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For any positive real number x, prove that there exists an irrational number y such that 0 is smaller than y and y is smaller than x.

Posted by Vaishnavi Dalvi 6 years, 2 months ago

CBSE > Class 10 > Mathematics

1 answers

Gaurav Seth 6 years, 2 months ago

x is positive real no

As we know square root of prime no is always irrational

So if x is prime no

Then √x will lie between 0 and x

So here there exists irrational

Now

If x is irrational

then its any fraction will also be irrational

So here y exists also

Now if x is not a prime and perfect square no

Then

As Between 0 and perfect square no except 1 there always exists a prime no ,also there exists square root of prime

Ex lets take x= 4

So 0 , 1 ,2,3,4

There exists 2,3 both prime

Also √2, √3 also lies between 0 and 4

So it would also give irrational no y

1Thank You

ANSWER

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