Q11. - Algebra 1

Question

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

$\sqrt{\frac{82}{20}}$

Step-by-Step Solution

Answer

The real number $\sqrt{\frac{82}{20}}$ belongs to the set of irrational numbers

Step-1. Apply the concept of the real number system.

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, $q\ne 0$.

Step-2. Examples of the real number system.

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 $\sqrt{2},\sqrt[3]{5},-\sqrt{3}$

Step-3. Analyze the given real number.

Consider the given real number $\sqrt{\frac{82}{20}}$

We can rewrite the given real number as:

$$\begin{gathered} \sqrt{\frac{82}{20}}=\sqrt{\frac{2\times 41}{2\times 10}} \\ =\sqrt{\frac{41}{10}} \end{gathered}$$

Now we will check each subset of real numbers:

1. Natural numbers: They are positive, countable and start from 1. So, $\sqrt{\frac{82}{20}}=\sqrt{\frac{41}{10}}$not being a counting object is not a natural number.
2. Whole numbers: They are natural numbers including 0. So, $\sqrt{\frac{82}{20}}=\sqrt{\frac{41}{10}}$ not being a natural number is not a whole number as well.
3. Integers: They are whole numbers that are both positive and negative. Since $\sqrt{\frac{82}{20}}=\sqrt{\frac{41}{10}}$ is not a whole number, it is not an integer as well.
4. Rational numbers: They can be written in the form $\frac{p}{q}$ where p and q are integers, $q\ne 0$. Since $\sqrt{\frac{82}{20}}$ cannot be written in the exact form, it is not 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, $\sqrt{\frac{82}{20}}$ cannot be written in the form $\frac{p}{q}$, it is an irrational number.

 

Therefore, the real number $\sqrt{\frac{82}{20}}$ belongs to the set of irrational numbers.