I've always found the fact that, α, β ∈ ℝ → α + β ∈ ℝ and α, β ∈ ℂ → α + β ∈ ℂ, that is if two numbers with the same essence exist than a sum of those two numbers will exist and share their essence. There is vast potential in this fact as it not only implies the the infinite nature of that essence, an infinite set, but the conditions in which a new set can be formed. We never receive a new set through and transformation of like essences and as such a new set can only be formed through the the combination of two dissimilar essences. In the construction of ℂ is is necessary to fundamentally change the algebraic structure of ℝ by changing changing the dimensionality of ℝ from ℝ to ℝ^2 thereby introducing the terms (α, β) and (γ, δ) which are elements of ℝ^2. The algebraic structure is produced through providing a vector addition and multiplication operation from which one can say (α, β) = α + βi.
The major breakthrough was not merely expanding operational capacity of some set to create a notion of inclusion but to also reformulate the algebraic structure of a set by also changing its dimensionality.
