twelve-year-old Einstein,

12 歲的愛因斯坦

and American President James Garfield have in common?

還有美國總統詹姆斯．加菲爾德的共通點是什麼？

They all came up with elegant proofs for the famous Pythagorean theorem,

他們都為畢氏定理想出巧妙的證明

the rule that says for a right triangle,

畢氏定理指的是一個直角三角形

the square of one side plus the square of the other side

一邊長的平方加上另一邊長的平方

is equal to the square of the hypotenuse.

等於斜邊長的平方

In other words, a²+b²=c².

也就是說，a²+b²=c²

This statement is one of the most fundamental rules of geometry,

這個闡述是幾何中最重要的基本規則之一

and the basis for practical applications,

也是實際應用的基礎

like constructing stable buildings and triangulating GPS coordinates.

像是建造堅固的建物，還有三角測量 GPS 座標

The theorem is named for Pythagoras,

這個定理是以畢達哥拉斯的名字來命名

a Greek philosopher and mathematician in the 6th century B.C.,

一位公元前 6 世紀的希臘哲學家兼數學家

but it was known more than a thousand years earlier.

但畢氏定理早在一千多年前就已為人所知

A Babylonian tablet from around 1800 B.C. lists 15 sets of numbers

公元前 1800 年左右，一個巴比倫泥板列出 15 組數字

that satisfy the theorem.

符合畢氏定理

Some historians speculate that Ancient Egyptian surveyors

一些歷史學家推測古埃及測量員

used one such set of numbers, 3, 4, 5, to make square corners.

使用 3、4、5 這組數字來製作直角

The theory is that surveyors could stretch a knotted rope with twelve equal segments

理論是，測量員拉開一條打了 12 等分繩結的繩索

to form a triangle with sides of length 3, 4 and 5.

來形成一個邊長 3、4、5 的三角形

According to the converse of the Pythagorean theorem,

根據畢氏定理來反推

that has to make a right triangle,

這必定是製作成直角三角形

and, therefore, a square corner.

因此有個直角

And the earliest known Indian mathematical texts

而已知最早的印度數學教科書

written between 800 and 600 B.C.

約在公元前 800 到 600 年之間出版

state that a rope stretched across the diagonal of a square

說明一條橫跨正方形對角的繩索

produces a square twice as large as the original one.

可以產生一個比原本的正方形大兩倍的正方形

That relationship can be derived from the Pythagorean theorem.

這樣的關係可以從畢氏定理推導出來

But how do we know that the theorem is true

但我們如何知道這個定理是正確的

for every right triangle on a flat surface,

在平面上所有的直角三角形都成立

not just the ones these mathematicians and surveyors knew about?

而不只限於那些數學家跟測量員知道的三角形呢？

Because we can prove it.

因為我們可以證明它

Proofs use existing mathematical rules and logic

證明過程要使用現有的數學規則與邏輯

to demonstrate that a theorem must hold true all the time.

來論證一個定理必須始終成立

One classic proof often attributed to Pythagoras himself

一個經常被認為是畢達哥拉斯自己寫的經典證明

uses a strategy called proof by rearrangement.

使用的策略是，重新排列來證明

Take four identical right triangles with side lengths a and b

拿四個完全相同的直角三角形，邊長為 a 跟 b

and hypotenuse length c.

還有斜邊長 c

Arrange them so that their hypotenuses form a tilted square.

將它們排列，使它們的斜邊形成一個傾斜的正方形

The area of that square is c².

這個正方形的面積就是 c²

Now rearrange the triangles into two rectangles,

現在重新排列這些三角形，變成兩個長方形

leaving smaller squares on either side.

在其餘兩邊留下比較小的正方形

The areas of those squares are a² and b².

這兩個正方形的面積分別是 a² 跟 b²

Here's the key.

重點來了

The total area of the figure didn't change,

這個圖形的總面積沒有改變

and the areas of the triangles didn't change.

三角形的總面積也沒有改變

So the empty space in one, c²

所以左邊正方形的空白面積 c²

must be equal to the empty space in the other,

一定等於另一邊的空白面積

a² + b².

a² + b²

Another proof comes from a fellow Greek mathematician Euclid

另一個證明來自希臘同胞，數學家歐幾里得

and was also stumbled upon almost 2,000 years later

也在大約 2,000 年後

by twelve-year-old Einstein.

被 12 歲的愛因斯坦偶然發現

This proof divides one right triangle into two others

這個證明是將一個直角三角形分割成兩個直角三角形

and uses the principle that if the corresponding angles of two triangles are the same,

並應用這個原則，如果這兩個三角形的對應角度相同

the ratio of their sides is the same, too.

它們邊長的比例也會相同

So for these three similar triangles,

所以這三個相似三角形

you can write these expressions for their sides.

可以寫出這樣的式子來表示它們的邊長關係

Next, rearrange the terms.

下一步，重新排列算式的形式

And finally, add the two equations together and simplify to get

最後，將兩個算式加起來，簡化後會得到

ab²+ac²=bc²,

ab²+ac²=bc²

or a²+b²=c².

或是 a²+b²=c²

Here's one that uses tessellation,

這個證明用到平面填充

a repeating geometric pattern for a more visual proof.

重複幾何圖形以得到視覺證明

Can you see how it works?

你能看出它如何證明嗎？

Pause the video if you'd like some time to think about it.

如果你需要時間想想，按下暫停吧

Here's the answer.

要公布答案了

The dark gray square is a²

深灰色的正方形是 a²

and the light gray one is b².

淺灰色的是 b²

The one outlined in blue is c².

藍色框線的是 c²

Each blue outlined square contains the pieces of exactly one dark

每一個藍色框線的正方形裡剛好包含一個深灰色正方形的碎片

and one light gray square,

跟一個淺灰色正方形的碎片

proving the Pythagorean theorem again.

再次證明畢氏定理

And if you'd really like to convince yourself,

如果你真的想說服你自己

you could build a turntable with three square boxes of equal depth

你可以做一個轉盤，還有三個相同深度的正方形容器

connected to each other around a right triangle.

容器相互連結，形成一個直角三角形

If you fill the largest square with water and spin the turntable,

如果你把最大的正方形容器注滿水，然後轉動轉盤

the water from the large square will perfectly fill the two smaller ones.

最大的正方形容器裡的水會剛好填滿另外兩個小正方形的容器

The Pythagorean theorem has more than 350 proofs, and counting,

畢氏定理的證明方法有超過 350 種，還在持續增加

ranging from brilliant to obscure.

從精采絕倫到晦澀難懂的都有

Can you add your own to the mix?

你有辦法想出一種證明加入它們嗎？

Did you enjoy this lesson?

喜歡這個課程嗎？

If so, please consider supporting our non-profit mission by visiting PATREON.COM/TEDED

喜歡的話，請前往 PATREON.COM/TEDED 支持我們的非營利使命