Name the set or sets of numbers to which each real number belongs.

$\sqrt{10.24}$

Real numbers include all the numbers except complex numbers and have the following five subsets:

1. Natural numbers: Includes counting objects and starting from 1.
2. Whole numbers: Includes the set of natural numbers along with 0.
3. Integers: Z = Includes numbers that are not fraction (positive and negative whole numbers)
4. Rational numbers: Includes the numbers which can be written in the form of $\frac{p}{q}$where p and q are integers, $q\ne 0$.
5. Irrational numbers: Includes numbers that cannot be written in the form of $\frac{p}{q}$where p and q are integers, $q\ne 0$.

1. Natural numbers:$\left\{1,2,3,4,...\right\}$
2. Whole numbers:$\left\{0,1,2,3,4,...\right\}$
3. Integers: Measurement of debts, temperatures, etc., fall under the set of integers $\left\{...,-3,-2,-1,0,1,2,3,...\right\}$
4. Rational numbers: If we cut a cake into equal pieces, then we may have a piece that represents a fraction like $\frac{5}{6},1.5\left(=\frac{3}{2}\right),\frac{6}{9},\frac{8}{3}$
5. Irrational numbers: The numbers that are square roots of positive rational numbers, cube roots of rational numbers, etc., such as $\sqrt{2},\sqrt[3]{5},-\sqrt{3}$

Consider the given real number$\sqrt{10.24}$

Rewrite the given real number using the square root property:

$$\begin{gathered} \sqrt{10.24}=\sqrt{\frac{1024}{100}} \\ =\sqrt{\frac{{32}^2}{{10}^2}} \\ =\frac{32}{10} \\ =\frac{16}{5} \end{gathered}$$

Now we will check each subset of real numbers:

1. Natural numbers: They are positive, countable, and start from 1. So, $\sqrt{10.24}=\frac{16}{5}$not being a counting object is not a natural number.
2. Whole numbers: They are natural numbers including 0. So, $\sqrt{10.24}=\frac{16}{5}$not being a natural number is not a whole number as well.
3. Integers: They are numbers that are not a fraction. Since $\sqrt{10.24}=\frac{16}{5}$is a proper fraction, it is not an integer.
4. Rational numbers: They can be written in the form $\frac{p}{q}$where p and q are integers, $q\ne 0$. Since $\sqrt{10.24}=\frac{16}{5}$is the exact form, it is a rational number.
5. Irrational numbers: They cannot be written in the form $\frac{p}{q}$where p and q are integers, $q\ne 0$. Since from the above point, $\sqrt{10.24}$can be written in the form $\frac{p}{q}$, it is not an irrational number.

Therefore, the real number $\sqrt{10.24}$belongs to the set of rational numbers.