Vachmi


What are Real Numbers?
A number whose square is non-negative is called a real number.

Real numbers include all rational numbers and all irrational numbers.
Rational numbers, as we know, include integers such as 6, -8 etc and fractions such as $\dfrac{7}{5}$.
Irrational numbers are the ones which can neither be expressed as a terminating decimal number nor as a repeating decimal number. e.g. 0.01001000100001...

Thus, every real number is either a rational number or an irrational number.

Real Numbers diagram
Properties of Real Numbers

Property

Example

Closure Property of Addition
The sum of two real numbers is always a real number.

$21 + 6 = 27$

Closure Property of Multiplication
The product of two real numbers is always a real number.

$21 * 6 = 27$

Commutative Property of Addition
$a + b = b + a$ is true for all real numbers.

$101 + 56 = 56 + 101$

Commutative Property of Multiplication
$a * b = b * a$ is true for all real numbers.

$21 * 45 = 45 * 21$

Distributive Property
Multiplication distributes over addition
$a * (b + c) = a*b + a*c$
$(a + b) * c = a*c + b*c$



$4 * (3 + 5) = 4*3 + 4*5$
$(6 + 7) * 2 = 6*2 + 7*2$

Density Property
Between any two real numbers, there exist infinitely many real numbers.

For example, there exist infinitely many real numbers between $\dfrac{3}{5}$ and $\dfrac{4}{5}$


Two more important points to be noted with respect to the real numbers are as follows

1. For every positive real number $x$, there exists $\sqrt{x}$ and $\sqrt{x}$ is also a positive real number.

2. For all positive real numbers $a$ and $b$,

(i)$\sqrt{ab}$ = $\sqrt{a} * \sqrt{b}$

(ii) $\sqrt{\dfrac{a}{b}}$ = $\dfrac{\sqrt{a}}{\sqrt{b}}$

Solved Examples

Please refer NCERT Real Numbers solved examples here