The mathematical secrets of Pascal’s triangle - Wajdi Mohamed Ratemi

Matematičke tajne Pascalovog trokuta - Wajdi Mohamed Ratemi

Prevoditelj: Tamara Rabuzin Recezent: Ivan Stamenković

This may look like a neatly arranged stack of numbers,

Ovo možda izgleda samo kao uredno složen stog brojeva,

but it's actually a mathematical treasure trove.

ali zapravo je matematičko skriveno blago.

Indian mathematicians called it the Staircase of Mount Meru.

Indijski matematičari zvali su ga Stepenice planine Meru.

In Iran, it's the Khayyam Triangle.

U Iranu, to je Khayyamov trokut,

And in China, it's Yang Hui's Triangle.

a u Kini, Yang Huijev trokut.

To much of the Western world, it's known as Pascal's Triangle

Većini zapadnog svijeta poznat je kao Pascalov trokut

after French mathematician Blaise Pascal,

po francukom matematičaru Blaiseu Pascalu,

which seems a bit unfair since he was clearly late to the party,

što je ipak malo nepravedno, jer očito nije izmislio ništa novo,

but he still had a lot to contribute.

ali ipak je i on dao svoj doprinos.

So what is it about this that has so intrigued mathematicians the world over?

Ali zašto je toliko zaokupljao matematičare diljem svijeta?

In short, it's full of patterns and secrets.

Ukratko, pun je obrazaca i tajni.

First and foremost, there's the pattern that generates it.

Prvo i najvažnije, uzorak koji ga generira.

Start with one and imagine invisible zeros on either side of it.

Počnite s jedinicom i zamislite nevidljive nule s obje strane jedinice.

Add them together in pairs, and you'll generate the next row.

Zbrojite po dva broja, i generirat ćete slijedeći red.

Now, do that again and again.

Sada ponavljajte postupak.

Keep going and you'll wind up with something like this,

Nastavite i dobit ćete nešto poput ovog,

though really Pascal's Triangle goes on infinitely.

iako se Pascalov trokut nastavlja u beskonačnost.

Now, each row corresponds to what's called the coefficients of a binomial expansion

Svaki red odgovara nečemu naziva binomni koeficijenti

of the form (x+y)^n,

raspisa (x+y)^n,

where n is the number of the row,

gdje je n broj reda,

and we start counting from zero.

ako krećemo brojiti od nule.

So if you make n=2 and expand it,

Na primjer ako raspišemo izraz za n=2,

you get (x^2) + 2xy + (y^2).

dobit ćemo (x^2)+2xy+(y^2).

The coefficients, or numbers in front of the variables,

Koeficijenti, ili brojevi ispred varijabli,

are the same as the numbers in that row of Pascal's Triangle.

jednaki su brojevima u odgovarajućem redu Pascalovog trokuta.

You'll see the same thing with n=3, which expands to this.

Isto možete vidjeti i za n=3, što se raspisuje ovako.

So the triangle is a quick and easy way to look up all of these coefficients.

Trokut je dakle brz i jednostavan način za pronalaženje koeficijenata.

But there's much more.

Ali to nije sve.

For example, add up the numbers in each row,

Na primjer, zbrojite brojeve u svakom redu,

and you'll get successive powers of two.

i dobit ćete uzastopne potencije od dva.

Or in a given row, treat each number as part of a decimal expansion.

Ili u bilo kojem redu, gledajte svaki broj kao dio decimalnog zapisa.

In other words, row two is (1x1) + (2x10) + (1x100).

Drugim riječima, drugi red je (1x1) + (2x10) + (1x100).

You get 121, which is 11^2.

Rješenje je 121, što je 11^2.

And take a look at what happens when you do the same thing to row six.

Pogledajte što će se dogoditi kada napravite isto u šestom redu.

It adds up to 1,771,561, which is 11^6, and so on.

Rješenje je 1 771 561, što je 11^6, i tako dalje.

There are also geometric applications.

Postoje i geometrijske primjene.

Look at the diagonals.

Pogledajte dijagonale.

The first two aren't very interesting: all ones, and then the positive integers,

Prve dvije nisu posebno zanimljive: samo jedinice, a zatim pozitivni brojevi;

also known as natural numbers.

poznatiji kao prirodni brojevi.

But the numbers in the next diagonal are called the triangular numbers

Ali brojevi u slijedećoj dijagonali zovu se trokutasti brojevi

because if you take that many dots,

jer ako uzmete toliko točkica,

you can stack them into equilateral triangles.

možete ih naslagati u jednakostranične trokute.

The next diagonal has the tetrahedral numbers

Slijedeća dijagonala ima tetraedne brojeve

because similarly, you can stack that many spheres into tetrahedra.

jer se navedeni broj sfera može naslagati u tetraedar.

Or how about this: shade in all of the odd numbers.

Ili primjerice: zasjenčajte sve neparne brojeve.

It doesn't look like much when the triangle's small,

To nije posebno zanimljivo kada je trokut mali,

but if you add thousands of rows,

ali ako se dodaje tisuće redova,

you get a fractal known as Sierpinski's Triangle.

dobije se Sierpinskijev fraktal.

This triangle isn't just a mathematical work of art.

Ovaj trokut nije samo matematičko umjetničko djelo.

It's also quite useful,

On je i koristan,

especially when it comes to probability and calculations

posebice u područjima vjerojatnosti i računanju

in the domain of combinatorics.

u području kombinatorike.

Say you want to have five children,

Ako primjerice želite imati petero djece

and would like to know the probability

i želite znati koja je vjerojatnost

of having your dream family of three girls and two boys.

da dobijete kako ste sanjali: tri djevojčice i dva dječaka.

In the binomial expansion,

U zapisu pomoću potencije binoma,

that corresponds to girl plus boy to the fifth power.

to odgovara djevojčici + dječaku na petu potenciju.

So we look at the row five,

Pa pogledajmo peti red,

where the first number corresponds to five girls,

gdje prvi broj odgovara pet djevojčica,

and the last corresponds to five boys.

a posljednji pet dječaka.

The third number is what we're looking for.

Mi tražimo treći broj.

Ten out of the sum of all the possibilities in the row.

Deset kroz zbroj svih mogućnosti u tom redu.

so 10/32, or 31.25%.

pa je to, 10/32, ili 31.25%.

Or, if you're randomly picking a five-player basketball team

Ili, ako nasumično izabirete peteročlanu košarkašku momčad

out of a group of twelve friends,

iz skupine od 12 prijatelja,

how many possible groups of five are there?

koliko mogućih grupa od pet osoba postoji?

In combinatoric terms, this problem would be phrased as twelve choose five,

Jezikom kombinatorike, ovaj problem izražen je kao 12 povrh 5,

and could be calculated with this formula,

i računa se pomoću ove formule,

or you could just look at the sixth element of row twelve on the triangle

ili jednostavno možete pogledati šesti element dvanaestog reda u trokutu

and get your answer.

i dobit ćete odgovor.

The patterns in Pascal's Triangle

Obrasci u Pascalovom trokutu

are a testament to the elegantly interwoven fabric of mathematics.

dokaz su elegantnog ispreplitanja djelova matematike.

And it's still revealing fresh secrets to this day.

Sve njegove tajne još nisu otkrivene.

For example, mathematicians recently discovered a way to expand it

Na primjer, matematičari su nedavno otkrili način kako ga proširiti

to these kinds of polynomials.

na ovu vrstu polinoma.

What might we find next?

Što ćemo naći slijedeće?

Well, that's up to you.

To je na vama.