# Hyperreal and Surreal Numbers

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These are the two lesser known number systems, with confusing names.

Hyperreal numbers originated from what we now call “non-standard analysis”. The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The term “hyper-real” was introduced by Edwin Hewitt in 1948. In non-standard analysis the concept of continuity and differentiation is defined using infinitesimals, instead of the epsilon-delta methods. In 1960, Abraham Robinson showed that infinitesimals are precise, clear, and meaningful.

Following is a relevant Numberphile video:

Surreal numbers, on the other hand, is a fully developed number system which is more powerful than our real number system. They share many properties with the real numbers, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they also form an ordered field. The modern definition and construction of surreal numbers was given by John Horton Conway in  1970. The inspiration for these numbers came from the combinatorial game theory. Conway’s construction was introduced in Donald Knuth‘s 1974 book Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness.

In his book, which takes the form of a dialogue, Knuth coined the term surreal numbers for what Conway had called simply numbers. This is the best source to learn about their construction. But the construction, though logical, is non-trivial. Conway later adopted Knuth’s term, and used surreals for analyzing games in his 1976 book On Numbers and Games.

Following is a relevant Numberphile video:

Many nice videos on similar topics can be found on PBS Infinite Series YouTube channel.

# Intra-mathematical Dependencies

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Recently I completed all of my undergraduate level maths courses, so wanted to sum up my understanding of mathematics in the following dependency diagram:

I imagine this like a wall, where each topic is a brick. You can bake different bricks at different times (i.e. follow your curriculum to learn these topics), but finally, this is how they should be arranged (in my opinion) to get the best possible understanding of mathematics.

As of now, I have an “elementary” knowledge of Set Theory, Algebra, Analysis, Topology, Geometry, Probability Theory, Combinatorics and Arithmetic. Unfortunately, in India, there are no undergraduate level courses in Mathematical Logic and Category Theory.

This post can be seen as a sequel of my “Mathematical Relations” post.

# 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 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].

# Area of Rectangle

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When we learn to find area enclosed by a curve, we are told to divide area in rectangular elements.

Taken from pp. 361 of Mathematics (Part – II), Class XII textbook, NCERT

But, how do we always know the value of area of rectangle? In this post, I will try to prove this well known fact in the spirit of Euclid and Cauchy.

Let’s define:

• Boundary: A boundary is that which is an extremity of anything.
• Figure: A figure is that which is contained by any boundary or boundaries.
• Triangular region : A triangular region is a figure which is the union of a triangle and its interior. Also the sides of the triangle are called edges of the region and vertices of the triangle are called vertices of the region.
• Polygonal region: A polygonal region is a plane figure which can be expressed as the
union of a finite number of triangular regions, in such a way that if two of the
triangular regions intersect, their intersection is an edge or a vertex of each of them.
• Square region: It is the union of a square and its interior.

The structure in our geometry is

$[\mathcal{S},\mathcal{L}, \mathcal{P}, d, m, \alpha]$

where $\mathcal{S}$ is the set of points, $\mathcal{L}$ is the set of lines, $\mathcal{P}$ is the set of planes, $d$ is distance (a function satisfying first three properties of metric function), $m$ is angular measure (a function defined for angles, with real numbers as values of the function, satisfying following postulates) and $\alpha$ is the area function satisfying following postulates:

1. $\alpha$ is a function $\mathcal{R} \rightarrow \mathbb{R}$, where $\mathcal{R}$ is the set of all polygonal regions and $\mathbb{R}$  is the set of all real numbers.
2. For every polygonal region R, $\alpha(R) > 0$.
3. If two triangular regions are congruent, then they have the same area.
4. If two polygonal regions intersect only in edges and vertices (or do not intersect at all), then the area of their union is the sum of their areas.
5. If a square region has edges of length 1, then its area is 1.

Proposition 1. If a square has edges of length $1/q$ ($q$ a positive integer), then its area is $1/q^2$.

Proof. A unit square region can be decomposed into $q^2$ square regions, all with the same edge $1/q$ as

Then all smaller square have the same area $A$ (divide each square into triangles using diagonals and then use Postulate 3 to prove that all of them have same area). Therefore $1 = q^2A$ (from Postulate 4) and $A = 1/q^2$.

