We discuss the properties of real numbers including integer, rational and irrational numbers. This class of numbers plays a key role in algebra as well as in real analysis. For example, the density of rational numbers in the set of real numbers is the key to proving many known mathematical results.

A concise summary of properties of real numbers

We note that the modern real analysis is mainly based on the set of real numbers. In fact, this set enjoys very important properties that can be translated to the other general sets. Let us discover together this magic set.

The set of natural numbers, positive integers

We use natural numbers every day in real life. These numbers are 1,2,3, … For example to count money, to count the population of a city or a county. These numbers are called natural numbers and we have collected them in a set of the form \begin{align*} \{1,2,3,4,\cdots,1000,\cdots\}. \end{align*} The complete natural numbers set is the following set \begin{align*} \{0,1,2,3,4,\cdots,1000,\cdots\}. \end{align*} We have only added 0 to the first natural numbers set. For simplicity, this set will be denoted by $\mathbb{N}$. Natural numbers are called positive numbers ”think of a positive balance in a bank account”. One may wonder why we added zero to the set of natural numbers. Really, zero is very important in all of the algebra ”and of course in real life”.

The zero is the transition point from positive to negative ”if your bank account is exhausted, your bank tells you that you have 0 dollars, and sometimes the bank allows you to withdraw another sum of money so you will be debited it i.e. a negative balance, so to go from positive to negative, you have to go through zero”.

We recall that if $n$ is a natural number, then \begin{align*} \underset{n\;\text{times}}{\underbrace{1+1+\cdots+1}}=n. \end{align*}