Question

What are non-zero rational numbers?

Answer 1

Hint: We will use the concepts of rational and irrational numbers to find the solution to this problem. We will discuss all types of numbers in mathematics including rational numbers too. We will look at some examples and some properties of rational and also irrational numbers.

Complete step by step answer:
In mathematics, there are many types of numbers. They are: -
- Natural numbers - \[\left( {1,2,3,4,......\infty } \right)\]
- Whole numbers - \[\left( {0,1,2,3,4,......\infty } \right)\]
- Integers - \[\left( { - \infty ,....., - 4, - 3, - 2, - 1,0,1,2,3,4,......\infty } \right)\]
- Rational numbers and
- Irrational numbers.
Let us know about rational and irrational numbers. Numbers which can be written in the form of \[\dfrac{p}{q}\left( {q \ne 0} \right)\] are called as rational numbers, where \[p{\text{ and }}q\] are co-primes. In this fraction form, the denominator should not be equal to zero.

In other words every rational number can be written in \[\dfrac{p}{q}\] form. All the natural numbers, whole numbers and also integers come under rational numbers. For example, take the numbers \[ - 3{\text{ and 6}}\]. These can be written in fractional form as \[\dfrac{{ - 3}}{1}{\text{ and }}\dfrac{6}{1}\]. So, we can conclude that, all the natural numbers come under whole numbers, whole numbers come under integers and integers come under rational numbers.So, we can also conclude that zero also comes under integers as it can be written as \[\dfrac{0}{1}\].

All the rational numbers excluding zero are known as non-zero rational numbers.
So, we can write the range of non-zero rational numbers as \[\left( { - \infty ,\infty } \right) - \left\{ 0 \right\}\] or \[R - \left\{ 0 \right\}\]. The numbers which cannot be written in \[\dfrac{p}{q}\] form are called irrational numbers.Generally, these are non-terminating non-repeating decimal numbers.Some examples of irrational numbers are \[\sqrt 2 ,12.827.....\] and so on.

Note: Co-primes are the numbers which have ‘1’ as Greatest Common Divisor (GCD) only. Some examples of co-primes are \[\left( {6,13} \right)\], \[\left( {100,107} \right)\] and so on. We define the value of \[\pi \] as \[\dfrac{{22}}{7}\]. Though we write it in fraction form, still, it is an irrational number. Its value is \[3.1415926.......\]