Have Different Definitions!

Some people (not me) say that whole numbers can also be negative, which makes them exactly the same as integers.

And some people say that zero is NOT a whole number. So there you go, not everyone agrees on a simple thing!

1 英语原意

复数：Complex Number

2 英文解释

2.1 complex adjective

1) involving a lot of different but related parts

由许多相关的不同部分组成的；构造复杂的

2) difficult to understand or find an answer to because of having many different parts

难懂的，费解的; 错综复杂的

2.2 What is a Complex Number?

A Complex Number is a combination of a Real Number and an Imaginary Number.

When we combine a Real Number and an Imaginary Number we get a Complex Number:

Examples: 1 + i , 39 + 3i, 0.8 − 2.2i ,−2 + πi

Complex does not mean complicated.It means the two types of numbers, real and imaginary, together form a complex, just like a building complex (buildings joined together).

3 中文解释

形如a+bi（a、b均为实数）的数为复数，其中，a被称为实部，b被称为虚部，i为虚数单位。复数通常用z表示，即z=a+bi，当z的虚部b＝0时，则z为实数；当z的虚部b≠0时，实部a＝0时，常称z为纯虚数。

1 英语原意

虚数：Imaginary number

2 英文解释

2.1 What is a Imaginary number?

The number whose square results in negative results is called an Imaginary number.

虚数，平方等于-1，即 i²=-1。

Imaginary numbers are numbers that can be written as a multiple of the imaginary unit i, where i is defined as the square root of -1. Examples include 2i, −3i, and 5i.

In simple words, the square root of negative numbers is called an imaginary number. They are called imaginary numbers as we cannot associate them with any real-life examples.

They are represented by “i” and are pronounced as iota at its value is,

i = √-1

2.2 History and Naming

The concept of imaginary numbers was first introduced in the 16th century by Italian mathematician Rafael Bombelli. Initially, the notion was met with skepticism and confusion. They got their name because they were deemed “impossible” and “imaginary,” as they couldn’t be represented on the number line of real numbers.

虚数的概念最早是由意大利数学家拉斐尔·邦贝利（Rafael Bombelli）在16世纪提出的。最初，这个概念遭到了怀疑和困惑。他们之所以得名，是因为它们被认为是“不可能的”和“虚构的”，因为它们不能在实数的数字线上表示。

Imaginary Numbers were once thought to be impossible, and so they were called "Imaginary" (to make fun of them).

虚数曾经被认为是不可能的，因此它们被称为“虚数”（取笑它们）。

But then people researched them more and discovered they were actually useful and important because they filled a gap in mathematics ... but the "imaginary" name has stuck.

但后来人们更多地研究了它们，发现它们实际上是有用和重要的，因为它们填补了数学的空白......但“虚构”的名字一直存在。

And that is also how the name "Real Numbers" came about (real is not imaginary).

这也是“实数”这个名字的由来（实数不是虚构的）。

1 英语愿意

实数：Real number

2 英文解释

2.1 What is a Real number?

A real number is any number that can be placed on a number line or expressed as in infinite decimal expansion. In other words, a real number is any rational or irrational number, including positive and negative whole numbers, integers, decimals, fractions, and numbers such as pi (π) and Euler’s number (e).

实数是可以放在数字线上或以无限十进制展开表示的任何数字。换句话说，实数是任何有理数或无理数，包括正负整数、整数、小数、分数和数字，例如 pi （π） 和欧拉数 （e）。

In contrast, an imaginary number or complex number is not a real number. These numbers contain the number i, where i² = -1.

相反，虚数或复数不是实数。这些数字包含数字 i，其中 i² = -1。

Real numbers are represented by the capital letter “R” or double struck typeface ℝ. The real numbers are an infinite set of numbers.

2.2 Why are they called "Real" Numbers?

Because they are not Imaginary Numbers.

因为它们不是虚数。

The Real Numbers had no name before Imaginary Numbers were thought of. They got called "Real" because they were not Imaginary. That is the actual answer!

