Castel - A whole new set of numbers that humans can't actually understand it's existence. Here well see what is a 'Castel' number and 'Casteline' numbers, how are they different from each other and what makes them unique from other set of numbers (ie, Real numbers, Imaginary numbers, Complex numbers).
[[heading|Introduction]]
Before dealing with this refer Natural, Whole, Integer, Rational, Irrational, Real, Imaginary and Complex numbers.
[[heading|Casteline Numbers]]
A Casteline number is denoted by (\mathbb{L}). Set of all Casteline number is infinite, unlike Castel numbers. If you grab a look at Real and Complex numbers you will get an idea how they both are differentiated. Real numbers happen in real and can be represented in real world. Complex numbers are not real and purely virtual and imaginary to represent in this real world. Every number with odd times (\sqrt[]{-1}) is Complex and with even times are Real.
Can you think of a situation beyond these cases? Do they actually exists?. Answer : Yes, they do exists. Namely Castel and Casteline.
Every Castel number are in the set of Casteline, except few, but every Casteline number need not be in the set of every Castel numbers.
[[heading|Definition]]
A very simple explanation defines both Castel and Casteline as " Any number, which is not rational, reaches infinity in it's simpler approach or touches the edge of 'Zero' in its simpler or complex approach is known as a Castel/Casteline number."
A Castel number is represented by (\mathbb{T}).
[[start-example]] ###For example
(\pi) is an irrational number and so real. ie, (\pi \in \mathbb{R}). You would now assume that (\dfrac{1}{\pi} \in R). But this is not the case.
(\dfrac{1}{\pi} \in L). By definition (\pi) cannot be a Castel(ine) number as pi is not either reaching infinity or zero. If you manage to estimate or approximate (\pi) then you end up with either (3) or (4). But take the case of (\pi) inverse.
(\dfrac{1}{\pi} = \dfrac{1}{3.1415\ldots} = \dfrac{\underbrace{1000\ldots0}{\text{n-terms}}}{\underbrace{31415\ldots}{\text{n+1 terms}}} \approx 0) [[end-example]]
[[heading|Castel and Casteline numbers]]
It's easy to understand Castel(ine) numbers than explaining. Look at the following examples.
||Number||Castel||Casteline|| ||(\dfrac{1}{\infty}) ||Yes ||Yes || ||(\dfrac{1}{\pi})||No || Yes|| ||(\dfrac{1}{0})||Yes||No|| ||(\dfrac{0}{0})||No ||Yes|| ||(\pi)||No||No|| ||(e)||No||No|| ||(\dfrac{e}{\pi})||No||No|| ||(\dfrac{\pi}{e})||No||No|| ||(\dfrac{1}{e})||Yes||Yes|| ||(\pi^i)||No||Assumed to be|| ||(\sin\left(\dfrac{1}{0}\right))||Yes||No|| ||(\sin\left(\dfrac{1}{\infty}\right))||Yes||Yes|| ||(\sin\left(\dfrac{0}{0}\right))||Yes||Yes||
- If something approaches infinity or assumed to do so then it is a Castel number. If a number directly reaches Zero, in a simple way, then also it is a Castel number.
- If something approaches Zero or assumed to by a complex method then it is said to be Casteline.
- A Castel number cannot contain any Imaginary or Complex numbers.
- A Casteline may contain a Complex or Imaginary number.
[[heading|Notes]]
- (\mathbb{L} \in \mathbb{T})
- (\mathbb{T} \not\in \mathbb{C}, \mathbb{I})
- (\mathbb{L} \in \mathbb{C}, \mathbb{I})
- (f(x) \rightarrow 0 \implies \mathbb{T} ~~ \boxed{\text{Simple way}})
- (f(x) \rightarrow \infty \implies \mathbb{T} ~~ \boxed{\text{Any way, mostly simple. If done in a complex way assumed to be Castel}})
- (f(x) \rightarrow 0 \implies \mathbb{L} ~~ \boxed{\text{Complex way}})
- (f(x) \rightarrow 0, \infty \text{(either or both)} \implies \mathbb{L} ~~ \boxed{\text{Any way}})