Name: 3秒算出任何數字的平方根！超狂數學密技 (Square Root of Any Number in 3 seconds - Crazy Math Trick)
Uploaded: 2026-01-06T02:53:29.000Z
Duration: 5 min 44 s

動名詞

What if I ask you to quickly tell me in five seconds what will be the value of square root of 81 without using a calculator?

如果我請你在五秒內，不用計算機，快速告訴我 81 的平方根是多少？

未來簡單式

You will laugh at me and say it doesn't require a calculator or even five seconds.

你可能會笑我，說這根本不需要計算機，甚至不用五秒。

情態助動詞 A2

But now what if I ask you to find the square root of 87 without using a calculator and within 5 seconds you will be like, wait a second, it's not a perfect square.

但現在，如果我請你在五秒內，不用計算機，算出 87 的平方根，你可能會說，等一下，它不是完全平方數。

And that's where things get tricky, right?

這時候事情就變得很棘手了，對吧？

Today in this video I am going to show you an awesome trick to estimate square roots of even a number that is not a perfect square at a lightning fast speed.

今天這支影片，我要教大家一個超讚的技巧，讓你閃電般的速度估計出，即使不是完全平方數的數字的平方根。

However, at the end of this video I will show you that at some places you can.

不過，影片最後我會告訴你，在某些地方是可以的。

You have to be careful while applying this trick.

使用這個技巧時，你必須小心。

Let us start with square root of 87 itself.

我們就從 87 的平方根本身開始吧。

As a first step, you need to write the squares of all numbers from 1 to 10 like this.

作為第一步，你需要像這樣寫出 1 到 10 所有數字的平方。

Now just find a number whose square is just less than this value, which is 9 in this case.

現在找出一個數字，它的平方正好小於這個值，這個例子中是 9。

So we know that our answer will be nine plus something and we need to find this something which will be a fraction.

所以我們知道答案會是九點多，而我們需要找出這個「多」的部分，它會是一個分數。

要做到這一點，就取這個數字 87。

Then subtract it with this squared number 81 to get 6 and write it in the numerator.

然後用這個平方數 81 去減它，得到 6，並將它寫在分子。

Then take this number nine and double it to get 18 and put it in the denominator.

然後取這個數字 9，將它乘以 2 得到 18，並將它放在分母。

This will become 1 over 3, which is roughly 0.33.

這會變成 1 除以 3，大約是 0.33。

Actual value of the square root of 87 is 9.327, which is super duper close to our approximation.

87 的平方根實際值是 9.327，跟我們估計的非常非常接近。

First let us find a number whose square is just less than this value, which is 6 in this case.

首先，找一個數字，它的平方剛好小於這個值，這裡就是 6。

So we know that our answer will be 6 plus a fraction.

所以我們知道答案會是 6 加上一個分數。

Then take this number 39 and subtract it with this squared number 36 to get 3 and write it in the numerator.

然後用 39 減掉這個平方數 36，得到 3，並將它寫在分子。

Then take this number six and double it to get 12 and put it in the denominator.

然後取這個數字 6，將它乘以 2 得到 12，並將它放在分母。

This will become 1 over 4, which is roughly 0.25.

這會變成 1 除以 4，大約是 0.25。

Actual value of the square root of 39 is 6.245, which is super duper close to our approximation.

39 的平方根實際值是 6.245，跟我們的近似值非常非常接近。

Now find square root of 6964 is just less than 69 and thus it will be 8 plus a fraction which will be 69 minus 64 or 5 over 8 times 2 or 16 which is 8.3125 and the actual value is 8.307 which is very very close and super fast.

現在找 6964 的平方根，比 69 小一點，所以會是 8 加上一個分數，這個分數是 69 減 64 或 5 除以 8 乘以 2 或 16，也就是 8.3125，實際值是 8.307，非常非常接近，而且速度超快。

It will be 5 plus 27 minus 25 or 2 over 5 times 2 or 10 which is 5.2 and the actual value is 5.196.

會是 5 加上 27 減 25 或 2 除以 5 乘以 2 或 10，也就是 5.2，實際值是 5.196。

Now let me show you where you have to be careful in order to use this trick.

現在我來告訴你，要怎麼運用這個技巧才不會出錯。

Let us try to find the square root of 2 using this trick we get 1 plus 2 minus 1 or 1 over 1 times 2 or 1.5 which is definitely not very close to our answer as the actual value is 1.414.

我們來試試看用這個技巧找 2 的平方根，得到 1 加上 2 減 1 或 1 除以 1 乘以 2 或 1.5，這肯定跟我們的答案差很多，因為實際值是 1.414。

So what to do now we will repeat it with this estimate of 1.5 its square is 2.25.

那我們現在該怎麼辦呢，我們就用這個 1.5 的估計值來重複一次，它的平方是 2.25。

Write 1.5 plus 2 minus 2.25 will be minus 0.25 over 2 times 1.5 or 3.25 is also equal to 1 over 4 and this will be 1.5 minus 1 over 12 or 1.417 which is a more accurate approximation of the square root of 2.

寫下 1.5 加上 2 減掉 2.25 會是負 0.25 除以 2 乘以 1.5 或 3.25，也等於 1 除以 4，這會是 1.5 減掉 1 除以 12 或 1.417，這是一個更精確的 2 的平方根近似值。

But you know what, if you want a more accurate answer then keep on repeating this until you are satisfied.

但你知道嗎，如果你想要更精確的答案，就一直重複這個步驟直到你滿意為止。

I will definitely not take credit for finding this square root method instead this method was developed by the ancient Babylonians, one of the earliest known civilizations who lived in Mesopotamia or the modern day Iraq.

我絕對不會把這個平方根方法的功勞攬在自己身上，這個方法其實是古代巴比倫人發展出來的，他們是居住在美索不達米亞，也就是現代伊拉克，最早的已知文明之一。

It was later rediscovered and generalized by Sir Isaac Newton in the 17th century as part of his Newton Raphson method which is an iterative approach to finding roots of a function F of x using this formula which involves calculus and it will be out of scope of this video to dive into its mathematical derivation.

後來在 17 世紀，它被艾薩克·牛頓爵士重新發現並推廣，成為他牛頓-拉弗森方法的一部分，這是一種尋找函數 F(x) 的根的迭代方法，使用這個公式，其中涉及微積分，而深入探討它的數學推導將超出本影片的範圍。

Now if this video gets 20,000 likes then I will make another banger trick video like this one.

現在，如果這部影片能獲得 2 萬個讚，我將會製作另一部像這樣超讚的技巧影片。

If you enjoy my videos and want to support my channel, consider becoming a Patreon as it helps me create more awesome content for you.

如果你喜歡我的影片並想支持我的頻道，可以考慮成為 Patreon，這能幫助我為你創作更多很棒的內容。

Also you can support my channel by joining our community and becoming a member.

你也可以透過加入我們的社群並成為會員來支持我的頻道。

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3秒算出任何數字的平方根！超狂數學密技 (Square Root of Any Number in 3 seconds - Crazy Math Trick)

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