What is the difference between Rational and Irrational numbers?

Rational numbers are the numbers that are written in the form of p/q, where p and q are the integer co-prime numbers and Irrational numbers are the numbers that can not be written in the form of p /q. All types of numbers, natural numbers, whole numbers, and integers are Rational numbers, and all unresolved roots,π,e, etc are Irrational numbers.

Natural Numbers: All those numbers which are used to count things are known as Natural numbers, the set of natural numbers is {1,2,3,4,5,6……….}. Natural numbers are Rational Numbers because we can write them in the form of P/Q

${\frac{1}{1},\frac{2}{1},\frac{3}{1},\frac{4}{1},........}$

Whole Numbers: All those numbers which are complete numbers are known as Whole numbers, the set of Whole numbers is {0,1,2,3,4,5,6……….} . Whole numbers are Rational Numbers because we can write them in the form of P/Q

$\frac{0}{1},{\frac{1}{1},\frac{2}{1},\frac{3}{1},\frac{4}{1},........}$

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What is the difference between Rational and Irrational numbers?

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Integers: Integers is the set of all the complete numbers in minus and positive including 0 also. If surface of the earth is 0 then height above the ground is shown by positive integers and depth below the surface of the earth is shown by negative integers.

All the integers are rational numbers because we can write them in the form of p/q, as example.

$.........-\frac{4}{1},\frac{}{}-\frac{3}{1},-\frac{2}{1},-\frac{1}{1},\frac{0}{1},{\frac{1}{1},\frac{2}{1},\frac{3}{1},\frac{4}{1},........}$

All the fractions that exist between the integers are also rational numbers. There are two types of fractional numbers on the basis of decimal expansion.

Type of rational number on the basis of decimal expansion:

On the basis of the decimal expansion, the rational numbers are of two kinds

(a) Finite Decimal (b) Infinite Decimal

Finite Decimal: The fraction p/q (p and q are co-prime no’s) when converted into decimal expansion, we get a quotient and remainder 0.as an example

$\frac{1}{2}=0.5,\frac{28}{5}=5.6,\frac{7}{10}=0.7$

These types of decimal expansion are known as finite decimals. We can observe a fraction p/q (p and q are co-prime no’s) to be a finite decimal if factors of q are in the form of 2ⁿ× 5ⁿ. So, if there is any other prime factor except to 2 and 5 of q then the decimal expansion of p/q can’t be finite decimal.

Infinite Decimal: The fraction p/q(p and q are co-prime no’s) when converted into decimal expansion,we get a quotient and remainder non-zero.as an example

$\frac{1}{3}=0.3333....,\frac{8}{7}=1.142857142857......$

We can observe a fraction p/q (p and q are co-prime no’s) to be an infinite decimal if factors of q are not in the form of 2ⁿ× 5ⁿ. So, if there is any other prime factor except 2 and 5 of q then the decimal expansion of p/q is infinite decimal.

Irrational number: All the infinite decimals that are non-recurrent decimals are known as irrational numbers. As an example, the decimal expansions of√2,√3,√5 , and π are infinite non-recurrent decimal expansions so these are irrational numbers. The decimal expansion such as 2.804005606006….are known as irrational numbers since after the decimal the numbers are not repeated.

√2 = 1.41421356…..,√6= 2.4494897…….

What is the difference between rational and irrational number?

The features of rational numbers and irrational numbers are following.

The sum of two irrational numbers may be an irrational number or a rational number, as an example (√2+ 4), (π + 2) are irrational numbers and √2 + (-√2) = 0
The product of an irrational number to a rational number is an irrational number, as an example, 2√5,2π are irrational number.
The product of two irrational number may be a rational or irrational number, as an example √2×–√2=-2,√2×√3 = √6
The product of two identical irrational numbers may be rational or irrational, as an example √2×√2 = 2, $\boldsymbol{\sqrt[3]{2}\times \sqrt[3]{2}=2^{2/3}=\sqrt[3]{2^{2}}=\sqrt[3]{4}}$
The division of two irrational numbers can be rational or irrational, as an example 2√2/3√2= 2/3 , 2√2/√3 etc.

What is the difference between rational and irrational number?

Is π an irrational number, what are rational, irrational, and surds numbers?

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π is written in the form of 22/7 then why is it an irrational number:π is the ratio between circumference and diameter, one of them either circumference or diameter is an irrational number, so π is called an irrational number, the value 22/7 of π we use in the calculation is actually the nearest rational value of π, therefore 22/7 is a rational number but π is an irrational number.

Questions related to Rational numbers and Irrational numbers

Click here to get the answers to the following questions: Number System (all types of numbers used in maths)

Q1.Find whether the following are finite decimal or infinite decimal

$\boldsymbol{\left ( i \right )\frac{20}{14}\: \left ( ii \right )\frac{140}{125}\: \left ( iii \right )\frac{225}{64}}$

Q2.Write the following rational number in ascending form

$\boldsymbol{\frac{2}{5},\frac{3}{4},\frac{7}{6},\frac{9}{7}}$

Q3. Write the following irrational number in acending number

$\boldsymbol{\sqrt[3]{3},\sqrt[4]{6},\sqrt{5},\sqrt{2},\sqrt[5]{12}}$

Click here:What are Surds and how to compare them?

