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Difference Between Rational and Irrational Numbers (With Table)

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:


 

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

Parameter of ComparisonRational NumberIrrational Number
DefinitionA number which can be expressed as a ratio of two integersA number which cannot be written as a ratio of two integers, that means “no ratio”
ExamplesPerfect squares such as 4, 9, 16, 25, and so onSurds such as √5, √11
How ExpressedIn both fraction and decimal formOnly in decimal form
Whether can be written in fraction?YesNo
Relation with integersAll integers are rational numbers
Example:
6 = 6/1, so 6 is a rational number.
Irrational numbers cannot be integers
Pattern of Decimalsa) 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 denominatorThe 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.

Rational Numbers
 

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).

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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 5th 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.

Irrational Numbers

Main Differences Between Rational and Irrational Numbers

  1. 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.    
  2. A rational number can be simplified to any fraction of an integer, whereas an irrational number cannot be so simplified.
  3. 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.
  4. 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.
  5. 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…..

 

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.

Word Cloud for Rational and Irrational Numbers

 

References

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