# Building Mathematics

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Let’s talk about the work of a mathematician. When I was young (before highschool), I used to believe that anyone capable of using mathematics is a mathematician. The reason behind this was that being a mathematician was not a job for people like Brahmagupta, Aryabhatta, Fermat, Ramanujan (the names I knew when I was young). So by that definition, even a shopkeeper was a mathematician. And hence I had no interest in becoming a mathematician.

Then, during highschool, I came to know about the mathematics olympiad and was fascinated by the “easy to state but difficult to solve” problems from geometry, combinatorics, arithmetic and algebra (thanks to AMTIVipul Naik and Sai Krishna Deep) . I practiced many problems in hope to appear for the exam once in my life. But that day never came (due to bad education system of my state) and I switched to physics, just because there was lot of hype about how interesting our nature is (thanks to Walter Lewin).

In senior school I realised that I can’t do physics, I simply don’t like the thought process behind physics (thanks to Feynman). And luckily, around the same time, came to know what mathematicians do (thanks to Uncle Paul). Mathematicians “create new maths”. They may contribute according to their capabilities, but no contribution is negligible. There are two kinds of mathematicians, one who define new objects (I call them problem creators) and others who simplify the existing theories by adding details (I call them problem solvers). You may wonder that while solving a problem one may create bigger maths problems, and vice versa, but I am talking about the general ideologies. What I am trying to express, is similar to what people want to say by telling that logic is a small branch of mathematics (whereas I love maths just for its logical arguments).

A few months before I had to join college (in 2014), I decided to become a mathematician. Hence I joined a research institute (clearly not the best one in my country, but my concern was just to be able to learn as much maths as possible).  Now I am learning lots of advanced (still old) maths (thanks to Sagar SrivastavaJyotiraditya Singh and my teachers) and trying to make a place for myself, to be able to call myself a mathematician some day.

I find all this very funny. When I was young, I used to think that anyone could become a mathematician and there was nothing special about it. But now I everyday have to prove myself to others so that they give me a chance to become a mathematician. Clearly, I am not a genius like all the people I named above (or even close to them) but I still want to create some new maths either in form of a solution to a problem or foundations of new theory and call myself a mathematician. I don’t want it to end up like my maths olympiad dream.

# 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].

# Imaginary Angles

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You would have heard about imaginary numbers and most famous of them is $i=\sqrt{-1}$. I personally don’t like this name because all of mathematics is man/woman made, hence all mathematical objects are imaginary (there is no perfect circle in nature…) and lack physical meaning. Moreover, these numbers are very useful in physics (a.k.a. the study of nature using mathematics). For example, “time-dependent Schrödinger equation

$\displaystyle{i \hbar \frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat H \Psi(\mathbf{r},t)}$

But, as described here:

Complex numbers are a tool for describing a theory, not a property of the theory itself. Which is to say that they can not be the fundamental difference between classical and quantum mechanics (QM). The real origin of the difference is the non-commutative nature of measurement in QM. Now this is a property that can be captured by all kinds of beasts — even real-valued matrices. [Physics.SE]

For more of such interpretation see: Volume 1, Chapter 22 of “The Feynman Lectures in Physics”. And also this discussion about Hawking’s wave function.

All these facts may not have fascinated you, but the following fact from Einstein’s Special Relativity should fascinate you:

In 1908 Hermann Minkowski explained how the Lorentz transformation could be seen as simply a hyperbolic rotation of the spacetime coordinates, i.e., a rotation through an imaginary angle. [Wiki: Rapidity]

Irrespective of the fact that you do/don’t understand Einstein’s relativity, the concept of imaginary angle appears bizarre. But, mathematically its just another consequence of non-euclidean geometry which can be interpreted as Hyperbolic law of cosines etc. For example:

$\displaystyle{\cos (\alpha+i\beta) = \cos (\alpha) \cosh (\beta) - i \sin (\alpha) \sinh (\beta)}$

$\displaystyle{\sin (\alpha+i\beta) = \sin (\alpha) \cosh (\beta) + i \cos (\alpha) \sinh (\beta)}$

Let’s try to understand what is meant by “imaginary angle” by following the article “A geometric view of complex trigonometric functions” by Richard Hammack. Consider the complex unit circle  $U=\{z,w\in \mathbb{C} \ : \ z^2+w^2=1\}$ of $\mathbb{C}^2$, in a manner exactly analogous to the definition of the standard unit circle in $\mathbb{R}^2$. Apparently U is some sort of surface in $\mathbb{C}^2$, but it can’t be drawn as simply as the usual unit circle, owing to the four-dimensional character of $\mathbb{C}^2$. But we can examine its lower dimensional cross sections. For example, if  $z=x+iy$ and $w=u+iv$ then by setting $y = 0$ we get the circle $x^2+u^2=1$ in x-u plane for v=0 and the hyperbola $x^2-v^2 = 1$ in x-vi plane for u=0.

