Chapter 1, Problem 18SGR

To determine

To name: the sets of numbers to which the given number belongs.

Answer to Problem 18SGR

The given number belongs to the Set of Rational numbers $\left(\mathbb{Q}\right)$ , and set of real numbers $\left(ℝ\right)$ .

Explanation of Solution

Given information:

A number is given as $1.\overline{3}$ .

Concept used:

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

Calculation:

Consider the number $1.\overline{3}$ .

The number can be written as

  $1.\overline{3}=\mathrm{1.33333...}$

The decimal expression is repeating.

The above number is a Rational number, as well as a real number.

Hence, the given number belongs to the Set of Rational numbers $\left(\mathbb{Q}\right)$ , and set of real numbers $\left(ℝ\right)$ .