Proposition 2. If a square has edges of rational length $p/q$, then its area is $p^2/q^2$.

Proof. Such a square can be decomposed into $p^2$ squares, each of edge $1/q$ as:

If $A$ is its area, then

$A = p^2 \times \frac{1}{q^2} = \frac{p^2}{q^2}$

Proposition 3.[Peter Lawes] If a square has edges of length $a$, then its area is $a^2$.

Proof. Given a square $S_a$ with edges of length $a$. Given any rational number $p/q$,  let $S_{p/q}$ be a square of edge $p/q$, with an angle in common with $S_a$, as:

Then, $\frac{p}{q} < a$ hence $S_{p/q}$ lies inside $S_a$. For some real number $s$ (by using Postulate 4) we get:

$\alpha(S_{p/q}) + s = \alpha(S_{a})$

$\Rightarrow \alpha(S_{p/q}) < \alpha(S_{a})$

$\Rightarrow \frac{p^2}{q^2}< \alpha(S_{a})$

$\Rightarrow\frac{p}{q} < \sqrt{\alpha(S_{a})}$

But, selection of $\frac{p}{q}$ being arbitrary, the upper-bound should be unique. Since there exists a unique supremum of the set consisting all possible side lengths of smaller square in $\mathbb{R}$, we can claim:

$a = \sqrt{\alpha(S_{a})}$

We can prove this claim by following the proof of statement: $\sup\{x \in \mathbb{R} : 0 \leq x, x^2 < 2\} = \sqrt{2}$. Hence:

$a^2=\alpha(S_a)$

Theorem: Area of rectangle is equal to the product of length of any two adjacent sides.

Proof. Given a rectangle of base $b$ and altitude $h$, we construct a square of edge $b + h$, and decompose it into squares and rectangles as:

Then from Postulate 4, we get:

$(b+h)^2 = 2A + A_1 +A_2$

$b^2 + 2bh + h^2=2A +h^2+b^2$

$2bh = 2A$

$bh = A$

REFERENCE:

Moise, Edwin (1990). Elementary Geometry from an Advanced Standpoint. Addison-Wesley Pub. Co.

Thanks to Dr. Shailesh Shirali for pointing out this beautiful book.

# So many Integrals – II

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As promised in previous post, now I will briefly discuss the remaining two flavors of Integrals.

Stieltjes Integral

Stieltjes

In 1894, a Dutch mathematician, Thomas Stieltjes, while solving the moment problem, that is, given the moments of all orders of a body, find the distribution of its mass, gave a generalization of the Darboux integral.

Let $P : a = x_0 < x_1 < x_2<\ldots < x_n = b$, $n$ being an integer, be a partition of the interval $[a, b]$.

For a function $\alpha$, monotonically increasing on $[a,b]$, we write:

$\Delta \alpha_i = \alpha(x_i) - \alpha(x_{i-1})$

Let $f$ be a bounded function defined on an interval $[a, b],\quad a, b$ being real numbers. We define the sum

$S_P = \sum_{i=1}^n f(t_i)\Delta \alpha_i, \quad \overline{S}_P = \sum_{i=1}^n f(s_i)\Delta \alpha_i$

where $t_i,s_i \in [x_{i-1} , x_i]$ be such that

$f(t_i) = \text{sup} \{ f(x) : x \in [x_{i-1}, x_{i}]\}$,

$f(s_i) = \text{inf} \{ f(x) : x \in [x_{i-1}, x_{i}]\}$

If the $\text{inf}\{S_P\}$ and $\text{sup}\{\overline{S}_P\}$ are equal, we denote the common value by  $\int_{a}^{b} f(x) d\alpha(x)$ and call it Steiltjes integral of $f$ with respect to $\alpha$ over $[a,b]$.

Lebesgue Integral

Lebesgue

Let me quote Wikipedia article:

The integral of a function f between limits a and b can be interpreted as the area under the graph of f. This is easy to understand for familiar functions such as polynomials, but what does it mean for more exotic functions? In general, for which class of functions does “area under the curve” make sense? The answer to this question has great theoretical and practical importance.

In 1901, a French mathematician, Henri Léon Lebesgue generalized the notion of the integral by extending the concept of the area below a curve to include functions with uncountable discontinuities .

