Numbers - The Basic Introduction

If mathematics was a language, logic was the grammar, numbers should have been the alphabet.

There are many types of numbers we use in mathematics, but at a broader aspect we may categorize them in two categories:

1. Countable Numbers
2. Uncountable Numbers

The numbers which can be counted in nature are called Countable Numbers and the numbers which can not be counted are called Uncountable Numbers.

Well, this is not the correct way to classify the bunch of types of numbers. We have some formal names for special types of numbers, like Real numbers, Complex Numbers, Rational Numbers, Irrational Numbers etc.. We shall discuss these non-interesting numbers (let me say them non-interesting) at first and then some interesting numbers(those numbers are really interesting to learn). Although in this post I have concisely described the classification, I will rigorously discuss them later.
Let me start this discussion with the memorable quote by Leopold Kronecker:

"God created the natural numbers, and all the rest is the work of man."

What does it mean? What did Kronecker think when he made this quote? Why is this quote true? ---First part of this article is based on this discussion.
Actually, he meant to say that all numbers, like Real Numbers, Complex Numbers, Fractions, Integers, Non-integers etc. are made up of the numbers given by God to the human. These God Gifted numbers are actually called Natural Numbers. Natural Numbers are the numbers which are used to count things in nature.

Eight pens, Eighteen trees, Three Thousands people etc. are measure of natural things and thus 'Eight', 'Eighteen', 'Three Thousands' etc. are called natural numbers and we represent them numerically as '8', '18', '3000' respectively. So, if 8, 18, 3000 are used in counting natural things, are natural numbers. Similarly, 1, 2, 3, 4, and other numbers are also used in counting things ---thus these are also Natural Numbers.

Let we try to form a set of Natural Numbers. What will we include in this set?

1? (yes!).
2? (yes).
3? (yes).
1785? (yes)
...and so on.

This way, after including all elements we get a set of natural numbers {1, 2, 3, 4, 5, ...1785, ..., 2011,....}. This set includes infinite number of elements. We represent this set by Borbouki's capital letter N, which looks like $ \mathbb{N}$ or bold capital letter N , where N stands for NATURAL. We will define the set of all natural numbers as:

$ \mathbb{N} := \{ 1, 2, 3, 4, \ldots, n \ldots \}$ .

It is clear from above set-theoretic notation that $ n$ -th element of the set of natural numbers is $ n$ .
In general, if a number $ n$ is a natural number, we right that $ n \in \mathbb{N}$ .
Please note that some mathematicians (and Wolfram Research) treat '0' as a natural number and state the set as $ \mathbb{N} :=\{0, 1, 2, \ldots, n-1, \ldots \}$ , where $ n-1$ is the nth element of the set of natural numbers; but we will use first notion since it is broadly accepted.

Now we shall try to define Integers in form of natural numbers, as Kronecker's quote demands. Integers (or Whole numbers) are the numbers which may be either positives or negatives of natural numbers including 0.
Few examples are 1, -1, 8, 0, -37, 5943 etc.
The set of integers is denoted by $ \mathbb{Z}$ or $ \mathbf{Z}$ (here Z stands for 'Zahlen', the German alternative of integers). It is defined by
$ \mathbb{Z} := \{ \pm n: n \in \mathbb{N} \} \cup \{0\}$
i.e., $ \mathbb{Z} := \{\ldots -3, -2, -1, 0, 1, 2, 3 \ldots \}$ . Kronecker's quote was therefore, later modified as

"God created the integers, and all the rest is the work of man."

Now please go a step back and again consider the statement of Kronecker. One may ask that how could we prepare the integer set $ \mathbb{Z}$ by the set $ \mathbb{N}$ of natural numbers?

The construction of $ \mathbb{Z}$ from $ \mathbb{N}$ is motivated from the requirement that every integer can be expressed as difference of two positive integers (i.e., Natural Numbers). Let $ a,b,c,d \in \mathbb{N}$ and a relation ρ is defined on $ \mathbb{N} \times \mathbb{N}$ by $ (a,b) \rho (c,d)$ if and only if $ a+d = b+c$ . The relation ρ is an equivalence relation and the equivalence classes under ρ are called integers and defined as $ \mathbb{Z} := \mathbb{N} \times \mathbb{N} /\rho$ . Now we can define set of integers by an easier way, as $ \mathbb{Z}:= \{a-b; \ a,b \in \mathbb{N}\}$ . Thus an integer is a number which can be produced by difference of two or more natural numbers. And similarly as converse definition, positive integers are called Natural Numbers.

