Q64. - Algebra 2

Question

Name the sets of numbers to which each number belongs.

 

$-4.\overline{2}$

Step-by-Step Solution

Answer

$-4.\overline{2}$ belongs to Rational (Q) and Real (R).

Step 1 - Define real numbers.

A number that can be stated as an infinite decimal expansion. 

 

Real numbers are numbers that include both rational and irrational numbers.

Step 2 - State the explanation.

All numbers that is whole numbers, integers, natural number, rational and irrational numbers are real numbers.

 

Here, the number is $-4.\overline{2}$.

 

Natural numbers are $\left\{1,2,3,...\right\}$ is not a natural number.

 

Whole numbers are $\left\{0,1,2,3,...\right\}$. Then $-4.\overline{2}$ is not a whole number.

 

Integers are $\left\{...,-3,-2,-1,0,1,2,3,...\right\}$. Then $-4.\overline{2}$ is not an integer.

 

Since $-4.\overline{2}$ can be expressed as $\frac{m}{n}$ form that is:

 

Let $x=-4.222$                                                                                                 …… (1)

 

Since one decimal is given, so multiply both sides of equation 1 by 10.

 

That is $10x=-42.222$                                                                                       …… (2)

 

Subtract equation 1 from equation 2,

 

$\begin{array}{c}9x=38\\ x=\frac{38}{9}\end{array}$

 

then it is a rational number.

 

As it is rational number then it is a real number.

Step 3 - State the conclusion.

Therefore, the set of numbers to which $-4.\overline{2}$ belongs is:

 

Rational (Q) and Real (R).