The cross-section of complex unit circle (defined by z^2+w^2=1 for complex numbers z and w) with the x-u-vi coordinate space (where z=x+iy and w=u+iv) © 2007 Mathematical Association of America

These two curves (circle and hyperbola) touch at the points ±o, where o=(1,0) in $\mathbb{C}^2$, as illustrated above. The symbol o is used by Richard Hammack because this point will turn out to be the origin of complex radian measure.

Let’s define complex distance between points $\mathbf{a} =(z_1,w_1)$ and $\mathbf{b}=(z_2,w_2)$ in $\mathbb{C}^2$ as

$\displaystyle{d(\mathbf{a},\mathbf{b})=\sqrt{(z_1-z_2)^2+(w_1-w_2)^2}}$

where square root is the half-plane H of $\mathbb{C}$ consisting of the non-negative imaginary axis and the numbers with a positive real part. Therefore, the complex distance between two points in $\mathbb{C}^2$ is a complex number (with non-negative real part).

Starting at the point o in the figure above, one can move either along the circle or along the right-hand branch of the hyperbola. On investigating these two choices, we conclude that they involve traversing either a real or an imaginary distance. Generalizing the idea of real radian measure, we define imaginary radian measure to be the oriented arclength from o to a point p on the hyperbola, as

If p is above the x axis, its radian measure is $\beta i$ with $\beta >0$, while if it is below the x axis, its radian measure is $\beta i$ with $\beta <0$. As in the real case, we define $\cos (\beta i)$ and $\sin (\beta i)$ to be the z and w coordinates of p. According to above figure (b), this gives

$\displaystyle{\cos (\beta i) = \cosh (\beta); \qquad \sin (\beta i) = i \sinh (\beta)}$

$\displaystyle{\cos (\pi + \beta i) = -\cosh (\beta); \qquad \sin (\pi + \beta i) = -i \sinh (\beta)}$

Notice that both these relations hold for both positive and negative values of $\beta$, and are in agreement with the expansions of  $\cos (\alpha+i\beta)$  and $\sin (\alpha+i\beta)$  stated earlier.

But, to “see” what a complex angle looks like we will have to examine the complex versions of lines and rays. Despite the four dimensional flavour, $\mathbb{C}^2$ is a two-dimensional vector space over the field $\mathbb{C}$, just like $\mathbb{R}^2$ over $\mathbb{R}$.

Since a line (through the origin) in $\mathbb{R}^2$ is the span of a nonzero vector, we define a complex line in $\mathbb{C}^2$ analogously. For a nonzero vector u in $\mathbb{C}^2$, the complex line $\Lambda$ through u is span(u), which is isomorphic to the complex plane.

In $\mathbb{R}^2$, the ray $\overline{\mathbf{u}}$ passing through a nonzero vector u can be defined as the set of all nonnegative real multiples of u. Extending this to $\mathbb{C}^2$ seems problematic, for the word “nonnegative” has no meaning in $\mathbb{C}$. Using the half-plane H (where complex square root is defined) seems a reasonable alternative. If u is a nonzero vector in $\mathbb{C}$, then the complex ray through u is the set $\overline{\mathbf{u}} = \{\lambda u \ : \ \lambda\in H\}$.

Finally, we define a complex angle is the union of two complex rays $\overline{\mathbf{u}_1}$ and $\overline{\mathbf{u}_2}$ .

I will end my post by quoting an application of imaginary angles in optics from here:

… in optics, when a light ray hits a surface such as glass, Snell’s law tells you the angle of the refracted beam, Fresnel’s equations tell you the amplitudes of reflected and transmitted waves at an interface in terms of that angle. If the incidence angle is very oblique when travelling from glass into air, there will be no refracted beam: the phenomenon is called total internal reflection. However, if you try to solve for the angle using Snell’s law, you will get an imaginary angle. Plugging this into the Fresnel equations gives you the 100% reflectance observed in practice, along with an exponentially decaying “beam” that travels a slight distance into the air. This is called the evanescent wave and is important for various applications in optics. [Mathematics.SE]