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.
Properties of Real Numbers
Property |
Example |
---|---|
Closure Property of Addition |
$21 + 6 = 27$ |
Closure Property of Multiplication |
$21 * 6 = 27$ |
Commutative Property of Addition |
$101 + 56 = 56 + 101$ |
Commutative Property of Multiplication |
$21 * 45 = 45 * 21$ |
Distributive Property |
|
Density Property |
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) $\sqrt{\dfrac{a}{b}}$ = $\dfrac{\sqrt{a}}{\sqrt{b}}$