Exponential growth: How folding paper can get you to the Moon

紙を折って月に行く方法

翻訳: Tomoshige Ohno 校正: Yasushi Aoki

(音楽)

１枚の紙は 何回 折ることができるでしょう？

How many times can you fold a piece of paper?

聖書に使われているような

Assume that one had a piece of paper that was very fine,

薄い紙が１枚あったとしましょう

like the kind they typically use to print the Bible.

実際 シルクのように見えます

In reality, it seems like a piece of silk.

計算を進めるため

To qualify these ideas,

let's say you have a paper

紙の厚さを1000分の1センチとしましょう

that's one-thousandth of a centimeter in thickness.

10のマイナス３乗センチ

That is 10 to the power of minus three centimeters,

which equals .001 centimeters.

つまり0.001cmです

またその紙は 新聞紙のように

Let's also assume that you have a big piece of paper,

十分に大きいものとしましょう

like a page out of the newspaper.

Now we begin to fold it in half.

これを半分に折っていくと

何回折れると思いますか？

How many times do you think it could be folded like that?

And another question:

もう１つの疑問は

If you could fold the paper over and over, as many times as you wish,

紙を好きなだけ何回でも 折っていけるとしたら

say 30 times,

例えば30回折ったときに その紙の厚さは どのくらいになるでしょう？

what would you imagine the thickness of the paper would be then?

先に進む前に

Before you move on,

I encourage you to actually think about a possible answer to this question.

答えを自分で考えてみるよう おすすめします

ではやってみましょう この紙を１回折ると

OK.

After we have folded the paper once,

厚さは1000分の2センチになります

it is now two thousandths of a centimeter in thickness.

もう１度半分に折ると

If we fold it in half once again,

the paper will become four thousandths of a centimeter.

1000分の4センチになります

With every fold we make, the paper doubles in thickness.

１回折るごとに 紙の厚さは２倍になります

繰り返し半分に 折り続けていくと

And if we continue to fold it again and again,

10回折った後

always in half, we would confront the following situation

after 10 folds.

どうなっているかというと

Two to the power of 10,

厚さは２の10乗

meaning that you multiply two by itself 10 times,

つまり２を10回 掛け合わせたものですが

1000分の1024センチになります

is one thousand and 24 thousandths of a centimeter,

which is a little bit over one centimeter.

1センチ強です

さらに紙を半分に 折り続けたとしたら

Assume we continue folding the paper in half.

どうなるでしょう？

What will happen then?

If we fold it 17 times,

17回折ったところで

we'll get a thickness of two to the power of 17,

厚さは２の17乗

which is 131 centimeters,

つまり131cmになり

and that equals just over four feet.

４フィートを越えます

If we were able to fold it 25 times,

25回折ることができたとしたら

then we would get two to the power of 25,

２の25乗倍の厚さ

which is 33,554 centimeters,

つまり33,554cmとなり

1,100フィートを 越えてしまいます

just over 1,100 feet.

エンパイアステートビルと ほぼ同じ高さです

That would make it almost as tall as the Empire State Building.

ちょっと立ち止まって 振り返ってみましょう

It's worthwhile to stop here and reflect for a moment.

Folding a paper in half, even a paper as fine as that of the Bible,

聖書の紙と同じくらい 薄い紙でも

25 times would give us a paper almost a quarter of a mile.

25回折ると 厚さが 約 3分の1 kmにもなります

What do we learn?

何が分かるでしょう？

This type of growth is called exponential growth,

このタイプの増え方を 指数的増加といい

and as you see, just by folding a paper

ご覧の通り 紙を折っていくだけで

we can go very far, but very fast too.

とても早くとても遠くまで行くのです

まとめです 紙を25回折ると

Summarizing, if we fold a paper 25 times,

厚さは約 3分の1 kmになり

the thickness is almost a quarter of a mile.

30回で厚さは10 kmに及び

30 times, the thickness reaches 6.5 miles,

飛行機の平均高度と 同じくらいになります

which is about the average height that planes fly.

40 times, the thickness is nearly 7,000 miles,

40回で厚さは ほぼ 11,000 km

or the average GPS satellite's orbit.

GPS衛星軌道までの 平均距離になり

48 times, the thickness is way over one million miles.

48回だと厚さは 何百万kmにもなります"

地球と月の間の距離は

Now, if you think that the distance between the Earth and the Moon

38万km程度なので

is less than 250,000 miles,

聖書の紙１枚から始めて

then starting with a piece of Bible paper

and folding it 45 times, we get to the Moon.

45回折ることで 月まで届きます

さらにもう１回折ると

And if we double it one more time,

地球に戻ってきます

we get back to Earth.

先生: アドリアン・パエンザ ナレーション: アドリアン・パエンザ アニメーション: TED-Edチーム