Question

What is the set of numbers to which $\dfrac{8}{3}$ belongs?

Answer 1

Hint: We know that there are a lot of sets of numbers from the set of natural numbers to the set of complex numbers. Also, we know that the number of the form $\dfrac{p}{q}$ where $p$ and $q$ are integers is a rational number.

Complete step-by-step solution:
We are asked to find in which set the given number $\dfrac{8}{3}$ belongs to.
As we know, we can list out different sets of numbers.
We know that the whole numbers from $1$ to infinity are called natural numbers. There the numbers we use for counting. We call the set that include these numbers the set of natural numbers, $\mathbb{N}.$
We call the set of numbers that include the elements of $\mathbb{N},$ negative of these elements and zero the set of integers, $\mathbb{Z}.$
Now, we know that the numbers of the form $\dfrac{p}{q}$ where $p$ and $q$ are integers with $q\ne 0$ are called the rational numbers. We refer to the set of these number the set of rational numbers, $\mathbb{Q}.$
We know that $8$ is an integer and $3$ is also an integer with $3\ne 0$
And we can see that the given fraction cannot be reduced to an integer. So, clearly, it cannot be included in the set of integers.
So, we can say that this fraction is a rational number and for that reason this number is an element in the set of rational numbers, $\mathbb{Q}.$
Hence $\dfrac{8}{3}$ belongs to $\mathbb{Q}.$

Note: We know that the set of rational numbers is a subset of the set of real numbers $\mathbb{R}.$ And so, we can say that the given number and every rational number are real numbers and thus belong to the set of real numbers. Since, the set of real numbers is a subset of the set of complex numbers $\mathbb{C},$ the given number belongs to the set of the set of complex numbers, $\mathbb{C}.$