A Formal Introduction to Complex Numbers
A Mathematically rigorous introduction to Complex Numbers
To understand why complex numbers are mathematically relevant, we need to understand what are algebraic fields.
Introduction to an Algebraic Field
In abstract algebra, a field is a fundamental algebraic structure characterized by a set equipped with two binary operations: addition and multiplication. These operations satisfy specific axioms that generalize the arithmetic of familiar number systems like the rational, real, and complex numbers.
Formally put, we call an ordered tuple (𝔽, +, ×) a field if we have that all elements of 𝔽 respect the following operational rules:
Examples
- Rational Numbers : ℚ
- Real Numbers : ℝ
The Special Field Extension
Fields are very common in mathematics, and they model in an algebraic way the set ℝ with the usual addition and multiplication, we denote the field of ℝ as the ordered pair (ℝ, +, ×).
One might ask, is there any way to extend the field ℝ to another field 𝔽 such that ℝ ⊆ 𝔽? There is, and in fact there is only one which is what we call complex numbers, and we denote it by ℂ.
The mathematical reason to study complex numbers C is that C is the unique field extension of R. Furthermore, we have that there is no field extension to C (check for Field Theory). So, in a sense, C is the furthest we can go in field extensions.
How can we construct ℂ ?
To construct ℂ we use the following algebraic structure ℂ = ℝ[i] which is defined as
where i is a formal element i ∈ ℂ such that i² = -1.
Useful Properties of ℂ
Proposition: The set ℂ is a field.
Check our definition step by step to see the veracity of the statement.
Corollary: If 0 ≠ z ∈ ℂ then we have that z has a unique inverse.
Let z = a + bi under our hypothesis and define
note first that zz^{-1} = 1. For the uniqueness, note that if we have another w = c +di such that zw = 1, then we have that
this yields the following system of equations
which turns out to be solved as
that is why the corollary is true.
Theorem: Every non-constant polynomial p(z) with complex coefficients has at least one root in ℂ.
This statement is justified by many methods, however I would argue that using Analytic Methods comes natural, specifically using Liouville’s Theorem from complex analysis that we will talk in the future. When a field has a result of that type, then we say that a field is Algebraically closed, which is quite a big thing in field theory.
The Complex Plane
There is a particular nice way to describe the complex numbers ℂ, as we can make the following isomorphism
where addition in ℝ² is defined as (a, b) + (c, d) = (a+c, b+d) and the multiplication is defined as (a, b)×(c, d) = (ac-bd, ad + bc), which we will enable us to represent the complex plane as the graphic plane:
Next Story
In the next story we will talk about holomorphic functions, they are a very special kind of functions that in a sense mimic differentiable functions in ℝ.
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