After Integers, we head to rational numbers. Say it again-- 'ratio-nal numbers' --numbers of ratio.

A rational number $ \frac{p}{q}$ is defined as a ratio of an integer p and a non-zero integer q. (Well that is not a perfect definition, but as an introduction it is great for understanding.) The set of rational numbers is defined by $ \mathbb{Q}$ .
Once integers are formed, we can form Rational (and Irrational numbers: numbers which are not rational ) using integers.
We consider an ordered pair $ (p,q):=\mathbb{Z} \times (\mathbb{Z} \setminus \{0 \})$ and another ordered pair $ (r,s):=\mathbb{Z} \times (\mathbb{Z} \setminus \{0\})$ and define a relation ρ by $ (p,q) \rho (r,s) \iff ps=qr$ for $ p,q,r,s \in \mathbb{Z}, \ q, r \ne 0$ . Then ρ is an equivalence relation of rationality, class (p,q). The set $ \mathbb{Z} \times (\mathbb{Z} \setminus \{0\})/\rho$ is denoted by $ \mathbb{Q}$ (and the elements of this set are called rational numbers).
In practical understandings, the ratio of integers is a phrase which will always help you to define the rational numbers. Examples are $ \frac{6}{19}, \ \frac{-1}{2}=\frac{-7}{14}, \ 3\frac{2}{3}, \ 5=\frac{5}{1} \ldots$ . Set of rational numbers includes Natural Numbers and Integers as subsets.
Consequently, irrational numbers are those numbers which can not be represented as the ratio of two integers. For example $ \pi, \sqrt{3}, e, \sqrt{11}$ are irrationals.

The set of Real Numbers is a relatively larger set, including the sets of Rational and Irrational Numbers as subsets. Numbers which exist in real and thus can be represented on a number line are called real numbers. As we formed Integers from Natural Numbers; Rational Numbers from Integers, we can form the Real numbers by Rational numbers.
The construction of set $ \mathbb{R}$ of real numbers from $ \mathbb{Q}$ is motivated by the requirement that every real number is uniquely determined by the set of rational numbers less than it. A subset $ L$ of $ \mathbb{Q}$ is a real number if L is non-empty, bounded above, has no maximum element and has the property that for all $ x, y \in \mathbb{Q}, x < y$ and $ y \in L$ implies that $ x \in L$ . Real numbers are the base of Real Analysis and detail study about them is the case of study of Real analysis.
Examples of real numbers include both Rational (which also contains integers) and Irrational Numbers.

The square root of a negative number is undefined in one-dimensional number line (which includes real numbers only) and is treated to be imaginary. The numbers containing or not containing an imaginary number are called complex numbers.
Some very familiar examples are $ 3+\sqrt{-1}, \sqrt{-1} =i, \ i^i$ etc. We should assume that every number (in lay approach) is an element of a complex number. The set of complex numbers is denoted by $ \mathbb{C}$ . In constructive approach, a complex number is defined as an ordered pair of real numbers, i.e., an element of $ \mathbb{R} \times \mathbb{R}$ [i.e., $ \mathbb{R}^2$ ] and the set as $ \mathbb{C} :=\{a+ib; \ a,b \in \mathbb{R}$ .
We verified Kronecker's quote and shew that every number is sub-product of positive integers (natural numbers) as we formed Complex Numbers from Real Numbers; Real Numbers from Rational Numbers; Rational Numbers from Integers and Integers from Natural Numbers. //
Now we reach to explore some interesting kind of numbers. There are millions in name but few are the follow:
Even Numbers: Even numbers are those integers which are integral multiple of 2. $ 0, \pm 2, \pm 4, \pm 6 \ldots \pm 2n \ldots$ are even numbers.

