# Introduction of Numbers for non-mathematician

## Introduction
**Numbers govern the whole universe**. People counting objects using natural numbers, describing distance using by real numbers. Even for geometry, Renatus Cartesius established the analytic geometry describing positions of a point by coordinates, which is a tuple of numbers. Numbers and equations a so powerful so physics use equations to describe motion and action of objects, data analysists establish models to predict the future trend. Everything in our universe, are essentially measured by numbers.

## What are types of numbers?
People use numbers to understand the universe. However, mathematician develop Number Theory to understand numbers. Up to now people has discovered all kinds of numbers including natural numbers, integers, rational numbers, real numbers, complex numbers. All of them can be described as **objects that could use addition, subtraction, multiplication, division for calculation**. Whenever we have a collection of calculable objects with those four operators, we call them **numbers**. Mathematician give those kind of collection a name -- **field**. A number is an object in a **field**. For example, the collection of real numbers called the *real number field*. The study of basic properties of each number field, is called the **Algebraic Number Theory**.

## The name card of number types -- L function
To study number fields, we should specify *which number fields are we talking about*. If number fields are residents in a big country, mathematicians associates an **ID card**, the **L-function**, to each residents in this country. The **L-function** for the real number field is the well-known Gamma function, which plays an important role in statistics. Gamma functions are also the birthplace of the most important real numbers such as $\pi$ and $e$, which plays an **important role** in **probabilities and statistics in our world**. The **L-function** for the rational numebrs is the Riemann Zeta function, which could describe the solution of many integer-based problems such as prime distribution and probability questions regarding to integers. In fact, if we are living in a world described by other number fields, then our statistics and probability theory will be reformulated by quantities essentially coming from the **L-function** of that number field. Theories dealing with those problems, is called the **Analytic Number Theory**.

## Number types with its desendents -- Perspective of L-function
The L-function has a lot of information of number fields. For example, Rational numbers give birth to our real numbers, but at the same time, it also create so-called **p-adic** numbers. In one word, **p-adic** numbers are brothers of our real numbers and they have the same mother -- rational numbers. This fact has reflected on their name card, the $L$-function. The $L$-function for the rational numbers are given by the $\xi$-function defined by
$$
\xi(x)=\frac12\pi^{-\frac s2}s(s-1)\Gamma(\frac s2)\left(1-\frac1{2^s}\right)^{-1}\left(1-\frac1{3^s}\right)^{-1}\left(1-\frac1{5^s}\right)^{-1}\left(1-\frac1{7^s}\right)^{-1}\cdots
$$
where numbers in the infinite product $2,3,5,7,\cdots$ run through all prime numbers. This $L$-function is a product of $L$-functions of its children. For example, the $L$-function of real numbers

$$
\frac12\pi^{-\frac s2}s(s-1)\Gamma(\frac s2)
$$

The $L$-function of p-adic numbers for each prime $p$ is
$$
\left(1-\frac1{p^s}\right)^{-1}.
$$
In real numbers, we can calculate integrations and derivatives summing up infinite sums, talking about probability distributions. In p-adic numbers, we also have the same notion. Explicitly, p-adic numbers are base p-numbers with finitely many decimal digit but possibly infinitely many integer digits. This makes the sum $1+2+4+8+16+\cdots$ converges to $-1$ in 2-adic numbers but the same infinite sum does not make sense in real numbers. In fact, given any infinite series of rational numbers, they could be converge and diverge in different types of numbers. It is true that if one infinite sum converges in a type of number, it must diverge in another type. No infinite sum could make all children happy. If a infinite sum of rational numbers converges in real numbers, it must diverge in some p-adic numbers. If a infinite sum converge for all p-adic numbers, it must diverge in real numbers.

## Numbers with Geometry
It might be a suprise to see **numbers are essentially linked with geometry**. This idea were firstly introduced in the **analytic geometry**, where numbers are used in coordinates to describe positions of a point. Now let us think about a geometric problem to understand real and complex numbers: *how many points are there in the intersections of two circles of radius one?* Let's start with two circles intersecting exactly at two points. Then we move one of them slowly and continously. As we are moving one circle away from another in the plane, the two intersection points are running towards each other until it collapse to one single point and finally disappears. This process tell us the intersection points of two circle could be $2$, $1$ or $0$. Now let's do another experiment, at each moment, we link two centers by a segment, then draw a line perpendicular to the segment through the midpoint of the segment. By our construction, this line exists no matter where those two circle is. But this line can also be defined by linking intersection points of two circles. In other words, the intersection point of two circles always exists because the line passing through them is always **visible** to us. We should understand that when two points collapse to each other, they are two points located at the same location. When there are no points in intersections, we should think those two points are **invisible** to us. Using analytic geometry, we see the coordinates of the insisible points are **complex numbers**, the reason for invisibility is the non-existence of the position described by coordinates with $\sqrt{-1}$. In short, coordinates of visible points are in the real numbers field, while coordinates of invisible points are in the complex numbers field. Generally speaking, we shuld think that **each field associates to a type of point. The possition of the point can only be described by numbers in this field.** The upgraded theory of analytic geometry, where we take all those kinds of points into consideration, is called the **Algebraic Geometry**.

## Symmetry of Numbers
Complex coordinated points are invisible to us, but why are the line linking through them always visible? The reason is **symmetry of numbers**. Equations with complex numbers have a certain **relfection action** described by interchanging $\sqrt{-1}$ with $-\sqrt{-1}$. Explicitly, this means whenever we have a eqaution with $\sqrt{-1}$ appears, changing $\sqrt{-1}$ everywhere by $-\sqrt{-1}$ will not change the equality. This could be think as looking the equations in a **mirrow** and you got another equation. In this point of view, real numbers are **symmetric numbers** which means they appears as the same thing in the mirrow. In other words, **non-real complex numbers are ansymmetric numbers**. Points are visible if they are in a symmetric location, while they become invisible if this symmetry breaks up. If two circles intersects at two points, then reflecting their coordinates do not change their position. Since their coordinates are symmetric, they are visible. For two circles intersects at nowhere, the reflection of the coordinates interchanges the two intersection points, therefore they become invisible. But note that the line linking them does not affect, because reflection of the coordinates of those two points only interchanges those two points. In the process they determine the same line. Since the line have not been changed in the process, it is symmetric, and become a **visible** line. In one word, visibility and invisibility are governed by symmetries. The study of symmetries, is called the **Representation Theory**.

In one word, different number fields describes different types of points. Each types of points has their own symmetries. Are numbers just a **phenomenon of symmetries**? Can types of symmetries detemine types of numbers? The study of numbers by understanding its symmetries, is called the **Langlands program**.

## Visualizing Number Symmetry by Geometric Objects
The only way to visualize symmetries of numbers, is to consider points coordinated by those numbers. Describing the symmetry of points is impossible until we put them into a geometric objects. Just like the invisible intersections of two far-away-cirlces. As we mentioned, those complex coordinated points are invisible because certain coordinate-reflection change their position. Points reflects itself in a strange way, which exceeded our ability of understanding. To trying to understand, we use symmetries of the original graph. The picture of two same-sized circles has two symmetric axis. The graph remain unchanged if we reflect it by the **center line** defined by adjoing the center of the circle, or by the **plumb line** in the middle of them. The **center line** reflection has the same effect as the coordinate reflection of invisible complex-coordinated points. Therefore the reflection of complex numbers can be described by realizing them as the intersections of two circles, and then reflecting the whole graph by the **center line**. The theory of constructing geometric objects to visualize symmetries of numbers, is called the **Arithmetic Geometry**.