在虚数出现之前，实数是没有名字的。它们被称为“真实”，因为它们不是虚构的。这才是真正的答案！

1 英语愿意

正态分布 Normal distribution

2 英文解释

2.1 Normal distribution noun [ C or U ]   MATHEMATICS   specialized

an arrangement of data in which most values are near the centre of the range and gradually become fewer towards each end, shown in a graph as the shape of a bell.

正态分布

例句：

• The resulting variable has a nearly normal distribution.
• Normal distributions consistently occur only when they involve random events and a large sample.
  正态分布只有在涉及随机事件和大样本时才会持续出现。

1 英语愿意

矢量：Vector

2 英文解释

2.1 vector noun [C] (CALCULATION)  PHYSICS

something physical such as a force that has size and direction

矢量；向量

2.2 vector noun [C] (ANIMAL) BIOLOGY

an insect or animal that carries a disease from one animal or plant to another

（在动物或植物之间传播疾病的）昆虫媒介，传病媒介

• Mosquitoes are the vectors of malaria.
  蚊子是疟疾的传播媒介。

2.3 What is a vector?

A vector has magnitude (size) and direction:

The length of the line shows its magnitude and the arrowhead points in the direction.

1 英语原意

在国内，三角函数计算里有个诱导公式，其对应的英文时Induction formula，Induction 的意思有就任、接纳会员、就职仪式、诱导、引发、归纳、（电磁）感应的意思，放在三角函数的计算公式里，怎么也说不过去，国内的诱导公式（Induction formula）应该是错误的写法，正确的写法应该叫简化公式（Reduction formula）。

在国外，没有“诱导公式”，只有“reduction formula”，是 reduction 而不是“induction”，reduction在数学里本来就有“简化”的意思，那么这套公式应该直接被翻译为“简化公式”，很显然，国内将induction和reduction混淆了。

1 英语原意

有理数 rational number

2 rational number 英语原意

A Rational Number can be made by dividing an integer by an integer.

Example:

1.5 is a rational number because 1.5 = 3/2   (3 and 2 are both integers)

Here are some more examples:

1 英语原意

代数 Algebra

2 Algebra 英文含义

Algebra, a part of mathematics in which signs and letters represent numbers.

3 Algebra 英文解释

Where did the word "Algebra" and its underlying ideas come from? Algebra, a key branch of mathematics, has a rich history. The term 'algebra' comes from an Arabic word meaning 'restoration' or 'completion'. Significant contributions to algebra were made by Diophantus in Greece, Brahmagupta in India, and al-Khwarizmi in Baghdad, who is often credited with giving algebra its name. Created by Sal Khan.

Algebra is one of the oldest branches in the history of mathematics that deals with number theory, geometry, and analysis. The definition of algebra sometimes states that the study of the mathematical symbols and the rules involves manipulating these mathematical symbols. Algebra includes almost everything right from solving elementary equations to the study of abstractions. Algebra equations are included in many chapters of Maths, which students will learn in their academics. Also, there are several formulas and identities present in algebra.

1 英语原意

函数：Function

1 英语原意

直角三角形 Right triangle

1 英语原意

包络线：envelope

2 envelope的含义

An envelope is something that envelops, or wraps around. It usually refers to the thing you stick a stamp on and snail mail.

1 英语原意

共轭：conjugate

2. conjugate 英语解释

verb (used with object),con·ju·gat·ed, con·ju·gat·ing.

1) Grammar.

a. to inflect (a verb).

b. to recite or display all or some subsets of the inflected forms of (a verb), in a fixed order:

One conjugates the present tense of the verb “be” as “I am, you are, he is, we are, you are, they are.”

2. to join together, especially in marriage.

verb (used without object),con·ju·gat·ed, con·ju·gat·ing.

3) Biology. to unite; to undergo conjugation.

4) Grammar. to be characterized by conjugation:

The Latin verb esse does not conjugate in the passive voice.

adjective

5)joined together, especially in a pair or pairs; coupled.

6)Botany. (of a pinnate leaf) having only one pair of leaflets.

