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What is the difference between rational and irrational numbers?

  • Last Updated : 17 Aug, 2021

A number System is portrayed as a course of action of writing to represent the numbers. It is the numerical documentation for addressing amounts of a given set by using digits or symbols in a consistent manner. It gives an exceptional portrayal of each number and addresses the math and logarithmic construction of the figures. It additionally permits us to operate arithmetic operations like addition, subtraction, and division. The number the numeral addresses is called its value. 

Difference between Rational Numbers and Irrational Numbers

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In a number system, decimal numbers are the ones that are used mostly in mathematics. There are different terms introduced based on the characteristics shown by the numbers. For instance, numbers starting from 1 and going up to infinity are Natural numbers, numbers starting from 0 and going up to infinity are Whole numbers. Numbers that can be expressed in the form of p/q, where q≠ 0 are Rational numbers, numbers that cannot be represented in the p/q are Irrational numbers. Let’s learn about the difference between Rational and Irrational numbers,



Sl.No 

Rational Numbers 

Irrational Numbers 

1.Those numbers that can be expressed as a ratio of two numbers p and q where p and q are any integer and q is not equal to zero is called rational numbers i.e we can represent it in the (p/q) format.Those numbers that cannot be expressed as a ratio of two numbers p and q where p and q are any integer and q is not equal to zero is called rational numbers i.e we cannot represent it in the (p/q) format. 
2.Rational Numbers are either finite or are recurring in nature.Irrational Numbers are non-terminating as well as non-repeating in nature.
3.Both the numerator and denominator are integers, in which the denominator is not equal to zero.These cannot be written in fractional form. So no concept of numerator and denominator here.
4.These include perfect squares such as 4, 9, 16, 25, 36, 49, and so onThese include surds such as √2, √3, √5, and so on.
5.Example: 3/2 = 1.5, 3.6767 , 6, 9.31, 64, 0.66666, 3.25 etc. Example: √5, √11, π(Pi), etc.

Sample Problems

Question 1: Is Pi (π) a rational or an irrational number, explain why?

Answer:

Pi (π) is an irrational number as it is non-terminating and non-repeating in nature. However, in mathematics, in order to make calculations easier, pi is rounded off as 3.14 and is also represented in fraction form as 22/7.

Question 2: Which of the given numbers are rational and which are irrational?

  • 6
  • 3/2
  • √7
  • √25

Answer:

  • 6 ⇢ Rational number, terminating and non-repeating in nature.
  • 3/2 ⇢ Rational number, in the form p/q, and q≠0.
  • √7 ⇢ Irrational number, is the square root of a number that is not a perfect square.
  • √25 ⇢ Rational number, it is the square root of a perfect square and the value is 5.

Question 3: The square root of a perfect square is an irrational number. Is this statement true or false?

Answer:

No, the statement “The square root of a perfect square is an irrational number” is not true. The correct fact is that the square root of perfect squares is a rational number, for instance, √36 = 6, √64 = 8. Irrational numbers are the square roots of those numbers that are not perfect squares, for instance, √2, √3, etc.

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