I want to talk about the algebraic and analytic differences between real and complex numbers. Firstly, let’s have a look at following beautiful explanation by Richard Feynman (from his QED lectures) about similarities between real and complex numbers:
Before reading this explanation, I used to believe that the need to establish “Fundamental theorem Algebra” (read this beautiful paper by to learn about proof of this theorem) was only way to motivate study of complex numbers.
The fundamental difference between real and complex numbers is
Real numbers form an ordered field, but complex numbers can’t form an ordered field. [Proof]
Where we define ordered field as follows:
Let be a field. Suppose that there is a set which satisfies the following properties:
- For each , exactly one of the following statements holds: , , .
- For , and .
If such a exists, then is an ordered field. Moreover, we define .
Note that, without retaining the vector space structure of complex numbers we CAN establish the order for complex numbers [Proof], but that is useless. I find this consequence pretty interesting, because though and are isomorphic as additive groups (and as vector spaces over ) but not isomorphic as rings (and hence not isomorphic as fields).
Now let’s have a look at the consequence of the difference between the two number systems due to the order structure.
Though both real and complex numbers form a complete field (a property of topological spaces), but only real numbers have least upper bound property.
Where we define least upper bound property as follows:
Let be a non-empty set of real numbers.
- A real number is called an upper bound for if for all .
- A real number is the least upper bound (or supremum) for if is an upper bound for and for every upper bound of .
The least-upper-bound property states that any non-empty set of real numbers that has an upper bound must have a least upper bound in real numbers.
This least upper bound property is referred to as Dedekind completeness. Therefore, though both and are complete as a metric space [proof] but only is Dedekind complete.
In an arbitrary ordered field one has the notion of Dedekind completeness — every nonempty bounded above subset has a least upper bound — and also the notion of sequential completeness — every Cauchy sequence converges. The main theorem relating these two notions of completeness is as follows [source]:
For an ordered field , the following are equivalent:
(i) is Dedekind complete.
(ii) is sequentially complete and Archimedean.
Where we defined an Archimedean field as an ordered field such that for each element there exists a finite expression whose value is greater than that element, that is, there are no infinite elements.
As remarked earlier, is not an ordered field and hence can’t be Archimedean. Therefore, can’t have least-upper-bound property, though it’s complete in topological sense. So, the consequence of all this is:
We can’t use complex numbers for counting.
But still, complex numbers are very important part of modern arithmetic (number-theory), because they enable us to view properties of numbers from a geometric point of view [source].