Question

Name the sets of numbers to which each number belongs.

$-4.\overline{2}$

Short answer

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

Step-by-step solution

- 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.

- 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.

- State the conclusion.

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

Rational (Q) and Real (R).