i mean, this is basically what mathematicians actually do. let's go on a tour, shall we?
Complex numbers: a classic. what if there was a square root of -1? things actually work out with an i!
Quaternions:
"ok but what if there were THREE square roots of -1 instead? i, j, k"
"why would you do that"
"for MATH. also sorry but the commutative property died"
Octonions:
"SEVEN square roots of -1. also associativity is gone too"
"...how does that even happen?"
Sedenions:
"FIFTEEN SQUARE ROOTS this is glorious"
"does anyone use this in real life?"
"what are you, a physicist?"
Trigintaduonions:
"no. no we're not doing that."
Dual numbers:
"OK I have a new number. it's called epsilon, and epsilon squared is 0"
"so is epsilon just 0?"
"NO it's DIFFERENT and COOL"
p-adic numbers:
"what if we did decimal places, like, the other way? like. instead of a tenths and a hundredths place you do tens and hundreds?"
"oo, neat, so like you could do entirely new numbers going the other way like 1+13+13^2+13^3+13^4+..."
"actually that one's just -1/12"
"ok WHAT kind of hangups do mathematicians have about that number"
Ordinal/surreal numbers:
"what if infinity was a number? and we could do infinity plus 1 and and infinity plus 2 and all that"
"isn't infinity plus 1 just infinity?"
"nope! 1 plus infinity is just infinity though"
"that...why?"
Inaccessible cardinals:
"OK I made a new number. you know how the reals are bigger than the rationals? it's bigger than the rationals but smaller than the reals"
"neat! how do you make it?"
"i can't tell you that. it's a secret."
"...are you sure this is an actual number?"
"no, but i do know you can't prove it's not"
"what does that even mean??"
these numbers make...a lot of things more cursed. but sometimes they're useful! in some cases! sometimes! ok not the trigintaduonions but some of the others!