All The Numbers
Beyond the safe place of the rational numbers, we venture into a world filled with mystery!
"Irrational numbers", "numbers you can't write as fractions" can be divided into several subcategories.
Constructable numbers can still be written with a pen and paper, ruler and compass. These are numbers such as phi, the constant of the golden ratio, or square root of two, which can be found inside a square.
All of these are algebraic numbers. Algebraic numbers are those numbers that can be written as the solution to an operation, to an equation. One algebraic number that is not constructable with a pen and paper is the cube root of two.
Beyond the algebraic numbers, there are the transcendental numbers. Transcendental numbers are like transcendental meditation. They are an attempt by a math that did not yet know alli it could be to go beyond itself. These include pi, e, and the natural log of two.
The numberss mentioned so far, and any you will typically hear about, are often called computable numbers. They are way less than most numbers, for which you can't get a specific rule for calculating them.
We, and everything you have ever done in school, live on a small island of infinitely many numbers, surrounded by a vastly bigger infinity of all the "real" in the world.
"Real" is written in quotations because the numbers that are typically considered not real, called complex numbers, by the way, have been found integral in Physics. Yes.
But, back to the real numbers, we know a few of the non-computable numbers, such as Chaitin's number, which is the vague probability that a random computer program you write will come to a stop. There are lots of these constants depending on how you write a program. To figure them out, you would have to account for every single variable, and calculate and account for all things. They are not computable, in short.
Great mathematicians have tried, and managed to find a few of them, at least. And you can also make numbers that are influenced by tons of variables.
A number that contains everything in the universe. Most numbers should be like that.
It is already said that pi, e, square root of two, and so on, are such numbers. Your name as ASCII numbers corresponding to English alphabet characters, your phone number, the entire works of Shakespeare... All of these should be in pi, which is the ratio between a perfect circle's circumference and diameter.
We have never managed to prove that a number contains everything in the universe, although we have managed to make a few, such as 0.123456789101112131415... It really does... contain all possible numerical combinations.