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  • Suppose when you sit in a beer garden

    假設你坐在露天啤酒店裡,

  • beautiful weather, and you have ordered the beer,

    天氣非常好,然後你點了一杯啤酒,

  • the beer comes, you put it in a table, and then you touch the table

    啤酒來了,你把它放在桌上,可是你一碰桌子,

  • and the table is unstable and the beer is poured out.

    桌子不穩,啤酒就打翻了。

  • You are angry!

    你超級生氣!

  • It’s a four-legged table. The table is completely stable.

    這是一張四腳桌,桌子非常平穩,

  • The problem is the ground on which the table stands.

    問題卻出在供桌子站立的地面。

  • This is not flat and that’s why one leg is above the ground.

    地面不平,所以有一隻桌腳總是懸在半空中。

  • And then if you put your hands again on the table,

    如果再把你的手放到桌上,

  • it goes down and it’s the instability of the table.

    它又靠到地上了,所以它就是桌子不平的最大原因。

  • The moment solution is you take a sheet of paper.

    現在的解決辦法就是拿一張紙。

  • For example this paper is under the beer glass and put it under this leg

    舉例來說,拿啤酒杯墊來墊桌腳,

  • and for a while, it looks okay but after a few minutes, we are angry again because of this paper

    剛開始看起來很有用,但在幾分鐘後,我們會因為這張紙

  • is compressed a little bit and instability again.

    被壓扁了,讓桌子再次不平穩而生氣。

  • And we hate that.

    我們非常討厭那樣。

  • Mathematicians never have unstable tables.

    數學家絕不會有不平穩的桌子。

  • They know what to do.

    他們知道該怎麼做。

  • And what you do is very very simple.

    你要做的事情其實非常簡單。

  • Turn the table and start moving the table and try to turn it

    轉動桌子,移動桌子,然後試著轉動它,

  • so that you have a quarter of a turn and on the way of your turning,

    那麼你就會有一個四分之一的圓圈,而且當你在轉動的時候,

  • there will be a moment where it’s absolutely stable.

    一定會找到一個地方是絕對平穩的。

  • So youre just rotating like a rotator like rotating a disc?

    所以你就像一個旋轉軸一樣轉動,像轉動唱片那樣嗎?

  • Yes, I rotate the table like a disc and typically only a few centimeters are needed

    對,我就像轉唱片一樣轉動桌子,我只需要幾公分的距離,

  • and suddenly it’s stable and this is not by chance.

    一下子它就平穩了,而且這不是偶然發生。

  • This there it’s a mathematical proof that this will always happen.

    這有數學證明的,這常常會發生。

  • Youre gonna have to give me that proof now.

    你現在證明給我看。

  • I give you that proof now.

    我現在就證明給你看。

  • Here’s the ground and here this is the position of the four legs

    這是地面,然後這個是四隻腳的位置,

  • and we enumerate themthis is leg 1, this is 2

    我們把它們點出來,這是第一隻,這是第二隻,

  • this is 3, this is leg 4.

    第三隻,第四隻。

  • And suppose that leg 1 is above the floor whereas these three are fixed on the ground.

    假設第一隻腳懸在半空中,其他三隻卻都靠在地上。

  • Now, of course, if we put pressure on 1, then we still the instability.

    當然,如果我們用手壓在第一隻腳上,桌子還是不會平。

  • And now, we do the following:

    所以現在,我們要這樣做:

  • We measure the height of leg 1.

    測量第一隻腳的長度。

  • Remember, we always measure the height of leg 1.

    一定要記住我們是要量第一隻腳的長度。

  • So if you do that in time, then we get associate it to time T,

    加入時間因子,把它命名為T,

  • we associate height of leg 1.

    與腳1的長度作關聯。

  • So time is zero, we get some T=0,

    所以時間為零,我們就寫T=0。

  • we get some number say X>0.

    我們得到的數字就寫X>0。

  • Now nothing is happening, now let’s start moving

    現在什麼事情都還沒發生,開始轉動桌腳吧!

  • and we do that obviously in time and at each time, we measure the height of leg 1.

    我們一定要馬上做,而且每一次,我們都要量第一隻腳的長度。

  • And we turn it in this way.

    我們現在轉向這裡。

  • All we turn it so that we try to bring leg 1 to the position of leg 2.

    我們要轉桌腳,所以我們要把腳1轉到腳2的位置。

  • At each time, we measure the height of leg 1.

    每個時間我們都要測量腳1的長度。

  • So this gives the function f (t).

    所以這時候就有功能f(t)了。

  • For each time, T we measure.

    每一次我們都要算時間。

  • Here’s something important.

    這很重要。

  • It can happen that if we fix It all 2, 3 and 4,

    如果我們校正腳2、腳3、腳4,

  • this is all you remember, we fix 2, 3 and 4 and now it could happen that the height of leg 1 is negative.

