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如何证明2=1

Trick your math teacher

Good day, welcome to the Tecmath Channel.
你们好 欢迎来到《数学技巧》频道
What we’re gonna be doing in this video is
今天我们要做的是
I’m gonna prove to you that 2 is equal to 1.
证明2=1
Okay, so you might be thinking
好吧 你可能会想
there’s no way you can prove this.
这不可能
But sit back. I’ll show you.
但请大家先等会 看好
Alright, the first thing I’m gonna do is
我要做的第一件事是
I’m gonna use two letters to do this.
借助两个字母
They’re gonna be a and b here.
也就是这儿的a和b
And we’re gonna say these guys are equal to one another.
我们假设a=b
So a could be equal to 1
所以a等于1
but b could be equal to 1.
b也等于1
Alright, so let’s start off with this.
好的 让我们从这里开始
We’re gonna say that a is equal to b.
我们知道a等于b
Alright, I think we can agree on that one.
我想我们可以在这一点达成共识
The first thing I drew this equation
写好这个等式后
is I am going to multiply it by a.
首先两边同时乘以a
So if we were to do this,
这样的话
a times a is a²
a乘以a等于a²
and a times b is ab.
a乘以b等于ab
Now to both sides what we’re going to do
现在我们要对等式两边做的是
is we are going to take away b².
我们要让两边都减去b²
So maybe you want to do that and see what we get.
也许你会想这么做 看看我们会得到什么
If you get a² and take away b²,
如果你用a²减去b²
you get your a² take away b²
你会得到a²-b²
and this side you’re gonna get ab take away b² .
等式这一边你会得到ab-b²
The next thing I’m gonna do,
接下来我要做的事情
I’m gonna get you to do
我会让你去做的事情
is we are going to factorize this.
是我们要将它因式分解
Okay, we’re going to factorize this.
是的 我们要把等式两边因式分解
Alright, maybe you even attempt doing this.
也许你还尝试这样做了
What you’ll notice is this is
你会注意到这是个
the difference between two squares here.
平方差公式
So a² take away b² becomes the following
所以a²-b²变成下面这项
and becomes a take away b and a plus b.
变成(a-b)乘以(a+b)
And if you don’t believe that, expand it out.
如果你不相信 你可以将式子展开
That’s what…this becomes here.
上面就是这样子变成下面这个的
This one here you’re going to notice this sort of the equation
在这里 你将注意到这种等式
that we have b on both of these,
每一项里面都有b
so we can put a b here
所以我们可以把b提出来
and if I divide b into… ab here, I get a
如果我拿ab去除以b 我会得到a
and if I divide b into b² here, I get to be here.
如果我拿b²去除以b 我会得到b
Okay so, this is where this has been factorized.
好的 这就是等式分解后的样子
The next thing I’m gonna do is I am going to
下一件我要去做的事情是
divide both of these sides by the common factor
将等式的两边除以一个公因子
which is a take away b we can divide
这个公因子就是(a-b) 我们可以用来除
I’ll put that in brackets a take away b.
我把a-b放在括号中
So, what are you do that and see what you get
所以 你做了什么 看看你得到了什么
if you divide this side by a take away b,
如果你把这边除以(a-b)
you’re going to get a plus b
你会得到a+b
and on this side if you divide by a take away b,
在等式这一边 如果你拿它除以(a-b)
you’re going to get b.
你会得到b
Alright, so what do we know?
好吧 那我们知道什么?
We know that a is equal to b.
我们知道a等于b
So if a is equal to b, I could say a here is b.
所以如果a等于b 我可以说这里的a是b
Okay? They’re the same thing a is equal to b
好吗?a是b与a等于b是一个意思
so I’m gonna substitute b for a.
所以我要用b代替a
So b plus b is equal to b,
所以b加上b等于b
that means 2b is equal to b
这意味着2b等于b
and if you now divide both sides by b, what do you get?
如果现在将等式两边同时除以b 你会得到什么?
You’re gonna get this proof here
你会得到这个证明
that indeed 2 is equal to 1. Proven!
确实2等于1 证毕!
