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Is there a number system that's not base 10 that would be a "more perfect" representation or that would be better able/more inherently able to capture infinities? Is my question complete nonsense?
Different bases would have different things they cannot represent as a decimal, but no matter what base you can find something that isn't there.
For real world use base 12 is much nicer than base 10. However it isn't perfect. Circles are 360 degrees because base 360 is even nicer yet, but probably too hard to teach multiplication tables.