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

Ako vás prekladanie papiera môže dostať na Mesiac

Translator: Dana Retová Reviewer: Ivana Kopisova

(hudba)

Koľkokrát sa dá zložiť kus papiera?

How many times can you fold a piece of paper?

Predpokladajme, že máme kus veľmi tenkého papiera,

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

takého, na aký zvykli tlačiť Bibliu.

like the kind they typically use to print the Bible.

In reality, it seems like a piece of silk.

V skutočnosti vyzerá ako kus hodvábu.

To qualify these ideas,

Aby sme podložili túto predstavu,

let's say you have a paper

povedzme, že máme papier, ktorý má hrúbku jedna tisícina centimetra.

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

To je 10 na mínus tretiu centimetrov,

That is 10 to the power of minus three centimeters,

which equals .001 centimeters.

čo sa rovná 0,001 centimetra.

Predpokladajme tiež, že je to veľký papier,

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

asi ako novinová stránka.

like a page out of the newspaper.

Now we begin to fold it in half.

Teraz ho začneme prekladať na polovicu.

Čo myslíte, koľkokrát sa dá takto preložiť?

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

And another question:

A druhá otázka:

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

Ak by ste mohli preložiť papier znova a znova, koľkokrát chcete,

say 30 times,

povedzme 30-krát, akej hrúbky by asi bol potom?

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

Predtým, než pôjdeme ďalej,

Before you move on,

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

naozaj sa zamyslite nad možnou odpoveďou na túto otázku.

Ok, po tom, čo sme preložili papier raz,

OK.

After we have folded the paper once,

má teraz hrúbku dve tisíciny centimetra.

it is now two thousandths of a centimeter in thickness.

Ak ho preložíme znova napoly,

If we fold it in half once again,

the paper will become four thousandths of a centimeter.

papier bude mať štyri tisíciny centimetra.

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

Každým preložením sa hrúbka papiera zdvojnásobí.

A keď budeme takto pokračovať v prekladaní napoly,

And if we continue to fold it again and again,

always in half, we would confront the following situation

dostaneme sa do nasledovnej situácie

after 10 folds.

po 10 preloženiach.

Two to the power of 10,

Dva na desiatu,

meaning that you multiply two by itself 10 times,

to znamená, že dvojku vynásobíte ďalšími 10 dvojkami,

je tisíc a 24 tisícin centimetra,

is one thousand and 24 thousandths of a centimeter,

which is a little bit over one centimeter.

čo je trochu viac ako jeden centimeter.

Assume we continue folding the paper in half.

Predpokladajme, že budeme pokračovať v prekladaní papiera napoly.

What will happen then?

Čo sa stane?

If we fold it 17 times,

Ak ho preložíme 17-krát,

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

dosiahne hrúbku dva na sedemnástu,

which is 131 centimeters,

čo je 131 centimetrov,

and that equals just over four feet.

čiže vyše metra.

If we were able to fold it 25 times,

Ak by sme ho preložili 25-krát,

then we would get two to the power of 25,

dostali by sme dva na dvadsiatupiatu,

which is 33,554 centimeters,

čo je 33 554 centimetrov,

just over 1,100 feet.

čiže viac ako 330 metrov.

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

To je skoro tak vysoké ako Empire State Building.

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

Teraz sa na chvíľu zastavme a porozmýšľajme.

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

Preložením papiera napoly, i keď je tenký ako ten z Biblie,

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

25-krát, získame papier hrubý skoro tretinu kilometra.

What do we learn?

Čo sme zistili?

This type of growth is called exponential growth,

Tento typ rastu sa volá exponenciálny rast,

and as you see, just by folding a paper

a ako vidíte, iba prekladaním papiera

we can go very far, but very fast too.

sa dostaneme veľmi ďaleko, a zároveň veľmi rýchlo.

Aby sme to zhrnuli, ak preložíme papier

Summarizing, if we fold a paper 25 times,

25-krát, hrúbka bude skoro tretina kilometra.

the thickness is almost a quarter of a mile.

30-krát a hrúbka dosiahne viac než 10 km,

30 times, the thickness reaches 6.5 miles,

čo je zhruba výška, v ktorej lietajú lietadlá.

which is about the average height that planes fly.

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

40-krát a hrúbka je viac ako 11 000 km,

or the average GPS satellite's orbit.

alebo priemerná obežná dráha GPS satelitov.

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

48-krát a hrúbka je oveľa väčšia ako 1,5 milióna kilometrov.

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

Ak si teraz predstavíte, že vzdialenosť medzi Zemou a Mesiacom

is less than 250,000 miles,

je menej ako 400 000 km,

then starting with a piece of Bible paper

stačí začať s kusom papiera z Biblie

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

a preložiť ho 45-krát a dostaneme sa na Mesiac.

A ak to ešte raz znásobíme,

And if we double it one more time,

we get back to Earth.

dostaneme sa naspäť na Zem.

Lekcia od: Adrian Paenza Nahovoril: Adrian Paenza Animácia: tím TED-ED