Hey, what do you think of that, all right?
嘿 你怎么看?
I can prove it! I have found the impossible!
我可以证明这个 我发现了不可能
I have shown that 2 is equal to 1.
我已经证明了2等于1
Okay, but is it true?
好吧 但这是真的吗?
And straight away you’re gonna know intuitively
马上你就会直觉地知道
that 2 is not equal to 1.
2是不等于1的
Of course, 2 is not equal to 1.
当然 2不等于1
So why don’t you go through here and
所以你要仔细看看
see where I have made a bit of a fallacy, a bit of an error,
我在哪里犯了一个隐蔽的错误
a bit of a sneaky sort of bit amounts.
这个错误藏的有点深
So pause it see if we can go through and find it.
所以停下来看看我们能不能找到它
Okay, did you find out where that was?
好吧 你找到它在哪里了吗?
Alright. I’ll give you a bit of a hint of what’s going on here.
好吧 这个问题我会给你一点提示
It’s what’s now an asset division by zero fallacy, okay?
这是一个零除谬误 对吗?
Somewhere along here we have divided by zero
在这里的某个地方 我们除以了0
and as you know you can’t divide by zero, yeah that’s not possible.
如你所知 你不能除以0 这是不可能的
You do that on a calculator you getting there error.
你在计算器上执行此操作 会出现错误
Okay you can’t divide by zero.
好吧 你不能除以0
The universe just does not like that, okay?
这个宇宙不喜欢那样 好吗?
You can’t see how many nothings are in something.
你无法得知1能由多少个0组成
It just doesn’t make sense.
这完全没有道理
So where do we divide by zero here?
那我们在哪里除以了0呢?
At this step we multiply by a,
在这一步 我们乘了a
so we didn’t divide by zero.
所以我们没有除以0
And this step we took away b², okay?
在这一步 我们减去了b²对吗?
This one we factorize
这一步我们分解了等式
and we just rearrange the equation.
我们只是重新排列了方程
This one we divided by a take away b.
在这里我们除以了(a-b)
We did divide here
我们在这里用了除法
but did we divide by zero?
但是我们除以0了吗?
Well, let’s substitute in our values here and we’ll find out.
让我们在这里替换我们的值 我们会找出答案
So we’ve divided by…
所以我们除以了…
and a is equal to 1
a是等于1的
and B is equal to 1.
b是等于1的
So 1 take away 1
所以1减去1
and if you have a look here
如果你看看这里
that means we’re divided by zero at this step.
这意味着我们在这一步除以了0
As we go from this step to this step,
当我们从这一步到这一步时
we’re divided by zero.
我们除以了0
And that’s caused what’s known as a division by zero fallacy.
这产生了一个我们所知的零除谬误
Interesting, good for tricking people,
很有趣 很容易骗到人
but also something that you want to watch here for when you’re doing maths okay?
但也是你在做数学的时候想在这里学到的东西
if you’re playing around with algebra or anything like this,
如果你在玩代数之类的东西
it’s a really really common easy mistake to make.
这是一个很常见的容易犯的错误
There’s a few little fallacies, a few little mistakes
当人们使用代数的时候会犯
that people make when they’re doing algebra
一些小谬论 一些小错误
and so this is one of the reasons I’m showing you this video,
因此 这就是我做这个视频的原因之一
not just because it’s a boy’s phone I’m a little bit fun
不只是因为一个男孩打电话问 而是我也有一点感兴趣
but also I think it’s a it’s a good thing to know
同时因为我认为这是件值得去了解的事
because then you know to watch out for it okay
因为之后你就会知道要小心了
and you know it’s a source of error.
而且你知道这是错误的根源
So tell us what you thought of that, okay?
那么告诉我们你的想法 好吗?
Uh… anyway great to be making videos.
嗯…无论如何制作视频都是很棒的事
See you next time, bye.
下次见 拜拜

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视频概述

视频介绍了一个在代数中很容易犯的零除谬误,希望我们了解这个错误的根源并谨慎学习,不要再犯这个错误。

听录译者

HiyaTay

翻译译者

审核员

审核员PH

视频来源

https://www.youtube.com/watch?v=J46WZrSkCyw

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