Chapter 1.2, Problem 56HP

To determine

Whether an irrational number is a real number with in a radical sign.

Answer to Problem 56HP

An irrational number is a rational number with in a radical sign.

An irrational number is a positive real number with in a radical sign.

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	$\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{33},\mathrm{...}$

Consider the number $\sqrt{-11}$ .

Here, $-11$ is a real number.

But $\sqrt{-11}$ is not an irrational number.

But, $\sqrt{11}$ is an irrational number.