Odd Numbers: Odd numbers are those integers which are not integrally divisible by 2. $ \pm 1, \pm 3, \pm 5 \ldots \pm (2n+1) \ldots$ are all odd numbers.

Prime Numbers: Any number $ p$ greater than 1 is called a prime number if and only if its positive factors are 1 and the number $ p$ itself.
In other words, numbers which are completely divisible by either 1 or themselves only are called prime numbers. $ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 \ldots$ etc. are prime numbers or Primes. The numbers greater than 1, which are not prime are called Composite numbers.
Twin Primes: Consecutive prime numbers differing by 2 are called twin primes. For example 5,7; 11,13; 17,19; 29,31; ... are twin primes.

Pseudoprimes: Chinese mathematicians claimed thousands years ago that a number $ n$ is prime if and only if it divides $ 2^n -2$ . In fact this conjecture is true for $ n \le 340$ and false for upper numbers because first successor to 340, 341 is not a prime ($ 31 \times 11$ ) but it divides $ 2^{341}-2$ . This kind of numbers are now called Pseudoprimes. Thus, if n is not a prime (composite) then it is pseudoprime $ \iff n | 2^n-2$ (read as 'n divides 2 powered n minus 2'). There are infinitely many pseudoprimes including 341, 561, 645, 1105.

Carmichael Numbers or Absolute Pseudoprimes: There exists some pseudoprimes that are pseudoprime to every base $ a$ , i.e., $ n | a^n -a$ for all integers $ a$ . The first Carmichael number is 561. Others are 1105, 2821, 15841, 16046641 etc.

e-Primes: An even positive integer is called an e-prime if it is not the product of two other even integers. Thus 2, 6, 10, 14 ...etc. are e-primes.

Germain Primes: An odd prime p such that 2p+1 is also a prime is called a Germain Prime. For example, 3 is a Germain Prime since $ 2\times 3 +1=7$ is also a prime.
Relatively Prime: Two numbers are called relatively prime if and only their greatest common divisor is 1. In other words, if two numbers are such that no integer, except 1, is common between them when factorizing. For example: 7 and 9 are relatively primes and same are 15, 49.

Perfect Numbers: A positive integer n is said to be perfect if n equals to the sum of all its positive divisors, excluding n itself. For example 6 is a perfect number because its divisors are 1, 2, 3 and 6 and it is obvious that 1+2+3=6. Similarly 28 is a perfect number having 1, 2, 4, 7, 14 (and 28) as its divisors such that 1+2+4+7+14=28. Consecutive perfect numbers are 6, 28, 496, 8128, 33550336, 8589869056 etc.

Mersenne Numbers and Mersenne Primes: Numbers of type $ M_n=2^n-1; \ n \ge 1$ are called Mersenne Numbers and those Mersenne Numbers which happen to be Prime are called Mersenne Primes. Consecutive Mersenne numbers are 1, 3 (prime), 7(prime), 15, 31(prime), 63, 127.. etc.

Catalan Numbers: The Catalan numbers, defined by $ C_n = \dfrac{1}{n+1} \binom{2n}{n} = \dfrac{(2n)!}{n! (n+1)!} \ n =0, 1, 2, 3 \ldots$ form the sequence of numbers 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, ...

Triangular Number: A number of form $ \dfrac{n(n+1)}{2} \ n \in \mathbb{N}$ represents a number which is the sum of n consecutive integers, beginning with 1. This kind of number is called a Triangular number. Examples of triangular numbers are 1 (1), 3 (1+2), 6 (1+2+3), 10(1+2+3+4), 15(1+2+3+4+5) ...etc.

Square Number: A number of form $ n^2 \ n \in \mathbb{N}$ is called a square number.
For example 1 ($ 1^2$ ), 4 ($ 2^2$ ), 9($ 3^2$ ), 16 ($ 4^2$ )..etc are Square Numbers.

Palindrome: A palindrome or palindromic number is a number that reads the same backwards as forwards. For example, 121 is read same when read from left to right or right to left. Thus 121 is a palindrome. Other examples of palindromes are 343, 521125, 999999 etc.