Lebesgue defined his integral by partitioning the range of a function and summing up sets of x-coordinates belonging to given y-coordinates, rather than, as had traditionally been done, partitioning the domain.

Lebesgue himself, according to his colleague, Paul Montel, compared his method with paying off a debt: (see:pp. 803,  The Princeton Companion to Mathematics)

I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral.

A set $\mathcal{A}$ is said to be Lebesgue measurable, if for each set $\mathcal{E} \subset \mathbb{R}$ the Carathéodory condition:

$m^{*} (\mathcal{E}) = m^{*}(\mathcal{E} \cap \mathcal{A}) + m^{*}(\mathcal{E}\backslash \mathcal{A})$

is satisfied, where $m^{*}(\mathcal{A})$ is called outer measure and is defined as:

$m^{*}(\mathcal{A}) = \inf\sum\limits_{n=1}^\infty (b_n-a_n)$

where $\mathcal{A}$ is a countable collection of closed intervals $[a_n,b_n], a_n\leq b_n$, that cover $\mathcal{A}$.

The Lebesgue integral of a simple function $\phi(x) = \sum_{i=1}^n c_i \chi_{\mathcal{A}_i} (x)$ on $\mathcal{A}$, where $\mathcal{A}=\bigcup_{i=1}^{\infty} \mathcal{A}_{i}$, $\mathcal{A}_i$ are pairwise disjoint measurable sets and $c_1, c_2, \ldots$ are real numbers, is defined as:

$\int\limits_{\mathcal{A}} \phi dm = \sum\limits_{i=1}^{n} c_i m(\mathcal{A}_i)$

where, $m(\mathcal{A}_i)$ is the Lebesgue measure of a measurable set $\mathcal{A}_i$.

An extended real value function $f: \mathcal{A}\rightarrow \overline{\mathbb{R}}$ defined on a measurable set $\mathcal{A}\subset\mathbb{R}$ is said to be Lebesgue measurable on $\mathcal{A}$ if $f^{-1} ((c,\infty]) = \{x \in\mathcal{A} : f(x) > c\}$ is a Lebesgue measurable subset of $\mathcal{A}$ for every real number $c$.

If $f$ is Lebesgue measurable and non-negative on $\mathcal{A}$ we define:

$\int\limits_{\mathcal{A}} f dm = \sup \int\limits_{\mathcal{A}} \phi dm$

where the supremum is taken over all simple functions $\phi$ such that $0\leq \phi \leq f$.

The function $f$ is said to be Lebesgue integrable on $\mathcal{A}$ if it’s integral over $\mathcal{A}$ is finite.

The Lebesgue integral is deficient in one respect. The Riemann integral generalizes to the improper Riemann integral to measure functions whose domain of definition is not a closed interval. The Lebesgue integral integrates many of these functions, but not all of them.

# So many Integrals – I

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We all know that, area is  the basis of integration theory, just as counting is basis of the real number system. So, we can say:

An integral is a mathematical operator that can be interpreted as an area under curve.

But, in mathematics we have various flavors of integrals named after their discoverers. Since the topic is a bit long, I have divided it into two posts. In this and next post I will write their general form and then will briefly discuss them.

Cauchy Integral

Newton, Leibniz and Cauchy (left to right)

This was rigorous formulation of Newton’s & Leibniz’s idea of integration, in 1826 by French mathematician, Baron Augustin-Louis Cauchy.

Let $f$ be a positive continuous function defined on an interval $[a, b],\quad a, b$ being real numbers. Let $P : a = x_0 < x_1 < x_2<\ldots < x_n = b$, $n$ being an integer, be a partition of the interval $[a, b]$ and form the sum

$S_p = \sum_{i=1}^n (x_i - x_{i-1}) f(t_i)$

where $t_i \in [x_{i-1} , x_i]f$ be such that $f(t_i) = \text{Minimum} \{ f(x) : x \in [x_{i-1}, x_{i}]\}$

By adding more points to the partition $P$, we can get a new partition, say $P'$, which we call a ‘refinement’ of $P$ and then form the sum $S_{P'}$.  It is trivial to see that $S_P \leq S_{P'} \leq \text{Area bounded between x-axis and function}f$

Since, $f$ is continuous (and positive), then $S_P$ becomes closer and closer to a unique real number, say $kf$, as we take more and more refined partitions in such a way that $|P| := \text{Maximum} \{x_i - x_{i-1}, 1 \leq i \leq n\}$ becomes closer to zero. Such a limit will be independent of the partitions. The number $k$ is the area bounded by function and x-axis and we call it the Cauchy integral of $f$ over $a$  to $b$. Symbolically, $\int_{a}^{b} f(x) dx$ (read as “integral of f(x)dx from a to b”).

