Question

What is the set of numbers to which \[- \dfrac{54}{19}\] belong?

Answer 1

Hint: In this question, we need to find the set of numbers where the term \[- \dfrac{54}{19}\] belongs. First we need to be clear with the definition of different sets of numbers and based on the theories of different sets of numbers we can find on which set does the given term belong. Then, we need to observe the given number and need to find in which set this number belongs. We can proceed like this to get our answer.

Complete step-by-step answer:
Given, \[- \dfrac{54}{19}\]
Here we need to find the set of numbers where the term \[- \dfrac{54}{19}\] belongs.
There are many set of numbers namely irrational numbers, real numbers, integers, rational numbers, natural numbers and whole numbers
To find whether the term \[- \dfrac{54}{19}\] belongs to the set of irrational numbers, real numbers , integers, rational numbers, natural numbers or whole numbers we need to check if it satisfies the definition of each of these types of numbers.
1. Irrational number is nothing but a number that cannot be written in the form of a fraction with an integer in the numerator and an integer in the denominator and also irrational numbers do not include zero.
2. Real number is nothing but a set of numbers which consists of every number from \[- \infty\] to \[\infty\] including zero and fraction.
3. Integer is nothing but a set of the whole numbers and their opposites.
4. Rational number is nothing but a number that can be written as a fraction with an integer in the numerator and an integer in the denominator.
5. Natural numbers are known as the counting numbers which can take up values from \[1\] to \[\infty\] but excluding fractions .
6. Whole numbers are also known as the counting numbers which can take up values from \[0\] to \[\infty\] but excluding fractions.
Now on observing the given number, \[- \dfrac{54}{19}\] which is a fraction that has an negative integer in the numerator and an integer in the denominator. Therefore the given number can’t be an irrational number because it is opposite to the definition of the irrational number.
So, \[- \dfrac{54}{19}\] is not an irrational number and it is a rational number since it is similar to the definition of the rational number.
The number \[- \dfrac{54}{19}\] is a real number since it seems similar to the definition of the real number .
On simplifying the term \[- \dfrac{54}{19}\] , we get \[-2.8\] . Here \[-2.8\] is not a whole number . So it is not an integer.
Also \[- \dfrac{54}{19}\] is not a counting number. So it is not a natural number and also not a whole number.
Thus the number \[- \dfrac{54}{19}\] belongs to the set of real rational numbers.
Final answer :
The number \[- \dfrac{54}{19}\] belongs to the set of real rational numbers.

Note: The given number is the rational number. Rational numbers are the numbers which can be represented in the form of \[\dfrac{p}{q}\] where p\] and \[q\] are integers and \[q \neq 0\] . Mathematically, these numbers are denoted by the Latin capital letter \[Q\] . The rational numbers are included in the real numbers. Moreover, there are bigger sets than real numbers, that is the set of complex numbers which include all real and imaginary numbers.