Properties of the Real number


Properties w.r.t addition

$i)$  Closure Property

  • Any two real numbers are added together to form a real number.

i.e. $x$, $y\varepsilon$ $\mathbb{R}$ $\Rightarrow$ $x+y$ $\varepsilon$ $\mathbb{R}$

Example:

 $16$, $24\varepsilon \mathbb{R}$   $20+24$ $=40\varepsilon\mathbb{R}$

$7$, $\sqrt{5}$ $\varepsilon\mathbb{R}$  $7+$ $\sqrt {5}$ $=7+2.236…$ $=9.236…$ $\varepsilon \mathbb{R}$

$ii)$    Commutative Property

  • The Sum of any two real numbers in any order is the same.

$x+y$ $=y+x$, $\forall$ $x,y$ $\varepsilon \mathbb{R}$

Examples:
$a)$  $2.6+7.2$ $=9.8$ $=7.2+2.6$
$b)$    $\sqrt{2}$ $+\sqrt{3}$ $=\sqrt{3}$ $+\sqrt{2}$


    $iii)$    Associative Property

  • The sum of any three numbers in any order remains the same.

$x$ $+(y+z)$ $=(x+y)$ $+z$, $\forall$  $x,y,z$ $\epsilon\mathbb{R}$

Example:

$a)$  $5$ $+(7+8)$ $=20$ $=(5+7)$ $+8$

$b)$  $\sqrt{3}$ $+(\sqrt{6} +\sqrt{7})$ $=(\sqrt{3}+\sqrt{6})$ $+\sqrt{7}$


$iv)$    Additive Identity

  • There exists a unique real number $0$, called the additive identity such that

$x+0$ $=x$ $=0+x$    $\forall$ $x$ $\epsilon\mathbb{R}$

Examples

$a)$    $0.4+0$ $=0.4$ $=0+0.4$

$b)$    $\sqrt{2}+0$ $=\sqrt{2}$ $=0+\sqrt{2}$


$v)$    Additive Inverse

  • If the sum of two numbers is zero, then the numbers are called the additive inverse of each other.
  • The additive inverse of any number can be found by changing the sign of the number.

$x+x'$ $=0$ $=x'+x$    $\forall$ $x,x'$ $\epsilon \mathbb{R}$

e.g.

$a)$  $2+(-2)$ $=0(-2)+2$

$b)$  $2\sqrt{5}+(-2\sqrt{5})$ $=0$ $=(-2\sqrt{5})+(2\sqrt{5})$


Properties w.r.t. Multiplication


$i)$  Closure Property

  • Any two real numbers can be multiplied together to get a real number.

i.e. $x,y$ $\varepsilon \mathbb{R}$ $\Rightarrow$   $x.y$ $\varepsilon\mathbb{R}$

Example:

 $1)$  $0.6$, $0.4\varepsilon \mathbb{R}$   $\Rightarrow$    $(0.6)(0.4)$ $=0.24$ $\varepsilon\mathbb{R}$

$2)$  $\sqrt{2}$, $\sqrt{3}$ $\varepsilon\mathbb{R}$   $\Rightarrow$  $\left( \sqrt{2} \right)$ $\left( \sqrt{3} \right)$ $=\sqrt{6}$ $\varepsilon \mathbb{R}$


$ii)$   Commutative Property

  • The product of any two real numbers in any order the result will be the same

$xy$ $=yx$, $\forall$ $x,y$ $\varepsilon \mathbb{R}$

Examples:
$a)$  $\sqrt{2}$ $\times \sqrt{3}$ $=\sqrt{6}$ $=\sqrt{3}$ $\times \sqrt{2}$
$b)$  $0.2$ $\times 0.6$ $=0.12$ $=0.6$ $\times 0.2$


     $iii)$  Associative Property

  • The product of any three real numbers in any order remains the same

$x.(y.z)$ $=(x.y).z$, $\forall$  $x,y,z$ $\epsilon\mathbb{R}$

OR    $x(yz)$ $=(xy)z$

Example:

$a)$  $0.2$ $\times $ $\left( \sqrt{3}\times \frac{5}{7} \right)$ $=0.247$ $=\left( 0.2\times \sqrt{3} \right)$ $\times \frac{5}{7}$

$b)$  $\left( \sqrt{3}\times 4 \right)$ $\times \sqrt{6}$ $=16.97$ $=\sqrt{3}$ $\times \left( 4 \times \sqrt{6} \right)$


$iv)$  Multiplicative Identity

  • There exists a unique real number $1$, called the multiplicative identity such that

$x\times 1$ $=x$ $=1$ $\times x$    $\forall$ $x$ $\epsilon\mathbb{R}$

Examples

$a)$  $\sqrt{5}$ $\times 1$ $=\sqrt{5}$ $=1$ $\times $ $\sqrt{5}$

$b)$    $0.21502$ $\times 1$ $=0.21502=$ $1\times 0.21502$


$v)$   Multiplicative Inverse

  • If the product of two numbers is one, then the numbers are called the multiplicative inverse of each other.
  • The multiplicative inverse of any number can be found by changing the numerator into the denominator and the denominator into the numerator.

$x+x'$ $=0$ $=x'+x$     $\forall$ $x,x'$ $\epsilon \mathbb{R}$

e.g.

$a)$  $\frac{2}{\sqrt{3}}$ $\times$ $ \frac{\sqrt{3}}{2}$ $=1$ $=\frac{\sqrt{3}}{2}$ $\times $ $\frac{2}{\sqrt{3}}$

$b)$  $\frac{1}{2}$ $\times $ $\frac{2}{1}$ $=1$ $=\frac{2}{1}$ $\times $ $\frac{1}{2}$