How many ways are there to prove the Pythagorean theorem? - Betty Fei

譯者: Helen Chang 審譯者: S Sung

What do Euclid,

歐幾里得

twelve-year-old Einstein,

十二歲的愛因斯坦

and American President James Garfield have in common?

和美國詹姆斯 · 加菲爾德總統 有什麼共同點？

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

他們都為著名的畢達哥拉斯定理 想出優雅的證明

the rule that says for a right triangle,

定理說的是直角三角形

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

兩條直角邊的長度的平方加起來

is equal to the square of the hypotenuse.

等於斜邊長的平方

In other words, a²+b²=c².

換句話說，a² + b² = c²

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

這是幾何最重要的基本規則

and the basis for practical applications,

也是實際應用的基礎

like constructing stable buildings and triangulating GPS coordinates.

像是用於建造穩定的建築物 以及三角測量 GPS 坐標

The theorem is named for Pythagoras,

該定理被命名為畢達哥拉斯

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

公元前六世紀的希臘哲學家和數學家

but it was known more than a thousand years earlier.

但它在那之前一千年已為人知

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

公元前 1800 年左右的 巴比倫書寫板列出 15 組數字

that satisfy the theorem.

滿足這個定理

Some historians speculate that Ancient Egyptian surveyors

一些歷史學家推測古埃及的測量者

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

用了 3、4、5 這組數字做直角

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

理論是測量員拉一條 打了十二個等分結的繩索

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

來形成邊長分別是 3、4、5 的三角形

According to the converse of the Pythagorean theorem,

由畢達哥拉斯定理反推

that has to make a right triangle,

那必然是個直角三角形

and, therefore, a square corner.

因此有個直角

And the earliest known Indian mathematical texts

而已知最早的印度的數學教科書

written between 800 and 600 B.C.

寫在公元前 800 到 600 年之間

state that a rope stretched across the diagonal of a square

陳述橫跨正方形的對角線長度的繩索

produces a square twice as large as the original one.

所產生的正方形面積是原來的兩倍大

That relationship can be derived from the Pythagorean theorem.

這關係可以從畢達哥拉斯定理得出

But how do we know that the theorem is true

但是我們如何確認這定理

對於每個平面上的 直角三角形都成立

for every right triangle on a flat surface,

not just the ones these mathematicians and surveyors knew about?

不限於那些數學家、測量員 知道的三角形而已呢？

Because we can prove it.

因為我們可以證明這一點

Proofs use existing mathematical rules and logic

使用現有的數學規則和邏輯

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

來證明定理始終成立

One classic proof often attributed to Pythagoras himself

一個往往被歸功於 畢達哥拉斯本人的經典證明

uses a strategy called proof by rearrangement.

用的策略是重新排列

Take four identical right triangles with side lengths a and b

拿四個相同的直角三角形

邊長為 a、b 而斜邊長為 c

and hypotenuse length c.

Arrange them so that their hypotenuses form a tilted square.

把它們排列成斜邊形成一個正方形

The area of that square is c².

正方形的面積是 c²

Now rearrange the triangles into two rectangles,

把三角形重排形成兩個正方形

leaving smaller squares on either side.

兩邊各有一個較小的正方形

The areas of those squares are a² and b².

兩個正方形的面積各是 a² 和 b²

Here's the key.

關鍵在於

The total area of the figure didn't change,

圖形的總面積不變

and the areas of the triangles didn't change.

而且三角形的總面積也不變

So the empty space in one, c²

所以一個空間的面積 c²

must be equal to the empty space in the other,

必須等於另一個空間的面積

a² + b².

a² + b²

Another proof comes from a fellow Greek mathematician Euclid

來自希臘同胞 數學家歐幾里德的另一證明

and was also stumbled upon almost 2,000 years later

也在差不多兩千年後

by twelve-year-old Einstein.

被十二歲的愛因斯坦偶然發現了

This proof divides one right triangle into two others

這個證明把直角三角形分為兩個

and uses the principle that if the corresponding angles of two triangles

並運用這個原則

如果兩個三角形的對應角度相等

are the same,

the ratio of their sides is the same, too.

它們的邊長比例也必定相等

所以這三個相似三角形

So for these three similar triangles,

you can write these expressions for their sides.

可以用這樣的式子表示邊長的關係

Next, rearrange the terms.

接下來重新排列

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

最後，兩個式子相加並簡化成

ab²+ac²=bc²,

ab² + ac² = bc²

or a²+b²=c².

或 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.

重複幾何圖案以取得視覺證明

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.

如果你要一段時間想想，請暫停

Here's the answer.

答案是

The dark gray square is a²

深灰色的方塊是 a²

and the light gray one is b².

淺灰色的是 b²

The one outlined in blue is c².

藍色圍起來的是 c²

Each blue outlined square contains the pieces of exactly one dark

每個藍色輪廓的正方形 恰好包含一個深灰色方塊的碎片

and one light gray square,

和一個淺灰色正方形的碎片

proving the Pythagorean theorem again.

再次證明畢達哥拉斯定理

And if you'd really like to convince yourself,

如果你真要說服自己

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

可以做個轉盤，上面掛著 三個深度相等的方框框容器

connected to each other around a right triangle.

容器相互連接圍繞成直角三角形

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

如果你用水填滿最大的正方形

旋轉轉盤

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

大正方形的水將完美注滿兩個小的

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

畢達哥拉斯定理的證明超過 350 種

還在增加當中

ranging from brilliant to obscure.

精彩、晦澀的都有

Can you add your own to the mix?

你有自己的證明可以加入嗎？