Question

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

$-\frac{54}{19}$

Short answer

The real number $-\frac{54}{19}$belongs to the set of rational numbers.

Step-by-step solution

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$\frac{-54}{19}$

Now we will check each subset of real numbers:

1. Natural numbers: They are positive, countable, and start from 1. So, $\frac{-54}{19}$not being a positive number is not a natural number.
2. Whole numbers: They are natural numbers including 0. So, $\frac{-54}{19}$not being a natural number is not a whole number as well.
3. Integers: They are numbers that are not a fraction. Since $\frac{-54}{19}$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 $\frac{-54}{19}$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, can be written in the form $\frac{p}{q}$, it is not an irrational number.

Therefore, the real number $\frac{-54}{19}$belongs to the set of rational numbers.