Understanding the Complexities of Zero: Rational or Irrational?

# Understanding the Complexities of Zero: Rational or Irrational?

Vikram Singh
Assistant Manager - Content
Updated on May 14, 2024 15:51 IST

“Is zero a rational number? The great mathematical debate that divides experts. Discover the arguments for and against this mind-boggling puzzle and find out where you stand on the zero-rationality spectrum.”

If you are here, you are one of us who got stuck with the traditional definition of a Rational Number and confused with the question “Is 0 a Rational Number?”.

Don’t worry, you are in the right place. Through this blog, we will clear all your doubts.

In this article, we will start with the overview of the controversy surrounding whether 0 is a rational number or not, arguments for considering 0 as a rational number and 0 as an irrational number, and finally discuss the conclusion of what 0 should be considered.

## Overview of Controversy

The controversy surrounding whether 0 is a rational number or not, comes from two different schools of thought.

A lot of mathematicians are aligned with the formal definition of Rational Numbers, that states, “Any number is said to be a rational number if it can be expressed in the form of p/q, where p, q are integers, and q is not equal to 0”.

While some mathematicians, are not aligned with considering p, and q as an integer only, according to them, they should be non-zero integers. i.e., any number of the form p/q is said to rational number only p, and q are non-zero integers.

So, those who are aligned with the first school of thought, consider 0 as a rational number, whereas those who are aligned with the second school of thought, consider 0 as an irrational number.

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Confused till now!!, don’t worry.

Let’s discuss the formal definition of Rational Numbers and Irrational Numbers.

## What are Rational Numbers?

Any number that can be expressed in the form of p/q, where p and q are integers and q is not equal to 0 is known as a Rational Number.

The decimal expression of a rational number must be either terminated or repeated.

Example: 5 (5/1), 234 (234/1), 22/7, 0.25, 3.14, 1.333333333…..

## What is an Irrational Number?

In simple terms, any number that is not rational is irrational, i.e., numbers that can’t be expressed in the form of p/q or have non-terminating and non-repeating decimal representation are known as irrational numbers.

Example: pi, e (Euler number), 3.1415623….

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Now, it’s time to discuss the elephant in the room.

### Is 0 a Rational Number?

0 can be written in the form 0/1, i.e., in p/q form where p = 0 and q = 1 (not equal to zero).

Therefore, from the above definition of a rational number, 0 is a rational number.

Note: Apart from the argument over the traditional definition of rational numbers, there are two other major arguments in favour of considering 0 as an irrational number:

1. Division By Zero: As we know, the division of any integer by zero is undefined, which leads to the argument that 0 can’t be a rational number, since it can’t be the result of dividing one integer from another integer.
2. Inconsistent Rule: When we multiply any number by its reciprocal, we get 1. But in the case of 0 (0/0), we consider it as an inconsistent form.
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## Conclusion

In conclusion, from the definition of rational numbers and irrational numbers, it is clear that 0 is a rational number. Since it can be expressed in the form of p/q (0/1), where p (= 0) and q (= 1) are integers and q is not equal to zero.

Keep Learning!!

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## FAQs

Is 0 a rational number?

Yes, zero is a rational number, as it can be expressed in the form of p/q where p = 0, and q = 1.

What is a Rational Number?

Any real number that can be expressed as the ratio of two integers (p/q), where denominator can't be zero is called a rational number.

What is an Irrational number?

Any real number that can't be expressed in the form of p/q or the decimal expression is non-terminating or non-repeating is known as an irrational number.

Why is the classification of 0 as a rational number is controversial?

The controversy around 0 about rational numbers comes from its traditional definition, that states any number is said to be rational only if it can be expressed in the form of two non-zero integers.