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 *n*th 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.

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