System of Real Numbers (Natural Numbers, Whole Numbers, Rational Numbers, Irrational Numbers, Real Numbers)

System of Real Numbers

Natural Numbers

• The numbers $(1, 2, 3,..)$ which we use for counting particular objects are called natural numbers or counting numbers or positive integers.
• It is denoted by "$\mathbb{N}$"

    i.e $\mathbb{N}$ $=\{1, 2, 3, 4\}$

Whole Numbers

• The set of natural numbers will be whole numbers if we add "$0$" to it.
• It is denoted by "$\mathbb{W}$".

Example: $\mathbb{W}$ $=\{0, 1, 2, 3,...\}$

The concept of "nothing" or "having nothing" is represented by the number zero.
Nowadays, both a numeric symbol and a concept help us do calculus in solving complicated equations and are the basis of the computer.
Arabs introduced the symbol "$0$" for zero and then the set of whole numbers "$\mathbb{W}$" was discovered.

$\mathbb{N}$ $⊂\mathbb{W}$

Rational Numbers

• A rational number is a number of the form $\frac{a}{b}$ where $a$ and $b$ are integers and $b=0$, is called a rational number.
• It is denoted by "$\mathbb{Q}$". 
• The word quotient is taken from the word quoziente which is an Italian word that means "quotient" since every rational number can be expressed as a quotient or fraction $\frac{p}{q}$ of two co-prime numbers $p$ and $q$, $q\neq 0$.
• It was the first denoted by Peano in $1895$.
• The rational numbers were introduced by Egyptians $(1550 B.C)$.
  $$\mathbb{Q}=\{x|x=\frac{p}{q},\forall p,q\epsilon \mathbb{Z} \wedge q\neq 0\}$$

Examples: $0$ $\frac{3}{4}$ $2.243524....,$ $0.333...$

Irrational Numbers

• An irrational number cannot be expressed in the form of $p/q$ where $p$ and $q$ are integers and $q\neq 0$ is called an irrational number.
• Irrational numbers were discovered by Pythagoras.
• It is denoted by $I$ or $Q'$.

$\mathbb{I}$ $=\{x|x\neq \frac{p}{q},$ $\forall$ $\text{ } p,q\epsilon \mathbb{Z}$ $\text{ } \wedge \text{ } q\neq 0\}$

For example:

$\sqrt{2}$ $=1.4142...,$ $\sqrt{3}$ $=1.7323...,$ $\pi$, $\sqrt{5},$ $\sqrt{6},$ $\sqrt{7},$ $\frac{7}{\sqrt{5}},$ $\frac{\sqrt{3}}{5}....$

Real Numbers

• The union of the set of rational numbers and irrational numbers is the set of real numbers.
• It is denoted by "$\mathbb{R}$".
  
  

$\mathbb{R}$ $=\mathbb{Q}∪\mathbb{I}$  or  $\mathbb{R}$ $=\mathbb{Q}$ $∪\mathbb{Q'}$

OR $\mathbb{Q}$ $=\{x|x\epsilon \mathbb{Q}$ $\vee  x\epsilon \mathbb{Q}'\}$

$\mathbb{Q}$ and $\mathbb{Q}'$ are disjoint sets

The distinction between Rational and Irrational Numbers

Rational Numbers

$1)$ Terminating Decimals

• A decimal number that has a finite number of digits in its decimal part is called terminating. 

Example:  $202.04$, $0.0041$, $0.12304$, $102.206704802$

$2)$ Recurring Decimals

• A recurring decimal or periodic decimal is a decimal in which one or more digits are repeated independently.

Example:  $0.333...,$  $23.2444...$

Let's try to convert in the form of $p/q$

$0.333...$ also written as $\overline {0.3}$ and $23.2\overline{4}$

Let  $x=0.333...$   $(i)$

Multiplying by $10$ on both sides

$10x=3.333...$      $(ii)$

subtract $(i)$ from $(ii)$

$10x-x$ $=3.333...-0.333...$

$9x$ $=3$

$x=$ $\frac{3}{9}$ $=\frac{1}{3}$

$x$ $=\frac{1}{3}$

OR

$x$ $=0.333...$       Ans.

$23.2\overline{4}$

Let $x$ $=23.2444...$

Multiplying by $10$

$10x$ $=232.444...$       $(i)$

Now multiplying by $10$

$100x$ $=2324.444...$           $(ii)$

subtract $(i)$ from $(ii)$

$100x-10x$ $=2324.444...-232.444...$

$90x$ $=2092$

$x$ $=\frac{2092}{90}$ $=\frac{1046}{45}$

$x$ $=\frac{1046}{45}$

OR

$23.2\overline{4}$ $=\frac{1046}{45}$     Ans.

Irrational Numbers

• Non-Recuring and Non-terminating Decimal Fractions
• A non-recuring and non-terminating decimal fraction is a decimal fraction that neither terminates nor it is recurring.

Example:

$\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, $\sqrt{6}$, $\sqrt{7}$, $\sqrt{8}$, $\sqrt{10}$,   $\pi$

The value of $\pi$ correct to $5$ lac decimal places have been determined with the help of a computer.