Category: Mathematics

3n+1

Take a number. Any number. Done?

Okay, now, we’re gonna do the following thing: if it’s even, we divide it by two. If it’s not, we multiply it by three and add one. Then, we repeat the process again and again.

10, 5, 16, 8, 4, 2, 1, 4, 2, 1…

11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1…

Notice something? They all end in 4, 2, 1! In a loop!

Cool, huh?

Maybe not “real world” cool, but it’s certainly math cool.

The thing is, we don’t know whether it happens for every number or only with the ones we’ve tried so far.

Here is a calculator for this conjecture (which, by the way, is called Collatz Conjecture, guess why)

We think it always finishes in the same loop of three numbers: 1, 2, 4, 1, 2, 4… One way to prove it, then, would be finding a new loop.

But, if there is another loop, it would have to be at least one million steps long. So good luck trying to find it.

 

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Elegant Mathematics: Euler’s Identity

e to the power of π√(-1) equals -1.

What do you think? Easy, right?

This has been often called “The most beautiful mathematical formula” because of it’s shortness, the use of e, π, the square root of a negative number and -1.

But why? Why is it that way? Well…

Imagine addition as sliding a scale, instead of putting things together. Then, multiplication would be making the scale bigger or smaller.

Imaginary numbers (that is, the square root of negative numbers) would make the scale turn 90º counter clockwise, so they are located in a vertical line parallel to the real numbers

Now, in this weird system, imaginary powers of e have a strange property: they take points in the vertical line of the imaginary numbers and turn them into numbers in a circle of radius one.

Then, it’s easy to see: a circle’s length divided by its diameter is exactly π. Then, if you take a line of length π and fold it around the circle, the end point would be exactly in the minus one.

 

Elegant Mathematics: Square Root of Two

In my last post of the series “Elegant Mathematics” I talked about the Pythagoreans, a Greek cult that basically worshipped numbers. They thought that reality was made of integers and of ratios between integers (that is, 1/2, 5/67 and so on)

The thing is, they couldn’t find a ratio that expressed the length of the diagonal of a square of 1 unit each side, so one of them supposed that there was no such ratio. And he proved it. And, as the story goes, “they went on a boat and came back without him”

Yep. Kids, cults are bad. Don’t get into them.

Anyway, his proof was as it follows: suppose there is such a ratio. Then, a/b = √2

We can square both sides to get to (a^2)/(b^2) = 2

And then you can multiply both sides by b^2 and get to (a^2) = 2(b^2)

Therefore, a^2 is an even number, and, since uneven numbers squared only give uneven numbers, it’s safe to assume that a is even too, that is, a = 2*K, where K is an integer.

So we know that (2K)^2 = 2(b^2), and by operating, that 4(K^2)=2(b^2).

But wait! That means that both a and b are even, which contradicts our premise that a and b are coprimes.

Therefore, there is no fraction a/b that is equal to the square root of two.

This is called “proof by contradiction”, which means assuming what we want to prove false until you get to a contradiction.

It was also used to try to prove the importance of Euclid’s Fifth, without success, creating the Non-Euclidean Geometries.

 

PD:

I know I was supposed to submit this on Monday, but I forgot and it has been resting in Drafts since then. Sorry!

Elegant Mathematics: Pythagoras Theorem

Here’s the second post in the series “Elegant Mathematics”. Today, it’s all about Pythagoras, the greek philosopher and mathematician (which were, at the time, the same thing).

Pythagoras was a bit of a strange guy: he was the leader of what we would call a cult. His followers couldn’t eat meat, had to wear special clothes, couldn’t pee towards the sun nor “Wipe their bottom with a stick”.

The thing is, the Pythagoreans thought that the reality was made of numbers, and made many interesting contributions to the field of mathematics, being the most well-known, the Pythagoras Theorem.

This states, as everybody knows, that the square of the hypotenuse equals the sum of the square of the other two sides. It is usually shown like this:

H^2=a^2+b^2

Of course, this affirmation has to be proven in order to become a theorem.

And it can be proved this way:

Take this picture:

Now, the area of the big square is (a+b)^2, since each side is equal to (a+b). By extending it, you get that the big square’s area is (a^2+2ab+b^2).

The area of the small triangle is c^2, or the big square minus the triangles. The area of the triangles is ab/2, and there are four of them, so they sum 2ab

Therefore, the area of the small square is equal to c^2, or (a^2+2ab+b^2)-2ab, that is, a^2+b^2. Then, c^2=a^2+b^2.

Elegant Mathematics: Euclid’s Infinite Primes

As you know, primes are the numbers 2,3,5,7,11… That is, numbers that can only be divided by 1 or themselves.

Now, as you can see, those numbers get less common as they grow larger: between 1 and 10 there are 4, while there are only 168 between 1 and 1,000.

This made the first mathematicians wonder: is there a last prime?

Euclid took the challenge, and showed a surprisingly simple proof:

Imagine you had the whole set of prime numbers. This set would be finite. But, what would happen if you multiplied all those primes together and added one?

The new number can’t be divided by any of the previous primes, so it must be either prime, either a multiplication of two primes that are not included in the set.

Therefore, there are infinite primes.

This proof is often expressed like this:

Q=2*3*5*7*11*13…*P+1

Since then, mathematicians have found many other proofs, some of which will be treated here later, but they all proof the same and are not as simple as this one is.

Announcement: Elegant Mathematics

Hey, people!

I’ve realised that my blog is going too much to the artsy side of the force with all the writing about writing and stuff, and I think I’m not showing enough numbers, so I’ve decided to start a series called “Elegant Mathematics”, with a new post coming out every Monday.

They will be tagged as “Elegant Mathematics”, in case you want to see them all together.

Right now, the schedule is:

Euclid’s Infinite Primes Proof
Pythagoras’ theorem: Many different proofs
Square Root of 2
Euler’s Identity

I intended to make an ascending level of difficulty, but it’s possible that I find something else in between and that I decide to post it.

Anyway, thanks for your attention!