Making sense of irrational numbers - Ganesh Pai

Davanje smisla iracionalnim brojevima - Ganeš Pai

Prevodilac: Mile Živković Lektor: Milenka Okuka

Like many heroes of Greek myths,

Kao mnogi heroji grčkih mitova,

the philosopher Hippasus was rumored to have been mortally punished by the gods.

za filozofa Hipasusa pričalo se da su ga bogovi kaznili na smrt.

But what was his crime?

Ali šta je bio njegov zločin?

Did he murder guests,

Da li je ubio goste,

or disrupt a sacred ritual?

ili prekinuo sveti ritual?

No, Hippasus's transgression was a mathematical proof:

Ne, Hipasusov prestup je bila matematička teorema:

the discovery of irrational numbers.

otkriće iracionalnih brojeva.

Hippasus belonged to a group called the Pythagorean mathematicians

Hipasus je pripadao grupi zvanoj Pitagorini matematičari

who had a religious reverence for numbers.

koja je imala religijsko poštovanje prema brojevima.

Their dictum of, "All is number,"

Izreka "Sve je u brojevima"

suggested that numbers were the building blocks of the Universe

navodi da su brojevi gradivni materijali univerzuma

and part of this belief was that everything from cosmology and metaphysics

i deo ovog uverenja je da sve od kosmologije i metafizike

to music and morals followed eternal rules

do muzike i moralnih načela prate večna pravila

describable as ratios of numbers.

poznata kao proporcija brojeva.

Thus, any number could be written as such a ratio.

Stoga, bilo koji broj može biti napisan u takvoj proporciji

5 as 5/1,

5 kao 5/1

0.5 as 1/2

0,5 kao 1/2

and so on.

i tako dalje.

Even an infinitely extending decimal like this could be expressed exactly as 34/45.

Čak i beskonačne decimale kao ova mogu biti iskazane kao 34/45

All of these are what we now call rational numbers.

Svi ovi brojevi su oni koje mi zovemo racionalnim brojevima.

But Hippasus found one number that violated this harmonious rule,

Hipasus je pronašao jedan broj koji je ugrozio harmoniju,

one that was not supposed to exist.

jedan koji nije trebalo da postoji.

The problem began with a simple shape,

Problem je počeo jednostavnim oblikom,

a square with each side measuring one unit.

kvadrat koji je sa svake strane merio jednu jedinicu

According to Pythagoras Theorem,

Po Pitagorinoj teoremi,

the diagonal length would be square root of two,

dužina dijagonale je jednaka kvadratnom korenu broja dva,

but try as he might, Hippasus could not express this as a ratio of two integers.

ali koliko god pokušavao, Hipasus nije mogao da izrazi ovo

kao proporciju dva cela broja.

And instead of giving up, he decided to prove it couldn't be done.

I umesto odustajanja, odlučio je dokazati da je to neizvodivo.

Hippasus began by assuming that the Pythagorean worldview was true,

Hipasus je počeo pretpostavljajući da je Pitagorejski princip tačan,

that root 2 could be expressed as a ratio of two integers.

da koren broja dva može biti izražen kao odnos dva cela broja.

He labeled these hypothetical integers p and q.

Označio je ova dva hipotetička broja kao p i q.

Assuming the ratio was reduced to its simplest form,

Pretpostavljajući da je odnos sveden na osnovnu formu,

p and q could not have any common factors.

p i q nisu mogli da imaju nijedan zajednički delilac.

To prove that root 2 was not rational,

Da bi dokazao da koren broja dva nije racionalan,

Hippasus just had to prove that p/q cannot exist.

Hipasus je morao da dokaže da p/q ne može postojati.

So he multiplied both sides of the equation by q

Obe strane je pomnožio sa q

and squared both sides.

i stavio na kvadrat,

which gave him this equation.

čime je dobio ovu jednačinu.

Multiplying any number by 2 results in an even number,

Množenje bilo kog broja brojem 2 daje paran broj

so p^2 had to be even.

tako da p^2 mora da bude parno.

That couldn't be true if p was odd

To ne bi moglo biti istinito ako bi p bilo neparno

because an odd number times itself is always odd,

jer neparan broj pomnožen sobom uvek daje drugi neparan broj,

so p was even as well.

tako da je i p bilo parno.

Thus, p could be expressed as 2a, where a is an integer.

Stoga se p moglo izraziti kao 2a, gde je a delilac.

Substituting this into the equation and simplifying

Kada se ovo ubaci u jednačinu i pojednostavi

gave q^2 = 2a^2

dobija se q^2 = 2a^2.

Once again, two times any number produces an even number,

Još jednom, bilo koji broj pomnožen sa dva daje paran broj,

so q^2 must have been even,

tako da q^2 mora biti parno,

and q must have been even as well,

a q mora da bude parno,

making both p and q even.

tako da su i p i q parni.

But if that was true, then they had a common factor of two,

Ali kada bi to bilo tačno, oboje bi imali zajednički faktor dva,

which contradicted the initial statement,

što bi se kosilo sa prvobitnom izjavom

and that's how Hippasus concluded that no such ratio exists.

i tako je Hipasus zaključio da takav odnos ne postoji.

That's called a proof by contradiction,

To se zove dokaz kontradikcijom,

and according to the legend,

i kako navodi legenda,

the gods did not appreciate being contradicted.

bogovima se nije dopalo da im se protivreči.

Interestingly, even though we can't express irrational numbers

Zanimljivo je to da iako ne možemo da izrazimo iracionalne brojeve

as ratios of integers,

kao odnose delilaca,

it is possible to precisely plot some of them on the number line.

moguće je da se neki od njih precizno prikažu na grafikonu.

Take root 2.

Uzmite koren iz dva.

All we need to do is form a right triangle with two sides each measuring one unit.

Treba da napravimo trougao gde dve strane daju jednu jedinicu.

The hypotenuse has a length of root 2, which can be extended along the line.

Hipotenuza ima dužinu korena iz 2, koji se može produžiti.

We can then form another right triangle

Onda možemo da napravimo još jedan trougao

with a base of that length and a one unit height,

sa osnovom te dužine i visinom jedne jedinice

and its hypotenuse would equal root three,

i njegova hipotenuza bila bi jednaka korenu od tri,

which can be extended along the line, as well.

koji se takođe može produžiti duž linije.

The key here is that decimals and ratios are only ways to express numbers.

Ovde je ključ da su decimale i odnosi jedini način izražavanja brojeva.

Root 2 simply is the hypotenuse of a right triangle

Koren i dva je prosto hipotenuza pravog trougla

with sides of a length one.

sa stranicama dužine jedan.

Similarly, the famous irrational number pi

Slično tome, čuveni iracionalni broj pi

is always equal to exactly what it represents,

uvek je jednak tačno onome što predstavlja,

the ratio of a circle's circumference to its diameter.

odnosu prečnika i obima kruga.

Approximations like 22/7,

Približne vrednosti poput 22/7

or 355/113 will never precisely equal pi.

ili 355/113 nikada neće biti precizno jednake broju pi.

We'll never know what really happened to Hippasus,

Nikada nećemo znati šta se desilo Hipasusu,

but what we do know is that his discovery revolutionized mathematics.

ali znamo da je njegovo otkriće napravilo revoluciju u matematici.

So whatever the myths may say, don't be afraid to explore the impossible.

Šta god da mitovi kažu, ne bojte se da istražite nemoguće.