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2021年2月23日发(作者：黑猫)

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－A

What

is

mathematics

Mathematics

comes

from

man’s

social

practice,

for

example,

industrial

and

agricultural

production,

commercial

activities,

military

operations

and scientific and technological researches. And in turn, mathematics serves the

practice

and

plays

a

great

role

in

all

fields.

No

modern

scientific

and

technological

branches could be regularly developed without the application of mathematics.

数

学来源于人类的社会实践，比如工农业生产，商业活动，

军事行动和科学技术研究。反过

来，数学服务于实践，并在各个 领域中起着非常重要的作用。

没有应用数学，任何一个现

在的科技的分支都不能正常发展。

From

the

early

need

of

man

came

the

concepts

of

numbers

and

forms.

Then,

geometry

developed

out

of

problems

of

measuring

land

,

and

trigonometry came from problems of surveying . To deal with some more complex

practical

problems,

man

established

and

then

solved

equation

with

unknown

numbers ,thus algebra occurred. Before 17th century, man confined himself to the

elementary mathematics, . , geometry, trigonometry and algebra, in which only the

constants

are

considered.

很 早的时候，人类的需要产生了数和形式的概念，接着，测量

土地的需要形成了几何，

出于测量的需要产生了三角几何，

为了处理更复杂的实际问题，

人

类建立和解决了带未知参数的方程，从而产生了代数学，

17

世纪前，人类局限于只考虑常

数的初等数学 ，即几何，三角几何和代数。

The rapid development of industry in 17th

century

promoted

the

progress

of

economics

and

technology

and

required

dealing

with

variable quantities. The leap from constants to variable quantities brought about

two

new

branches

of

mathematics----analytic

geometry

and

calculus,

which

belong

to

the higher mathematics. Now there are many branches in higher mathematics, among

which are mathematical analysis, higher algebra, differential

equations, function

theory and so on. 17

世纪工业的快速发展推动了经济技术的进步，

< p>
从而遇到需要处理变

量的问题，从常数带变量的跳跃产生了两个新的数学分 支

-----

解析几何和微积分，他们都

属于高等数学，现在高等数学里面有很多分支，其中有数学分析，高等代数，微分方程，函

数论等。

Mathematicians study conceptions and propositions, Axioms, postulates,

definitions

and

theorems

are

all

propositions.

Notations

are

a

special

and

powerful

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tool

of

mathematics

and

are

used

to

express

conceptions

and

propositions

very

often.

Formulas ,figures and charts are full of different symbols.

Some

of the

best known

symbols of mathematics are the Arabic numerals 1,2,3,4,5,6,7,8,9,0 and the signs

of addition, subtraction , multiplication, division and equality.

数学家研究的是

概念和命题，公理，公设，定义和定理都是命题。符号是数学中一个特殊 而有用的工具，常

用于表达概念和命题。公式，图表都是不同的符号……..The

conclusions

in

mathematics

are

obtained

mainly

by

logical

deductions

and

computation.

For

a

long

period

of

the

history of mathematics, the centric place of mathematics methods was occupied by

the

logical

deductions.

Now

,

since

electronic

computers

are

developed

promptly

and

used

widely,

the

role

of

computation

becomes

more

and

more

important.

In

our

times,

computation is not only used to deal with a lot of information and data, but also

to

carry

out

some

work

that

merely

could

be

done

earlier

by

logical

deductions,

for

example, the proof of most of geometrical theorems.

数学结论主要由逻辑 推理和计算

得到，

在数学发展历史的很长时间内，逻辑推理一直 占据着数学方法的中心地位，

现在，由

于电子计算机的迅速发展 和广泛使用，

计算机的地位越来越重要，

现在计算机不仅用于处 理

大量的信息和数据，

还可以完成一些之前只能由逻辑推理来做 的工作，

例如，

大多数几何定

理的证明 。

1

－

B

Equation

An

equation

is

a

statement

of

the

equality

between

two

equal

numbers or number symbols. Equation are of two kinds---- identities and equations

of condition. An arithmetic or an algebraic identity is an equation. In such an

equation either the two members are alike. Or become alike on the performance of

the indicated operation.

等式是关于两个数或者数的符号相等的一种描述。等式有两种

－ 恒等式和条件等式。

算术或者代数恒等式是等式。

这种等式的两 端要么一样，

要么经过执

行指定的运算后变成一样。

< p>
An identity involving letters is true for any set of

numerical values of the letters in it. An equation which is true only for certain

values of a letter in it, or for certain sets of related values of two or more of

its

letters,

is

an

equation

of

condition,

or

simply

an

equation.

Thus

3x-5=7

is

true

for x=4 only; and 2x-y=0 is true

for

x=6

and

y=2 and

for

many other pairs of

values

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for x and y.

含有字母的恒等式对其中字母的任 一组数值都成立。一个等式若仅仅对其中

一个字母的某些值成立，

或对其中两个或着多个字母的若干组相关的值成立，

则它是一个条

件等式，简称方程。因此

3x-5=7

仅当

x=4

时成立，而

2x-y=0

，当

x=6,y=2

时成立，且对

x,

y

的其他许多对值也成立。

A

root

of

an

equation

is

any

number

or

number

symbol

which

satisfies

the

equation.

There

are

various

kinds

of

equation.

They

are

linear

equation,

quadratic equation, etc.

方程的根是满足方程的任意数或者数的符号。方程有很 多种，

例如：

线性方程，二次方程等。

To solve an equation means to find the value of the

unknown

term.

To

do

this

,

we

must,

of

course,

change

the

terms

about

until

the

unknown

term stands alone on one side of the equation, thus making it equal to something

on the other side. We then obtain the value of the unknown and the answer to the

question.

