Chapter 0.2, Problem 11E

To determine

To find: The set or sets of numbers to which real number $\sqrt{\frac{82}{20}}$ belong.

Answer to Problem 11E

The real number $\sqrt{\frac{82}{20}}$ is an irrational number.

Explanation of Solution

Given information:

The real number $\sqrt{\frac{82}{20}}$ .

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

Rewrite it as,

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

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 not 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 an irrational number.

Thus, the real number $\sqrt{\frac{82}{20}}$ is an irrational number.