Making sense of irrational numbers - Ganesh Pai

Prevoditelj: Tamara Rabuzin Recezent: Ivan Stamenković

Like many heroes of Greek myths,

Kao i mnoge heroje grčke mitologije,

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

priča se da su i filozofa Hipasusa bogovi kaznili smrću.

But what was his crime?

Ali koji je bio njegov zločin?

Did he murder guests,

Je li usmrtio goste

or disrupt a sacred ritual?

ili pak oskvrnuo sveti ritual?

No, Hippasus's transgression was a mathematical proof:

Ne, Hipasusov prijestup bio je matematički dokaz:

the discovery of irrational numbers.

otkriće iracionalnih brojeva.

Hippasus belonged to a group called the Pythagorean mathematicians

Hipasus je pripadao skupini zvanoj Pitagorejska škola

who had a religious reverence for numbers.

čiji članovi su prema brojevima imali vjersko štovanje.

Their dictum of, "All is number,"

Izreka "U biti svega je broj",

suggested that numbers were the building blocks of the Universe

sugerira da su brojevi građevne jedinice Svemira

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

a vjeruje se da sve, od kozmologije i metafizike

to music and morals followed eternal rules

do glazbe i morala, slijedi vječna pravila

describable as ratios of numbers.

koja se mogu prikazati pomoću omjera dva broja.

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

Dakle, svaki broj može se prikazati pomoću razlomka.

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 decimalni broj s beskonačnim zapisom poput ovog može se zapisati kao 34/45

All of these are what we now call rational numbers.

Takve brojeve zovemo racionalni brojevi.

But Hippasus found one number that violated this harmonious rule,

Ali Hipasus je pronašao jedan broj koji narušava ovaj sklad,

one that was not supposed to exist.

jedan koji ne bi trebao postojati.

The problem began with a simple shape,

Problem je započeo s jednostavnim oblikom,

a square with each side measuring one unit.

kvadratom s duljinom stranice 1.

According to Pythagoras Theorem,

Prema Pitagorinom teoremu,

the diagonal length would be square root of two,

duljina dijagonale je kvadratni korijen iz 2,

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

ali koliko god pokušavao, Hipasus ga nije mogao izraziti kao omjer dvaju brojeva.

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

Umjesto da odustane, odlučio je dokazati da to nije ni moguće učiniti.

Hippasus began by assuming that the Pythagorean worldview was true,

Pretpostavio je da je Pitagorejsko vjerovanje točno,

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

i da se korijen iz 2 može prikazati kao omjer dvaju brojeva.

He labeled these hypothetical integers p and q.

Označio je ta dva broja s p i q.

Assuming the ratio was reduced to its simplest form,

Pod pretpostavkom da je razlomak skraćen do kraja,

p and q could not have any common factors.

p i q ne mogu imati zajedničkih djelitelja.

To prove that root 2 was not rational,

Da bi dokazao da korijen iz 2 nije racionalan

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

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

So he multiplied both sides of the equation by q

Pomnožio je obje strane jednakosti s q

and squared both sides.

i kvadrirao obje strane,

which gave him this equation.

čime je dobio ovu jednadžbu.

Multiplying any number by 2 results in an even number,

Množenje obiju strana s 2 daje parni broj,

so p^2 had to be even.

pa p^2 mora biti paran.

That couldn't be true if p was odd

To nije moguće ako je p neparan

because an odd number times itself is always odd,

jer je neparan broj puta taj broj uvijek neparan,

so p was even as well.

pa je p također paran.

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

Dakle, p se može prikazati kao 2a, gdje je a neki broj.

Substituting this into the equation and simplifying

Uvrštavanjem ovog u jednadžbu i pojednostavljivanjem

gave q^2 = 2a^2

dobije se q^2=2a^2

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

Ponovno, dva puta neki broj daje paran broj,

so q^2 must have been even,

pa q^2 mora biti paran,

and q must have been even as well,

a q je također paran,

making both p and q even.

pa su onda i p i q parni.

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

Ali ako je to točno, onda oni imaju zajednički djelitelj 2,

which contradicted the initial statement,

što je u kontradikciji s pretpostavkom,

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

te je tako Hipasus zaključio da takav omjer ne može postojati.

That's called a proof by contradiction,

To se zove dokaz kontradikcijom,

and according to the legend,

i prema legendi,

the gods did not appreciate being contradicted.

bogovima se nije svidjelo da im se proturječi.

Interestingly, even though we can't express irrational numbers

Zanimljivo, iako ne možemo izraziti iracionalne brojeve

as ratios of integers,

pomoću razlomaka,

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

moguće je prikazati ih na brojevnom pravcu.

Take root 2.

Uzmimo korijen iz 2.

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

Sve što trebamo je nacrtati pravokutni trokut s katetama duljine 1.

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

Duljina hipotenuze je korijen iz 2, i može se nanijeti na brojevni pravac.

We can then form another right triangle

Zatim možemo nacrtati još jedan pravokutni trokut

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

s jednom katetom te duljine i jednom duljine 1,

and its hypotenuse would equal root three,

a hipotenuza će iznositi korijen iz 3,

which can be extended along the line, as well.

što se također može nanijeti na pravac.

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

Ključ je u tome da se može zapisati jedino u obliku razlomka ili decimalnog broja.

Root 2 simply is the hypotenuse of a right triangle

Korijen iz 2 jednostavno je duljina hipotenuze pravokutnog trokuta

with sides of a length one.

s katetama duljine 1.

Similarly, the famous irrational number pi

Slično, poznati iracionalni broj pi

is always equal to exactly what it represents,

uvijek je jednak točno onome što predstavlja,

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

a to je omjer opsega kruga i njegovog promjera.

Approximations like 22/7,

Aproksimacije poput 22/7

or 355/113 will never precisely equal pi.

ili 355/113 nikad neće biti jednake točno pi.

We'll never know what really happened to Hippasus,

Nikad nećemo saznati što se točno dogodilo Hipasusu,

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

ali znamo da je njegovo otkriće bilo revolucionarno za matematiku.

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

Dakle, bez obzira na mitove, nemojte se bojati istraživati nemoguće.