Chapter 0.2, Problem 9E

To determine

The set or sets of numbers to which real number $\sqrt{10.24}$ belong.

Answer to Problem 9E

The real number $\sqrt{10.24}$ is a rational number.

Explanation of Solution

Given information:

The real number $\sqrt{10.24}$ .

Formula used:

Real numbers are categorized as follows:

Natural number are the positive numbers (integer) which start from 1.

Whole number are positive numbers which start from 0.

Integers are numbers which is not a fraction (whole number).

Rational numbers are number which are in the form of $\frac{p}{q}$ where p and q are integers, $q\ne 0$ .

Irrational numbers are those number which cannot be expressed in the form of $\frac{p}{q}$ where $q\ne 0$ .

Calculation:

Consider the providedreal number $\sqrt{10.24}$ .

Rewrite it as,

  $\sqrt{10.24}=\sqrt{\frac{1024}{100}}$

Apply square root property and rewrite as,

  $\begin{array}{c}\sqrt{\frac{1024}{100}}=\sqrt{\frac{{32}^2}{{10}^2}}\\ =\frac{32}{10}\\ =\frac{16}{5}\end{array}$

Recall that real numbers are categorized as follows:

Natural number are the positive numbers (integer) which start from 1.

Therefore, it is not a natural number.

Whole number are positive numbers which start from 0.

Therefore, it is not a whole number.

Integers are numbers which is not a fraction (whole number).

Therefore it is not an integer.

Rational numbers are number which are in the form of $\frac{p}{q}$ where p and q are integers, $q\ne 0$ .

Therefore, it is a rational number.

Irrational numbers are those number which cannot be expressed in the form of $\frac{p}{q}$ where $q\ne 0$ .

Therefore, it is not an irrational number.

Thus, the real number $\sqrt{10.24}$ is a rational number.