ADM-201 dump PMP dumps pdf SSCP exam materials CBAP exam sample questions

如何来证明数学定理 – 译学馆
未登录,请登录后再发表信息
最新评论 (0)
播放视频

如何来证明数学定理

How to prove a mathematical theory - Scott Kennedy

证明是什么意思呢?
What is proof?
为什么它在数学中如此重要?
And why is it so important in mathematics?
证明为数学家,
Proofs provide a solid foundation for mathematicians
逻辑学家,统计学家,经济学家,建筑师,工程师
logicians, statisticians, economists, architects, engineers,
和许多其他学者去建立和测试他们的理论提供了坚实的基础。
and many others to build and test their theories on.
他们都是非常杰出的人物。
And they’re just plain awesome!
让我从头来谈
Let me start at the beginning.
我将给你介绍一个前辈:欧几里得
I’ll introduce you to a fellow named Euclid.
在这儿,”这儿正在看你,克里德。”
As in, ‘here’s looking at you, Clid.’
他生活在2300年前的希腊
He lived in Greece about 2,300 years ago,
被公认为几何学之父
and he’s considered by many to be the father of geometry.
如果你想要知道,该将你的几何粉丝邮件发给谁
So if you’ve been wondering where to send your geometry fan mail,
来自亚历山大的欧几里得,就是你要感谢的人
Euclid of Alexandria is the guy to thank for proofs.
欧几里得在发明或发展数学定律方面并不出名
Euclid is not really known for inventing or discovering a lot of mathematics
但是他改进了交流的方式,包括书写方式
but he revolutionized the way in which it is written,
表达方式和思考方式
presented, and thought about.
欧几里得着手于通过建立游戏规则的方式建立可视化数学模型
Euclid set out to formalize mathematics by establishing the rules of the game.
这些游戏的规则被称作公理
These rules of the game are called axioms.
一旦你理解了这些公理
Once you have the rules,
欧几里得说,你就可以利用这些公理去证明你想要证明的东西
Euclid says you have to use them to prove what you think is true.
如果你不能证明你是对的,那么你的定理或想法
If you can’t, then your theorem or idea
有可能是错的
might be false.
而且如果你的定理是错的,那么就任何定理在它之后出现并被使用的
And if your theorem is false, then any theorems that come after it and use it
就有可能是错的
might be false too.
就像一根错位的屋梁是如何降低整个屋子的高度一样
Like how one misplaced beam can bring down the whole house.
那么,所谓证明就是
So that’s all that proofs are:
使用已经建立好的公理(规则)去证明处于质疑中的一些定理的正确性
using well-established rules to prove beyond a doubt that some theorem is true.
之后你就可以像用砖盖房子一样,用这些定理
Then you use those theorems like blocks
构建数学的大厦
to build mathematics.
让我们通过一个例子检验一下
Let’s check out an example.
如果我想证明这两个三角形
Say I want to prove that these two triangles
是有这相同的大小和形状
are the same size and shape.
换言之,它们是全等的
In other words, they are congruent.
那么,一个方法去证明两三角形全等就是写一个证明过程
Well, one way to do that is to write a proof
这个证明过程展示了一个三角形的三边
that shows that all three sides of one triangle
和另一个三角形的三边完全相等
are congruent to all three sides of the other triangle.
那么,我们该怎么证明它呢?
So how do we prove it?
首先,我会写下我们知道的规则
First, I’ll write down what we know.
我们知道点M是边AB的中点
We know that point M is the midpoint of AB.
我们还知道边AC和边BC已经全等
We also know that sides AC and BC are already congruent.
综上,中点告诉了我们什么?
Now let’s see. What does the midpoint tell us?
幸运的是,我知道中点的定义
Luckily, I know the definition of midpoint.
它是一条线正中间的点
It is basically the point in the middle.
这意味着AM和BM的长度相等
What this means is that AM and BM are the same length,
如果M确实是AB的中点的话
since M is the exact middle of AB.
换言之,每个三角形底部的线全等
In other words, the bottom side of each of our triangles are congruent.
我将这步作为第二步
I’ll put that as step two.
很好!到目前为止我们有两对边全等了
Great! So far I have two pairs of sides that are congruent.
证明最后一对全等是容易的
The last one is easy.
第三条边在三角形的左边
The third side of the left triangle
是CM, 并且第三条边在三角形右边的是
is CM, and the third side of the right triangle is –
一样的,也是CM
well, also CM.
他们共享了一条边
They share the same side.
这理所当然是全等的
Of course it’s congruent to itself!
这个过程被叫做反射律
This is called the reflexive property.
每个东西都和自己本身是全等的
Everything is congruent to itself.
我将把这步作为第三步
I’ll put this as step three.
太好啦!我们刚刚证明了左边三角形的三条边
Ta dah! You’ve just proven that all three sides of the left triangle
与右边三角形的三条边全等
are congruent to all three sides of the right triangle.
因此,这两个三角形全等
Plus, the two triangles are congruent
根据三边全等三角形全等定理
because of the side-side-side congruence theorem for triangles.
当完成了这个证明,我想要做一个欧几里得做过的事
When finished with a proof, I like to do what Euclid did.
他用字母QED标记这证明的结束
He marked the end of a proof with the letters QED.
这是拉丁文’quid erat demonstrandum’
It’s Latin for ‘quod erat demonstrandum,’
它字面上的意思是
which translates literally to
什么被证明了
‘what was to be proven.’
但是我认为它的意思是,看我刚刚完成的
But I just think of it as ‘look what I just did!’
我能听到你思考的声音
I can hear what you’re thinking:
我为什么要学证明呢?
why should I study proofs?
一个原因是证明可以帮助你赢得任何论点
One reason is that they could allow you to win any argument.
亚伯拉罕·林肯,我们国家最伟大的领袖之一
Abraham Lincoln, one of our nation’s greatest leaders of all time
过去常在床头放一本欧几里得的《几何原本》
used to keep a copy of Euclid’s Elements on his bedside table
使他的头脑保持清醒
to keep his mind in shape.
另一个原因是你可以获得一百万美金
Another reason is you can make a million dollars.
你听我说
You heard me.
一百万美金
One million dollars.
这是马萨诸塞克雷数学研究所提供的价格
That’s the price that the Clay Mathematics Institute in Massachusetts
用于奖励证明出极难证明出的定理
is willing to pay anyone who proves one of the many unproven theories
而且这一行为被成为“千禧年的问题”
that it calls ‘the millenium problems.’
这些问题中一对在20世纪90年代和21世纪被证明出来
A couple of these have been solved in the 90s and 2000s.
但是在还研究所的声明和奖金之前
But beyond money and arguments,
证明在生活中无处不在
proofs are everywhere.
他们存在于建筑,艺术,计算机编程和网络安全之中。
They underly architecture, art, computer programming, and internet security.
如果没有一个人可以理解或者开始一个证明
If no one understood or could generate a proof,
我们就不能预测未来可发生的事情在我们的地球上
we could not advance these essential parts of our world.
最终,我们都知道谚语“证据在布丁中”(空谈不如实践的意思)
Finally, we all know that the proof is in the pudding.
布丁是十分美味的(所以证明也是美味的)。QED(证明结束)
And pudding is delicious. QED.

发表评论

译制信息
视频概述
听录译者

收集自网络

翻译译者

收集自网络

审核员

自动通过审核

视频来源

https://www.youtube.com/watch?v=S0DSM-EkQE8

相关推荐