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Are there any more types of numbers other than C?

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Depends on what are you willing to call a number, I guess.

There are things like quaternions, octonions, etc., which are constructed in a similar way to the complex numbers.

There are ordinal numbers and cardinal numbers, which we use to describe infinities.

There are p-adic numbers, which are a different way of extending rationals, instead of creating Reals.

I'm probably forgetting some things.
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An unfathomable number, yes.


Any "field" is probably going to satisfy you as a collection of numbers.


Now, bare in mind, the stranger and more obscure the number system, the less direct our way of representing it. There may be mathematical numbers that "don't seem like proper numbers" because of how we write them down, but this is a limitation of notation. Ugly numbers are just as numbery as numbers with dedicated symbols.
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The largest set of numbers are the surreal numbers

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