this post was submitted on 12 Mar 2025
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[–] Leate_Wonceslace@lemmy.dbzer0.com 14 points 3 days ago (5 children)

Okay, so I had a personal project for a long time that addressed the potential for an algebra that allowed for the multipicitive inverse of the additive identity.

In the context of the resulting non-associative algebra, 0/0=1, rather than 0.

For anyone wondering, the foundation goes as such: Ω0=1, Ωx=ΩΩ=Ω, x+Ω=Ω, Ω-Ω=Ω+Ω=0.

A fun consequence of this is the exponential function exp(x)=Σ((x^n)/n!) diverges at exp(Ω). Specifically you can reduce it to Σ(Ω), which when you try to evaluate it, you find that it evaluates to either 0 or Ω. This is particularly fitting, because e^x has a divergent limit at infinity. Specially, it approaches infinity when going towards the positive end and it approaches 0 when approaching the negative.

There's more cool things you can do with that, but I'll leave it there for now.

[–] CompassRed@discuss.tchncs.de 2 points 3 days ago

Interesting. I think it isn't unital either otherwise Ω=0.

0=Ω+Ω=Ω+ΩΩ=Ω(1+Ω)=ΩΩ=Ω

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