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.

 

Mind Palace

I’ll tell you a story. It happened a long, long time ago, when I was watching the wonderful program called “Sherlock”. And in one of the episodes (the second of the second season, in case there’s anybody interested) the detective talked about something called “mind palace”.

Immediately after the episode, I googled it up and found a lot about the “Method of Loci”, aka Mind Palace.

 

It’s a very powerful mnemonic technique that allows the user to absorbe great quantities of information quantum-fast by using creativity, free association and a little bit of reconstruction skills.

It goes as it follows: the human brain is brilliant at remembering places and shapes, and it can easily picture itself walking through his home, his parents’ home, or even a friend’s.

It’s also very good with strange things that get its attention. Therefore, you can combinate this two things to make a Mind Palace.

Picture yourself walking through your home (or any place, really). On your way, keep noticing things that are not the background: tables, beds, pictures hanging, plants… They will be called “pegs”

Done? Good.

The next part is getting a list of things to remember, like the following sequence: Chewing gum, pear, bag, german, silicium.

It’s a short list, but it’s just to grasp the idea.

Now, the process consist in making each of those into a ridiculous, catchy picture and associate it to each one of the pegs: your dinning table hanging from a chewing gum balloon in the middle of the room, a plant being eaten by a gigantic pear*, some guy taking the picture out of your wall and putting it into his bag, a gigantic pretzel and a gigantic C  (which reminds of Si, the symbol of Silicium)

* You cannot make the plant simply grow pears, that would not be strange and you will forget it easily

Now, to remind them, you only have to walk again in your Mind Palace, checking that everything is in order.

There are also some tricks to do it with numbers: a spear for one, a swan for two… There is even the Person-Action-Object system that professionals use to remember entire decks of cards, but it would be too long for this post.

Enjoy your new, improved memory!

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.