The rational numbers are dense in RR. This means that between any two real numbers a and b with a < b, there exists a rational number q such that a < q < b. Using this fact, establish that the irrational numbers are dense in RR as well.

BenoguigoliB 2021-01-23 Answered
The rational numbers are dense in R. This means that between any two real numbers a and b with a < b, there exists a rational number q such that a < q < b. Using this fact, establish that the irrational numbers are dense in R as well.
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2k1enyvp
Answered 2021-01-24 Author has 94 answers

We need to prove that Irrational numbers are dense in R .We are given that Rational numbers are dense in R , which means between any two real numbers there exist a rational number. Now, 2 is an irrational number .
Let, a and b be two real numbers, such that, a Since a,b,2 are real numbers therefore there division is also a real number
Therefore , a2 and b2 are also real numbers. We have,
a<ba2<b2
And we are given that between any two real numbers there exist a rational number .
Therefore , let c in Q be a rational number between a2 and b2a2<c<b2
We got,
a2<c<b2
Now , multiply by 2 throughout, we get,
2(a2)<2c<2(b2)a<c2<b
Now, product of a rational and an irrational number is an irrational number , therefore , since c in Q and 2RQ, therefore, c2
Let, c2=ma<m<b
and a and b are real numbers and m is irrational number .
Therefore, we get that between any two real numbers a and b, with a Therefore, Irrational numbers are dense in real numbers .
Answer: Irrational numbers are dense in R .

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