Suggested languages for you:

Americas

Europe

Q11.

Expert-verifiedFound in: Page 10

Book edition
Student Edition

Author(s)
Carter, Cuevas, Day, Holiday, Luchin

Pages
801 pages

ISBN
9780078884801

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

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

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

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

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

- Natural numbers: $\left\{1,2,3,4,...\right\}$
- Whole numbers: $\left\{0,1,2,3,4,...\right\}$
- Integers: Measurement of debts, temperatures, etc., fall under the set of integers $\left\{...,-3,-2,-1,0,1,2,3,...\right\}$
- Rational numbers: If we cut a cake into equal pieces, then we may have a piece that represents a fraction like $\frac{5}{6},1.5\left(=\frac{3}{2}\right),\frac{6}{9},\frac{8}{3}$
- 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 $\sqrt{\frac{82}{20}}$

We can rewrite the given real number as:

$\sqrt{\frac{82}{20}}=\sqrt{\frac{2\times 41}{2\times 10}}\phantom{\rule{0ex}{0ex}}=\sqrt{\frac{41}{10}}$

Now we will check each subset of real numbers:

- 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.
- 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.
- 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.
- 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. - 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.

94% of StudySmarter users get better grades.

Sign up for free