Q3. - Algebra 1

Question

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

$\sqrt{\mathbf{28}}$

Step-by-Step Solution

Answer

The real number $\sqrt{28}$ 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:

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, $q\ne 0$.

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 $\sqrt{28}$

We can rewrite the given real number using square root property as:

$$\begin{gathered} \sqrt{28}=\sqrt{7\times 4} \\ =\sqrt{7\times 2^2} \\ =2\sqrt{7} \end{gathered}$$

Now we will check each subset of real numbers:

1. Natural numbers: They are positive, countable, and start from 1. So, $\sqrt{28}$ not being a counting object is not a natural number.
2. Whole numbers: They are natural numbers including 0. So, $\sqrt{28}$ 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 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 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, $\sqrt{28}$ cannot be written in the form $\frac{p}{ q}$, it is an irrational number.

 

Therefore, the real number $\sqrt{28}$ belongs to the set of irrational numbers.