Q4.Write the following infinite recurrent decimal number into the form of p/q.

$\boldsymbol{\left ( i \right )3.\bar{5}}\: \boldsymbol{\left ( ii \right )2.\overline{1001}\; \left ( iii \right )5.22\overline{002}}$

Q5. Rationalize the denominator of the following fractions

$\boldsymbol{\left ( i \right )\frac{2}{\sqrt{5}}\: \left ( ii \right )\frac{3}{\sqrt[3]{2}}\: \left ( iii \right )\frac{10}{3\sqrt{2}-2\sqrt{3}}}$

Q6. Find the value of a and b

$\boldsymbol{\frac{2-\sqrt{5}}{2+\sqrt{5}}=a+b\sqrt{5}}$

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NCERT Solutions of Science and Maths for Class 9,10,11 and 12

NCERT Solutions for class 9 maths

Chapter 1- Number System	Chapter 9-Areas of parallelogram and triangles
Chapter 2-Polynomial	Chapter 10-Circles
Chapter 3- Coordinate Geometry	Chapter 11-Construction
Chapter 4- Linear equations in two variables	Chapter 12-Heron’s Formula
Chapter 5- Introduction to Euclid’s Geometry	Chapter 13-Surface Areas and Volumes
Chapter 6-Lines and Angles	Chapter 14-Statistics
Chapter 7-Triangles	Chapter 15-Probability
Chapter 8- Quadrilateral

NCERT Solutions for class 9 science

Chapter 1-Matter in our surroundings	Chapter 9- Force and laws of motion
Chapter 2-Is matter around us pure?	Chapter 10- Gravitation
Chapter3- Atoms and Molecules	Chapter 11- Work and Energy
Chapter 4-Structure of the Atom	Chapter 12- Sound
Chapter 5-Fundamental unit of life	Chapter 13-Why do we fall ill ?
Chapter 6- Tissues	Chapter 14- Natural Resources
Chapter 7- Diversity in living organism	Chapter 15-Improvement in food resources
Chapter 8- Motion	Last years question papers & sample papers

NCERT Solutions for class 10 maths

Chapter 1-Real number	Chapter 9-Some application of Trigonometry
Chapter 2-Polynomial	Chapter 10-Circles
Chapter 3-Linear equations	Chapter 11- Construction
Chapter 4- Quadratic equations	Chapter 12-Area related to circle
Chapter 5-Arithmetic Progression	Chapter 13-Surface areas and Volume
Chapter 6-Triangle	Chapter 14-Statistics
Chapter 7- Co-ordinate geometry	Chapter 15-Probability
Chapter 8-Trigonometry

CBSE Class 10-Question paper of maths 2021 with solutions

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CBSE Class 10 -Question paper of maths 2020 with solutions

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NCERT Solutions for Class 10 Science

Chapter 1- Chemical reactions and equations	Chapter 9- Heredity and Evolution
Chapter 2- Acid, Base and Salt	Chapter 10- Light reflection and refraction
Chapter 3- Metals and Non-Metals	Chapter 11- Human eye and colorful world
Chapter 4- Carbon and its Compounds	Chapter 12- Electricity
Chapter 5-Periodic classification of elements	Chapter 13-Magnetic effect of electric current
Chapter 6- Life Process	Chapter 14-Sources of Energy
Chapter 7-Control and Coordination	Chapter 15-Environment
Chapter 8- How do organisms reproduce?	Chapter 16-Management of Natural Resources

NCERT Solutions for class 11 maths

Chapter 1-Sets	Chapter 9-Sequences and Series
Chapter 2- Relations and functions	Chapter 10- Straight Lines
Chapter 3- Trigonometry	Chapter 11-Conic Sections
Chapter 4-Principle of mathematical induction	Chapter 12-Introduction to three Dimensional Geometry
Chapter 5-Complex numbers	Chapter 13- Limits and Derivatives
Chapter 6- Linear Inequalities	Chapter 14-Mathematical Reasoning
Chapter 7- Permutations and Combinations	Chapter 15- Statistics
Chapter 8- Binomial Theorem	Chapter 16- Probability

CBSE Class 11-Question paper of maths 2015

CBSE Class 11 – Second unit test of maths 2021 with solutions

NCERT solutions for class 12 maths

Chapter 1-Relations and Functions	Chapter 9-Differential Equations
Chapter 2-Inverse Trigonometric Functions	Chapter 10-Vector Algebra
Chapter 3-Matrices	Chapter 11 – Three Dimensional Geometry
Chapter 4-Determinants	Chapter 12-Linear Programming
Chapter 5- Continuity and Differentiability	Chapter 13-Probability
Chapter 6- Application of Derivation	CBSE Class 12- Question paper of maths 2021 with solutions
Chapter 7- Integrals
Chapter 8-Application of Integrals