# Teaching Mathematics

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One of the most challenging and rewarding thing associated with being a math enthusiast (a.k.a. mathematician) is an opportunity to share your knowledge about the not so obvious truths of mathematics. A couple of years ago, I tried to communicate that feeling through an article for high school students.

When I joined college, I tried to teach mathematics to some kids from financially not-so strong family. Since they had no exposure to mathematics, I had to start with  concepts like addition and multiplication of numbers. My experience can be summarized as the following stand-up comedy performance by Naveen Richard:

After trying for about a couple of months to teach elementary mathematics, I gave up and now I discuss mathematics only above the high school level. Last week I delivered a lecture discussing the proof of Poncelet’s Closure Theorem:

Whenever a polygon is inscribed in one conic section and circumscribes another one, the polygon must be part of an infinite family of polygons that are all inscribed in and circumscribe the same two conics.

I had spent sufficient time preparing the lecture, and believed that I was aware of all possible consequences of this theorem. But, almost half way through the lecture one person (Haresh) from the audience of 10 people, pointed out following fascinating consequence of the theorem:

If an n-sided polygon is inscribed in one conic section and circumscribed by the other one, then it must be a convex polygon and no other m-sided polygon (with m≠n) can be inscribed and circumscribed by this pair of conic sections.

This kind of insights by audience motivates me to discuss mathematics with others!

# In the praise of norm

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If you have spent some time with undergraduate mathematics, you would have probably heard the word “norm”. This term is encountered in various branches of mathematics, like (as per Wikipedia):

But, it seems to occur only in abstract algebra. Although the definition of this term is always algebraic, it has a topological interpretation when we are working with vector spaces.  It secretly connects a vector space to a topological space where we can study differentiation (metric space), by satisfying the conditions of a metric.  This point of view along with an inner product structure, is explored when we study functional analysis.

Some facts to remember:

1. Every vector space has a norm. [Proof]
2. Every vector space has an inner product (assuming “Axiom of Choice”). [Proof]
3. An inner product naturally induces an associated norm, thus an inner product space is also a normed vector space.  [Proof]
4. All norms are equivalent in finite dimensional vector spaces. [Proof]
5. Every normed vector space is a metric space (and NOT vice versa). [Proof]
6. In general, a vector space is NOT same a metric space. [Proof]

# Real vs Complex Plane

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Real plane is denoted by $\mathbb{R}^2$ and is commonly referred to as  Cartesian plane. When we talk about $\mathbb{R}^2$ we mean that $\mathbb{R}^2$ is a vector space over $\mathbb{R}$. But when you view $\mathbb{R}^2$ as Cartesian plane, then it’s not technically a vector space but rather an affine space, on which a vector space acts by translations, i.e. there is no canonical choice of where the origin should go in the space, because it can be translated anywhere. Cartesian Plane (345Kai at the English language Wikipedia [Public domain, GFDL or CC-BY-SA-3.0], via Wikimedia Commons)

On the other hand, complex plane is denoted by $\mathbb{C}$ and is commonly referred to as Argand plane. But when we talk about $\mathbb{C}$, we mean that $\mathbb{R}^2$ is a field (by exploiting the tuple structure of elements) since there is only way to explicitly define the field structure on the set $\mathbb{R}^2$ and that’s how we view $\mathbb{C}$ as a field (if you allow axiom of choice, there are more possibilities; see this Math.SE discussion). Argand Plane (Shiva Sitaraman at Quora)

So, when we want to bother about the vector space structure of $\mathbb{R}^2$ we refer to Cartesian plane and when we want to bother about the field structure of $\mathbb{R}^2$ we refer to Argand plane. An immediate consequence of the above difference in real and complex plane is seen when we study multivariable analysis and complex analysis, where we consider vector space structure and field structure, respectively (see this Math.SE discussion for details). Hence the definition of differentiation of a function defined on $\mathbb{C}$ is a special case of definition of differentiation of a function defined on $\mathbb{R}^2$.

# Real vs Complex numbers

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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: From Chapter 2 of the book “QED – The Strange Theory of Light and Matter” © Richard P. Feynman, 1985.

Before reading this explanation, I used to believe that the need to establish “Fundamental theorem Algebra” (read this beautiful paper by Daniel J. Velleman 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 $\mathbf{F}$ be a field. Suppose that there is a set $\mathcal{P} \subset \mathbf{F}$ which satisfies the following properties:

• For each $x \in \mathbf{F}$, exactly one of the following statements holds: $x \in \mathcal{P}$, $-x \in \mathcal{P}$, $x =0$.
• For $x,y \in \mathcal{P}$, $xy \in \mathcal{P}$ and $x+y \in \mathcal{P}$.

If such a $\mathcal{P}$ exists, then $\mathbf{F}$ is an ordered field. Moreover, we define $x \le y \Leftrightarrow y -x \in \mathcal{P} \vee x = y$.

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 $\mathbb{R}$ and $\mathbb{C}$ are isomorphic as additive groups (and as vector spaces over $\mathbb{Q}$) 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 $\mathcal{S}$ be a non-empty set of real numbers.

• A real number $x$ is called an upper bound for $\mathcal{S}$ if $x \geq s$ for all $s\in \mathcal{S}$.
• A real number $x$ is the least upper bound (or supremum) for $\mathcal{S}$ if $x$ is an upper bound for $\mathcal{S}$ and $x \leq y$ for every upper bound $y$ of $\mathcal{S}$ .

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 $\mathbb{R}$ and $\mathbb{C}$ are complete as a metric space [proof] but only $\mathbb{R}$ 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 $\mathbf{F}$, the following are equivalent:
(i) $\mathbf{F}$ is Dedekind complete.
(ii) $\mathbf{F}$ 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 $1+1+\ldots+1$ whose value is greater than that element, that is, there are no infinite elements.

As remarked earlier, $\mathbb{C}$ is not an ordered field and hence can’t be Archimedean. Therefore, $\mathbb{C}$  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].