Properties of the Real number (Closure Property, Commutative Property, Associative Property, Additive Identity, Additive Inverse, Multiplicative Identity, Multiplicative Inverse) w.r.to addition and multiplication

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 $ $\frac{1}{2}$