Riemann Integral

Riemann

Cauchy’s definition of integral can readily be extended to a bounded function with finitely many discontinuities. Thus, Cauchy integral does not require either the assumption of continuity or any analytical expression of $f$ to prove that the sum $S_p$ indeed converges to a unique real number.

In 1851, a German mathematician, Georg Friedrich Bernhard Riemann gave a more general definition of integral.

Let $[a,b]$ be a closed interval in $\mathbb{R}$. A finite, ordered set of points $P :\{ a = x_0 < x_1 < x_2<\ldots < x_n = b\}$, $n$ being an integer, be a partition of the interval $[a, b]$. Let, $I_j$ denote the interval $[x_{j-1}, x_j], j= 1,2,3,\ldots , n$. The symbol $\Delta_j$ denotes the length of $I_j$. The mesh of $P$, denoted by $m(P)$, is defined to be $max\Delta_j$.

Now, let $f$ be a function defined on interval $[a,b]$. If, for each $j$, $s_j$ is an element of $I_j$, then we define:

$S_P = \sum_{j=1}^n f(s_j) \Delta_j$

Further, we say that $S_P$ tend to a limit $k$ as $m(P)$ tends to 0 if, for any $\epsilon > 0$, there is a $\delta >0$ such that, if $P$ is any partition of $[a,b]$ with $m(P) < \delta$, then $|S_P - k| < \epsilon$ for every choice of $s_j \in I_j$.

Now, if $S_P$ tends to a finite limit as $m(P)$ tends to zero, the value of the limit is called Riemann integral of $f$ over $[a,b]$ and is denoted by $\int_{a}^{b} f(x) dx$

Darboux Integral

Darboux

In 1875, a French mathematician, Jean Gaston Darboux  gave his way of looking at the Riemann integral, defining upper and lower sums and defining a function to be integrable if the difference between the upper and lower sums tends to zero as the mesh size gets smaller.

Let $f$ be a bounded function defined on an interval $[a, b],\quad a, b$ being real numbers. Let $P : a = x_0 < x_1 < x_2<\ldots < x_n = b$, $n$ being an integer, be a partition of the interval $[a, b]$ and form the sum

$S_P = \sum_{i=1}^n (x_i - x_{i-1}) f(t_i), \quad \overline{S}_P =\sum_{i=1}^n (x_i - x_{i-1}) f(s_i)$

where $t_i,s_i \in [x_{i-1} , x_i]$ be such that

$f(t_i) = \text{sup} \{ f(x) : x \in [x_{i-1}, x_{i}]\}$,

$f(s_i) = \text{inf} \{ f(x) : x \in [x_{i-1}, x_{i}]\}$

The sums $S_P$ and $\overline{S}_P$ represent the areas and  $S_P \leq \text{Area bounded by curve} \leq \overline{S}_P$. Moreover, if $P'$ is a refinement of $P$, then

$S_p \leq S_{P'} \leq \text{Area bounded by curve} \leq \overline{S}_{P'} \leq \overline{S}_{P}$

Using the boundedness of $f$, one can show that $S_P, \overline{S}_P$ converge as the partition get’s finer and finer, that is $|P| := \text{Maximum}\{x_i - x_{i-1}, 1 \leq i \leq n\} \rightarrow 0$, to some real numbers, say $k_1, k_2$ respectively. Then:

$k_l \leq \text{Area bounnded by the curve} \leq k_2$

If $k_l = k_2$ , then we have $\int_{a}^{b} f(x) dx = k_l = k_2$.

There are two more flavours of integrals which I will discuss in next post. (namely, Stieltjes Integral and Lebesgue Integral)