To

solve

the

equation,

therefore,

means

to

move

and

change

the

terms

about

without making the equation untrue,

until only the unknown quantity is left on one

side ,no matter

which side.

解方程意味着求未知项的值，为了求未知项的值，当然必须

移项，

直到未知项单独在方程的一边，令其等于方程的另一边，从而求得未知项 的值，

解决

了问题。

因此解方程意味着 进行一系列的移项和同解变形，

直到未知量被单独留在方程的一

边，无论那一边。

Equation are of very great use. We can use equation in many

mathematical

problems.

We

may

notice

that

almost

every

problem

gives

us

one

or

more

statements

that

something

is

equal

to

something,

this

gives

us

equations,

with

which

we may work if we need it.

方程作用很大，可以用方程解决很多数学问题。注意到几乎

每一个问题都给出一个或多个关于一个事情与另一个事情相等的陈述，

这就给出了方 程，

利

用该方程，如果我们需要的话，可以解方程。

< p>

2

－

A

Why

study

geometry

Many

leading

institutions

of

higher

learning

have

recognized that positive benefits can be gained by all who study this branch of

mathematics. This is evident from the fact that they require study of geometry as

a

prerequisite

to

matriculation

in

those

schools.

许多 居于领导地位的学术机构承认，

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所有学习这个数学分支的人都将得到确实的受益，

许多学校把几何的学习作 为入学考试的先

决条件，从这一点上可以证明。

Geometr y

had

its

origin

long

ago

in

the

measurement

by

the Babylonians and Egyptians of their lands inundated by the floods of the Nile

River. The greek word geometry is derived from geo,

meaning “earth” and metron,

meaning

“measure”

.

As

early

as

2000

.

we

find

the

land

surveyors

of

these

people

re- establishing vanishing landmarks and boundaries by utilizing the truths of

geometry .

几何学起源于很久以前巴比伦人和埃及人测量他们被尼罗河洪水淹没的土地，

希腊语几何来源于

geo

，意思是”土地“，和

metron

意思是”测量“。公元前

2000

年之

前，我们发现这些民族的土地测量者利用几何知识重新确定消失了的土地标志和边界。

2

－

B

Some

geometrical

terms

A

solid

is

a

three-dimensional

figure.

Common

examples

of solids are cube, sphere, cylinder, cone and pyramid. A cube has six faces which

are

smooth

and

flat.

These

faces

are

called

plane

surfaces

or

simply

planes.

A

plane

surface has two dimensions, length and width. The surface of a blackboard or of a

tabletop is an example of a plane surface.

立体是一个三维图形，立 体常见的例子是

立方体，球体，柱体，圆锥和棱锥。立方体有

6

个面，都是光滑的和平的，这些面被称为平

面曲面或者简称为平 面。

平面曲面是二维的，

有长度和宽度，

黑板和桌子上面的面都是平面

曲面的例子。

2

－

C

三角函数于直角三角形的解

One

of

the

most

important

applications

of trigonometry is the solution of triangles. Let us now take up the solution to

right triangles. A triangle is composed of six

parts three sides

and three angles.

To

solve

a

triangle

is

to

find

the

parts

not

given.

A

triangle

may

be

solved

if

three

parts (at least one of these

is a side )

are

given. A right triangle

has one

angle,

the right angle, always given. Thus a right triangle can be solved when two sides,

or one side and an acute angle, are given.

三角形最重要的应用之一是解三角 形，现

在我们来解直角三角形。

一个三角形由

< br>6

个部分组成，

三条边和三只角。

解一个三角形就是

要求出未知的部分。

如果三角形的三个部分 （其中至少有一个为边）为已知，则此三角形就

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可以解出。直角三角形 的一只角，即直角，总是已知的。因此，如果它的两边，或一边和一

锐角为已知，则此直 角三角形可解。

9-A Introduction A large variety of scientific problems arise in which one tries

to

determine

something

from

its

rate

of

change.

For

example

,

we

could

try

to

compute

the

position

of

a

moving

particle

from

a

knowledge

of

its

velocity

or

acceleration.

Or a radioactive substance may be disintegrating at a known rate and we may be

required to determine the amount of material present after a given time.

大量 的

科学问题需要人们根据事物的变化率来确定该事物，

例如，< /p>

我们可以由已知速度或者加速度

来计算移动粒子的位置

< p>
.

又如，

某种放射性物质可能正在以已知的速度 进行衰变，

需要我们

确定在给定的时间后遗留物质的总量。

In

examples

like

these,

we

are

trying

to

determine

an

unknown

function

from

prescribed

information

expressed

in

the

form

of

an

equation

involving

at

least

one

of

the

derivatives

of

the

unknown

function

.

These

equations

are

called

differential

equations,

and

their

study

forms

one

of

the

most

challenging

branches of mathematics.

在类似 的例子中，我们力求由方程的形式表示的信息来确定未

知函数，

而这种方程至少包含了未知函数的一个导数。

这些方程称为微分方程，

< br>对其研究形

成了数学中最具有挑战性的一门分支。

The

study

of

differential

equations

is

one

part

of mathematics that, perhaps more than any other, has been directly inspired by

mechanics, astronomy, and mathematical physics.

微分方程的研究是数学的一部分，也

许比其 他分支更多的直接受到力学，天文学和数学物理的推动。

Its

history

began

in

the

17th

century

when

Newton,

Leibniz,

and

the

Bernoullis

solved

some

simple

differential

equations

arising

from

problems

in

geometry

and

mechanics.

These

early

discoveries, beginning about 1690, gradually led to the development of a lot of

“special tricks” for solving certain special kinds of differential equation. 微< /p>

分方程起源于

17

世纪，当时牛顿，莱布 尼茨，波努力家族解决了一些来自几何和力学的简

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