Irrational Numbers
Irrational Numbers
Irrational Numbers: An irrational number is a number which cannot be written in the form
, where p and q are both integers and
.Such as
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, , 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 is irrational.
Solution: Let us assume that is rational.
Squaring both sides, we get
...(1)
Therefore 2 is a factor of
Or we can say that 2 is factor of p
Let p= 2r
Putting this value in Equation (1)
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
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| Right Option : B | |||
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| Right Option : C | |||
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| Right Option : A | |||
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