**Rational vs Irrational Numbers**

The construct of *rational* and *irrational* numbers is in vogue from a very long time ago. The difference between a *rational* number and an *irrational* one is that a *rational* number can be stated in the form of a fraction such as a/b where both “a” and “b” are integers whereas an *irrational* number cannot be written in such way.

However, the above is not the only difference. A comparison between both the terms on certain parameters can shed light on subtle aspects:

Table of Contents

## Comparison Table Between Rational and Irrational Numbers (in Tabular Form)

Parameter of Comparison | Rational Number | Irrational Number |
---|---|---|

Definition | A number which can be expressed as a ratio of two integers | A number which cannot be written as a ratio of two integers, that means “no ratio” |

Examples | Perfect squares such as 4, 9, 16, 25, and so on | Surds such as √5, √11 |

How Expressed | In both fraction and decimal form | Only in decimal form |

Whether can be written in fraction? | Yes | No |

Relation with integers | All integers are rational numbers Example: 6 = 6/1, so 6 is a rational number. | Irrational numbers cannot be integers |

Pattern of Decimals | a) Can be recurring or repeating decimal Example: 1.26262626 (Repeated pattern is 26) b) Is finite in nature i.e. a terminating decimal Example: 1.25 (decimals are finite) | a) Non-repeating in nature Example: 1.4142135623....... (No repeated pattern) b) Is non-terminating i.e. has non-repeating (endless) digits to the right of decimal point Example: 0.4141141114.... (decimals are non-terminating) |

Relation with numerator and denominator | The numerator and denominator are integers and denominator is not zero. Example: 1/9 | Numerator and denominator cannot be integers. Example: 3/0 – Fraction with denominator zero |

## What are Rational Numbers?

A *rational* number (derived from the word “ratio”) is stated as a ratio of two numbers/integers. That is, it can be stated as a fraction. Example: a/b. Basically “a” is divisible by “b” and “b” is not “0”.

Common examples of *rational* numbers include 1/2, 1, 0.68, -6, 5.67, √4 etc. Think, for example, the number 4 which can be stated as a ratio of two numbers i.e. 4 and 1 or a ratio of 4/1. Similarly, 4/8 can be stated as a fraction and hence constitute a *rational* number.

A *rational* number can be simplified. The decimal expansion of a *rational* number terminates after a finite number of digits.

Also, a *rational* number portrays a continual finite sequence of digits again and again (i.e. repeated pattern). The set of all *rational* numbers is possible to count.

## What are Irrational Numbers?

A number that is not *rational* is known as *irrational*. *Irrational* numbers are based on the premise that on a line segment, not all numbers can be *rational* and there exist certain numbers which when taken together with *rational* numbers constitute a real number.

Popular instances include √2 (which translates to 1.414213..), π(pi) (which is 3.141592), e (euler’s number which is 2.7182…..), φ (Golden ratio which is 1.618033…..). Other examples would be surds such as √5, √7 and so on.

In simple terms, *an irrational* number is one that cannot be stated in the form of a fraction/ratio though it can be stated as a decimal. The decimal expansion of an *irrational* number is non-terminating (i.e. endless) and does not portray any pattern (i.e. no repetitions).

Since the non-terminating decimals have no repeated pattern it cannot be converted into a fraction and hence constitute an *irrational* number.

Though *irrational* numbers are not a part of regular life, they do subsist on the number line. In reality, if we consider there is a line and if we place the numbers 0 and 1, there will be an infinite *irrational* numbers between them.

The credit of discovering *rational* and *irrational* numbers goes to Greek mathematicians Pythagoras and Hippassus (one of his students) in 5^{th} century BC. It is believed that Hippassus who propounded the existence of *irrational* numbers may have been sentenced to death for his discovery as Pythagoreans did not believe in *irrational* numbers.

**Is “0” rational or an irrational number?**

“0” is *a rational* number as it belongs to a set of integers which are *rational* numbers.

**Are square roots of numbers rational?**

Not all. For example, √4 = 2 is *rational* whereas √5 is *irrational*. That means square roots of non-perfect squares represent *irrational* numbers.

**Main Differences Between Rational and Irrational Numbers**

- A
*rational*number is any number in mathematics such as a whole number, fraction, decimal, even negatives. An irrational number is one that is not a rational number. - A rational number can be simplified to any fraction of an integer, whereas an irrational number cannot be so simplified.
- In rational numbers, both numerator and denominator are whole numbers, and the denominator is not zero. However, irrational numbers cannot be expressed in a fraction.
- Common examples of
*rational*numbers include 3, 1, 0.65, 0.11 and also perfect squares like 9, 16, 25, 36 and so on. Examples of*irrational*number include √7, √5, √3 and so on. Basically they cannot be simplified further. - The rational number includes finite and repeating decimals. An example is 1.25. However, the decimal expansion of irrational numbers is infinite, non-repetitive and shows no pattern. An example is 1.4142…..

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## Conclusion

Fundamentally, if the number can be expressed in a fraction it is a *rational* number else *irrational*. If we add *a rational* and *irrational* number, we get a real number which represents “all the numbers” on the number line. The concept of a real number is used to measure quantities such as time, mass, energy, etc.

*Irrational* numbers (which some may think as nonsensical) constitute an essence in the fields of mathematics and physics in which finding an exact result and approximation is like passing or failing the test. Basically, *irrational* numbers “fill up” the spaces between *rational* numbers.

The best way to understand the difference between *rational* and *irrational* numbers is to see if we are able to measure or count the number in exact terms in our daily activity. If yes, it is *rational, *else *irrational*.

## Word Cloud for Difference Between Rational and Irrational Numbers

The following is a collection of the most used terms in this article on **Rational and Irrational Numbers**. This should help in recalling related terms as used in this article at a later stage for you.

## References

- https://spectrum.library.concordia.ca/973825/
- https://journals.aps.org/prb/abstract/10.1103/PhysRevB.14.2239