# Sum of squares

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In the past few posts, I have talked about representing integers as a sum of squares:

In this post, I would like to state Lagrange’s four-square theorem following section 6.4 of Niven-Zuckerman-Montgomery’s An introduction to the theory of number.

Firstly, by applying Hensel’s lemma to the result from the earlier post we get (Theorem 5.14):

Proposition: Let $a,b,c$ be arbitrary integers. Then the congruence $ax^2+by^2+cz^2\equiv 0\pmod{p}$ has a non-trivial solution modulo any prime $p$.

The theorem stated in the earlier post establishes that there is no need for any condition modulo primes p not dividing abc. The above proposition, application of Hensel’s lemma, just demonstrates it more explicitly by telling that the equation is solvable everywhere locally (i.e. modulo every prime).

Secondly, we need following result from Geometry of numbers (Theorem 6.21):

Minkowski’s Convex Body Theorem for general lattices: Let $A$ be a non-singular $n\times n$ matrix with real elements, and let $\Lambda = A\mathbb{Z}^n=\{A\mathbf{s}\in \mathbb{R}^n: \mathbf{s}\in \mathbb{Z}^n\}$ be a lattice. If $\mathcal{C}$ is a set in $\mathbb{R}^n$ that is convex, symmetric about origin $\mathbf{0}$, and if $\text{vol}(\mathcal{C})> 2^n |\det(A)|$, then there exists a lattice point $\mathbf{x}\in\Lambda$ such  that $\mathbf{x}\neq 0$ and $\mathbf{x}\in \mathcal{C}$.

Now we are ready to state the theorem (for the proof see Theorem 6.26):

Lagrange’s four-square theorem: Every positive integer $n$ can be expressed as the sum of four squares, $n=x_1^2+x_2^2+x_3^2+x_4^2$, where $x_i$ are non-negative integers.

# 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]