Till about $400$ years ago, the number line was believed to be $1$- dimensional. Then came the discovery of imaginary numbers which forced the number line to be re-defined as $2$-dimensional.
My question is:
Is there are any reason why there are no more dimensions of the number lines?
How many dimensions are there of the number line or are they infinite in number?
How would you define any more dimensions to the number lines?
I would appreciate any help that I can receive. Thanks in advance!
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We can define vector spaces to have as many dimensions as we might need for any given task. Are vectors "numbers" though? They are mathematical objects with algebraic properties that might make them "number like."
In many circumstances is it convenient to consider complex numbers as vectors. And sometimes it is convenient to use matrices to represent complex numbers.
But, most directly, Quaternions form a number system that is a 4D extension of the complex numbers. And if we have Quaternions, is that the end of the line? No, there are Octonions. However as we increase the dimension we lose algebraic properties. Quaterions are not commutative, and Octonions are neither commutative of associative.
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Sadly, I have not yet 50 rep to comment but still I want to add my 2 Cents: First of all, what is actually the "real number line"? There many ways to construct it, for instance one may start off with Peano Axioms of Natural Numbers and then enlarge the set of Natural Numbers to Integers. With the Integers, one can define the Rational Numbers by $\{p/q:p\in\mathbb Z\text{ and }q\in\mathbb N\}$ --- so aren't the Rational Numbers already some sort of 2-dimensional object?! (the "$p$-line" and the "$q$-line").
Another way of introducing the Real Numbers is (and that's the way I have learnt it) to start off with the Axioms of a Vector Space / Field to construct the Rational Numbers (and define the Integers and Natural Numbers as certain subsets).
Given these preliminaries, what do you understand under the term "dimension"? Dimension of a Vector Space? The answer to this question is crucial. Because one can construct the Complex Number in a similar way: One can enlarge the set of Real Numbers by allowing another "dimension" or one can directly define a set $\mathbb C$ by giving it the structure of a field with a certain addition and a certain multiplication and a certain topology.
That is, your question boils down to: HOW do we construct (possible) "extensions" of the Real Line? And are the different construction possibilities equivalent (in some way)?
By the way, there exist more extensions. For instance the Quaterions, which are some sort of "3d" Complex Numbers.
The number line is still one-dimensional.
The complex plane is two-dimensional.
You can have number systems with arbitrary dimensions, which does not invalidate or 'force re-definition' of lower-dimensional number systems.