The Golden Number

When someone says "Maths", maybe the most iconic number they can come up is pi, or rather the e number. But what if there's another number, mucho more important than this other two? Well, it actually exists, and it's called the phi number, also known as the golden ratio.

The value of this number is 1.6180339887 and it represents a proportion between two segments:

The usage of this numbers has been developed by years. Euclid was the first Mathematician who studied the divinal proportion in a regular pentagon and a golden rectangle. By cutting the gold rectangle, it creates a square and a smaller rectangle with the same aspect radio. The most remarkable ones in Mathemathics are Michael Maestlin, who published in 1597 a decimal approximation of the golden ratio, Fibonacci mentioned the numerical series, Johannnes Kepler created the Kepler triangles, a combination of phi along with the Theorem of Pythagoras, and Édouard Lucas gave us the Fibonacci sequence, a numerical sequence.

There are several applications for this peculiar number. Some of them are related to Architecture, painting, design(bookdesign, postcards, playing cards, posters, wide-screen televisions...), music, narute and optimization.

Let's go deeper into the number. This is the negative root of the number, known as the conjugate root

Taking it's absolute value and dividing one by it, we have a new number called golden ratio conjugate.

The golden ratio and the golden ratio conjugate can be expressed like this

Those two expression are used as a scale for every proportion. For example:

The major usages of the number is seen in triangles and pyramids, containing different symmetries.

Monty Hall

Imagine a game show where you’re the participant. You see in front of you three doors: One of them contains a car and the other two, a goat (I think is obvious what the prize is…). The presenter knows which door leads to the car and because he’s a generous guy, he offers you to choose a door. So, you choose one of the doors, but wait, don’t open it yet! The presenter seems to be more generous and opens one of the remaining doors, revealing a goat. Furthermore, you have the possibility of keeping your door, or choose the other one. At this point, is obvious to think the probability of having the car is 50 %, isn’t it? Well…. What if I told you… That statement is wrong?

This problem is known as the “Monty Hall” problem, asked in Parade Magazine in 1990, where Craig F. Whitaker sent a letter to Marilyn vos Savant’s column. The TV program problem’s is a reference to Let’s Make a Deal, a game show where they can’t change their boxes.

So here’s the thing: The presenter knows exactly where the car and the goats are, so he can’t eliminate the door which leads to the car. If the presenter wouldn’t know anything, then we have another probabilistic problem, but that’s not what we’re talking about. We’re talking about the presenter making his decision AFTER the player chooses a door.

When you have the three doors and choose one of them, the probabilities are 1/3 for the car and 2/3 for the goat. So it’s more probable that you first choose a goat and not the car. But either choosing the car or a goat the first time, there will be always a goat that can be eliminated. In that case, the probability of eliminating the car is 0.

So now, we have two doors, one leading to the prize and the one to the failure. Considering the fact that it’s more possible to choose a goat in the first choice, what happens when you change it? That your chance of obtaining the car is higher!

You have a simulator here in this link to contribute the Mathemathics. Keep calm and use logic!