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Irrational Numbers

Real Numbers I - Concepts

HCF and LCM Using Prime Factorisation

Class - 10th CBSE Subjects

Concept Explanation

Irrational Numbers

Irrational Numbers: An irrational number is a number which cannot be written in the form

$large frac{p}{q}$ , where p and q are both integers and $large qneq 0$ .Such as $large sqrt{2},sqrt{3};;and;;sqrt{5}$ are irrational numbers.

Addition: Sum of two positive irrational number is irrational.

Product of Rational and Irrational number is irrational.

Divisibility If a and b are two integers, $large aneq 0$ , and there exixts an integer q such that b=aq then we say that a divides b and write a | b.For example, let a=6, b=7 and p=3. Then p divides ab=42 and note that p | a that is 3 | 6 The lemma does not hold if we drop the condition that p is prime.

For example, let a = 6, b=14. Then 4 divides ab=6 X 14 = 84 but 4 divides neither 4 divide 6 nor 4 divide 14..

Alternatively,

Suppose a,b,c are integers and c>0, If HCF of (a,c)=1 and c | ab then c | b.

For example, if c=4, a=5 and b=24 then HCF of (a,c)=1, and c | ab and note that c | b.

Example: Prove that $large sqrt2$ is irrational.

Solution: Let us assume that $large sqrt2$ is rational.

$large So.; we; can; represent ;in ;the; form ;of;frac{p}{q}; where ;p;and;q;are;coprime.$

$large sqrt2=frac{p}{q}$

Squaring both sides, we get

$large 2=frac{p^2}{q^2}$

$large p^2=2q^2$ ...(1)

Therefore 2 is a factor of $large p^2$

Or we can say that 2 is factor of p

Let p= 2r

Putting this value in Equation (1)

$large (2r)^2=2q^2$

$large 4r^2=2q^2$

$large q^2=2r^2$

$large Hence; 2; is; a; factor; of;q^2$

Or 2 is factor of q

Or we can say that 2 is a factor of both p and q

But, according to our assumption p and q are coprime

As we have proved that 2 is a factor of p and q

So our assumpton is wrong.

Thus $large sqrt2 ;is ;irrational$

Sample Questions

(More Questions for each concept available in Login)

Question : 1

$sqrt5 ;;and;; sqrt7$ are irrational numbers. their product will be __________________
A. Integer	B. Rational	C. Irrational	D. Neither rational nor irrational
Right Option : C
View Explanation

Question : 2

$sqrt2$ is an irrational number . what will be $3sqrt2$ ?
A. rational	B. irrational	C. integer	D. neither rational nor irrational
Right Option : B
View Explanation

Question : 3

$sqrt2 and sqrt3$ are irrational numbers.their product will be
A. irrational	B. rational	C. integer	D. neither rational nor irrational
Right Option : A
View Explanation

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Language - English

Chapters

Polynomial I

Polynomials II

Pair of Linear Equations I

Pair of Linear Equations II

Real Numbers I

Real Numbers II

Coordinate Geometry I

Coordinate Geometry II

Quadratic Equations I

Quadratic Equations II

Arithmetic Progression I

Arithmetic Progression II

Introduction to Trigonometry I

Introduction to Trigonometry II

Surface Area and Volume I

Surface Area and Volume II

Areas Related to Circles

Some Applications of Trigonometry

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