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# 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

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

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

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

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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.

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

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

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

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

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

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

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

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 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

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### 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

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