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Prove that between any two rational numbers, there is an irrational number.
We will find an irrational number between two arbitrary rational numbers.
Answer: Between any two rational numbers, there is an irrational number is proved.
Follow the explanation given below.
Explanation:
Assume a and b to be two arbitrary rational numbers such that b > a. We claim c = a + (b - a)/√2 is an irrational number that lies between a and b.
1/√2 is an irrational number that lies between 0 and 1. This means c is also an irrational number.
0 < 1/√2 < 1
0 < (1/√2) (b - a) < (b - a) [Since b > a, b - a > 0]
a + 0 < a + 1/√2 (b - a) < a + (b - a)
a < a + (b - a)/√2 < b
This means c = a + (b - a)/√2 lies between a and b.
So, we found an irrational number between two arbitrary rational numbers.
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