    請記住,我們校正腳2、腳3、腳4可能會發現,腳1的長度成為負值。

  • Yeah because this will happen, if we now put leg 1 into position of leg 2,

    因為接下來我們會把腳1換到腳2的位置,

  • leg 2 to the position of leg 3, leg 3 to the position of leg 4

    腳2換到腳3,腳3換到腳4,

  • and leg 4 is at the position of leg 1.

    腳4換到腳1的位置。

  • But now, we remember that we fix the position of 2, 3 and 4.

    但現在,我們要記得已經校正過腳2、腳3和腳4的位置。

  • I fixed them on the ground.

    它們的長度都已經被修正到剛好可以靠在地上。

  • I keep them on the ground.

    它們都剛好靠在地上。

  • And since we did it here, at this position, this was above the ground,

    因為我們把它換到這裡,這個位置,所以它懸在半空中,

  • and now we force these three to be on the ground.

    現在,我們要把這三個腳都靠在地上。

  • That means this position has to be under the ground.

    那表示這個位置必須要在地平面下。

  • You see that before you fix these three, now you force these to go down,

    在你校正這三支腳的長度前,他們被迫往下,

  • and this is suddenly under the ground.

    突然,這隻腳就會埋入地底。

  • So it’s time 1, let’s suppose take time into 1 until at this position,

    所以現在是第一個時間,假設到這個位置才是第一個時間,

  • so it’s t=1, this height is negative.

    所以時間是1,這個長度不理想。

  • So now, we draw if I can get you a sheet of paper.

    現在,我來找張紙畫給你看。

  • Now we draw this curve so this is time = 0, this is time=1

    我們現在來畫曲線,這裡時間為0,這裡時間為1

  • Here we draw the height and at times zero, the height was something positive.

    我們把長度畫下來了,時間為0,長度可能是理想的。

  • And at time =1, the height was negative.

    時間為1,長度不理想。

  • So this is f (0), this is f (1).

    這是函數0,這是函數1

  • And now, at each time T, we get the position of the height and you see we get a curve

    每個時間,我們都能達到桌腳長度的位置,而且我們能到了一個曲線,

  • and it might go even up and down but in the very end, it has to end here.

    曲線可能會在高一點或低一點,但在最後一定會在這裡結束。

  • And now comes the famous theorem of Mathematics, the Intermediate Value Theorem

    而有名的數學定理─中間值定理就出現了,

  • which just says that if you have a continuous function which is positive here and negative here,

    這表示如果函數有連續功能,這裡是理想,這裡是不理想的,

  • they match the opposition here where it is 0.

    那他們就能在時間為0時達到正確的位置。

  • It could be multiple ~It could beyou can have fun with it, you turn your table further

    它可以是多樣化的,它可以,你會覺得它很有趣,你再繼續轉桌子,

  • and it might be in the 2nd position of this table.

    它可能會是這張桌子的第2個位置。

  • You don’t need that but it’s fun to try that out.

    你不需要第2個位置,但值得一試。

  • And if you are in the beer garden,

    而且如果你在露天啤酒店裡,

  • and if you do it the next time in the beer garden, you will easily fix the table.

    下次你到露天啤酒電石這樣試試,你會更容易修好桌腳高低不一致的問題。

  • And you will be pleased and you can taste it even better.

    你就會非常開心,也會腳的啤酒更好喝了。

  • I’ll do it all the time whenever I’m in the beer garden or even in a restaurant,

    每次我到露天啤酒店或是餐廳,我都會轉動桌子來測量。

  • often the ground is not flat, and I sit there with my friends

    通常地面都不平,所以我和我朋友坐在那裡,

  • and they are saying, “Ahh, let’s put this under.”

    他們都會說:「把這個店在桌腳下吧!」

  • I said, “Don’t do it!”

    我都會說:「不行!」

  • I move it just a little bit and they are obviously very surprised.

    我稍微移動桌子,他們就會非常驚訝。

  • And we do not change and it’s fix for the whole evening.

    其實我們並沒有做什麼事,一個晚上就修好了。

  • What if the tables are all lined-up or with special shape or something?

    那如果桌子排成一長列,或是其他特別的形狀呢?

  • Oh, that’s of courseMathematics is always theoretical so if you cannot move the table,

    當然可以,數學是一種理論,所以如果你不能移桌子,

Suppose when you sit in a beer garden

假設你坐在露天啤酒店裡,

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A2 初級 中文 英國腔 桌子 長度 位置 換到 平穩 啤酒

修理搖搖欲墜的桌子(有了數學) (Fix a Wobbly Table (with Math))

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    Go Tutor 發佈於 2021 年 01 月 14 日
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