Irrational Numbers


 
 
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;; sqrt7are irrational numbers. their product will be  __________________

Right Option : C
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Explanation
Question : 2

sqrt2  is an irrational number . what will be 3sqrt2  ?

Right Option : B
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Explanation
Question : 3

sqrt2 and sqrt3 are irrational numbers.their product will be 

Right Option : A
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Explanation
 
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