3 共轭的中文含义

共轭在数学、物理、化学、地理等学科中都有出现。 本意：两头牛背上的架子称为轭，轭使两头牛同步行走。共轭即为按一定的规律相配的一对。通俗点说就是孪生。在数学中有共轭复数、共轭根式、共轭双曲线、共轭矩阵等。

共轭，是数学中一个比较有逼格的词。轭，是牛拉车用的木头，同时拉一辆车的两头牛，就是 共轭 的关系。把这种关系引申到数学中，只要是成对的东西，又找不着一个更合适的叫法时，就常常称它们为 共轭，这个 轭 并不总能找到相应的数学概念。

轭，是人类非常早的一个发明创造，就是下图这玩意，牛用车轭，英语叫yoke。

牛用车轭具体使用起来，大致就是下图这个样子：

1. 英语原意

积分：integration

Integration is a way of adding slices to find the whole.

Integration can be used to find areas, volumes, central points and many useful things.

1 英语本意

互质数：Coprime (or Relatively Prime or Mutually Prime) numbers

have no common factors other than 1.

2 互质数的定义

互质数为数学中的一种概念，即两个或多个整数的公因数只有1的非零自然数。公因数只有1的两个非零自然数，叫做互质数。

互质数的两个数不一定都是质数，比如11和35是互质数，但35不是质数。

3 互质数的应用

为什么齿数要设计为互质数！教材只告诉我结果，却没告诉我原因 - CAD2D3D.com

1 英语原意

方程：Equation

2 英文含义

equation noun

a mathematical statement in which you show that two amounts are equal using mathematical symbols

方程式；等式

In the equation 3x - 3 = 15, x = 6.

在方程式 3x-3=15中，x=6。

3 英文解释

What is an Equation?

方程是什么？

An equation says that two things are equal. It will have an equals sign "=" like this:

一个方程式说两件事是相等的。它有一个等号“=”，类似下式：

x − 2 = 4

That equations says:

what is on the left (x − 2)  equals  what is on the right (4)

So an equation is like a statement "this equals that"

4 中文解释

方程（equation）是指含有未知数的等式。是表示两个数学式（如两个数、函数、量、运算）之间相等关系的一种等式，使等式成立的未知数的值称为“解”或“根”。求方程的解的过程称为“解方程”。

5 方程是怎么来的？

中文数学书上使用的“方程”一词是翻译词。是中国清代数学家李善兰（1811—1882）在1859年和英国传教士伟烈亚力合译（Alexander·Wylie,1815—1887）英国数学家奥古斯都·德·摩尔根（Augustus·De·Morgan，1806—1871）的著作《代数学》（又称《代数学原理》或《代数初步》）一书时，李、伟两人很注重数学名词的正确翻译，他们借用或创设了近四百个数学的汉译名词，许多名词至今沿用。其中,“equation”的译名就是借用了中国古代的“方程”一词。

李善兰在书中解释道：“并代数之几数名为式，两式之间作等号，谓之方程。”这样，“方程”一词首次意为“含有未知数的等式”。

1873年,中国近代早期的又一个西方科学的传播者华蘅芳（1833—1902）与英国传教士傅兰雅（John·Fryer，1839—1928）合译英国著名数学家约翰·沃利斯（John·Wallis，1616—1703）的著作《代数术》（又称《代数论文》）,他们则把“equation”首次翻译为“方程式”。他们的意思是,“方程”与“方程式”应该区别开来,方程是指“未知数的意思”，而方程式是指“今有未知数的等式”。华、傅的主张在很长时间里被广泛采纳。直到1934年,中国数学学会对名词进行一审查，确定“方程”与“方程式”两者意义相通。在广义上,它们是指一元n次方程以及由几个方程联立起来的方程组。狭义则专指一元n次方程。既然“方程”与“方程式”同义,那么“方程”就显得更为简洁明了了。

6 感想体会

方程，英文原意是Equation，由Equal变形而来，所谓的方程，本质就是等式，单纯看中文很难明白其中的意思，而看方程的英文单词Equation，则通俗易懂。

本来很简单的词语，翻译成中文之后就变得晦涩难懂，在我们学习的时候，看到中文专业术语之后，还要经过二次转换才能明白其中的意思，无形之中增加了学习的难度，这也是为什么很多人学不好数理化的原因之一。

参考文献：

• https://dictionary.cambridge.org/dictionary/english/equal
• https://www.mathsisfun.com/algebra/equations-solving.html