Chapter 1.2, Problem 57HP

To determine

Whether Closure Property of Multiplication applicable on irrational number.

Answer to Problem 57HP

Closure Property of Multiplication is not applicable on irrational number.

Explanation of Solution

For any real numbers the sets are tabulated below:

Notation	Name of set	Examples
$\mathbb{N}$	Set of Natural numbers	$1,2,3,4,5,\mathrm{...}$
$W=0+\mathbb{N}$	Set of Whole numbers	$0,1,2,3,4,5,\mathrm{...}$
$\mathbb{Z}$	Set of Integers	$0,1,-2,3,4,5,18,-\mathrm{9...}$
$\mathbb{Q}$	Set of Rational numbers	$0,1,-2,3,4,5,18,-9,\frac{1}{2},\frac{5}{7},-\frac{3}{4},\mathrm{...}$
$I$	Set of Irrational numbers	$\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{33},\mathrm{...}$

Property	Addition	Multiplication
Commutative	$a+b=b+a$	$a\cdot b=b\cdot a$
Associative	$\left(a+b\right)+c=a+\left(b+c\right)$	$\left(a\cdot b\right)\cdot c=a\cdot \left(b\cdot c\right)$
Identity	$a+0=0+a=a$ 0 is identity element for addition	$a\cdot 1=1\cdot a=a$ 1 is identity element for Multiplication
Inverse	$a+\left(-a\right)=\left(-a\right)+a=0$   $\left(-a\right)$ is inverse element for addition	$a\cdot \frac{1}{a}=\frac{1}{a}\cdot a=1$   $\frac{1}{a}$ is inverse element for Multiplication
Closure	$a+b$ is real number	$a\cdot b$ is real number
Distributive	$\begin{array}{l}a\left(b+c\right)=ab+bc\\ \left(b+c\right)a=ba+ca\end{array}$	$\begin{array}{l}a\left(b+c\right)=ab+bc\\ \left(b+c\right)a=ba+ca\end{array}$

Closure Property of Multiplication is not applicable on irrational number.

Consider the irrational numbers $\sqrt{11}$ and $\sqrt{11}$ .

Now, multiply the numbers as shown:

  $\begin{array}{c}\sqrt{11}\cdot \sqrt{11}=\sqrt{11\times 11}\\ =11\end{array}$

But, 11 is not an irrational number.

So, Closure Property of Multiplication is not applicable on irrational number.