How many irrational numbers can exist between two rational numbers ?
As there are infinite numbers between two rational numbers and also there are infinite rational numbers between two rational numbers. So number of irrational numbers between the numbers should be infinite or something finite which we cannot tell ? Or there exist some specific irrational which could be told by some methods ?
for eg. irrational numbers between 2 and 3 are 5^(1/2),7^(1/2) etc…
If there exists only some specific irrationals then why so?
[Math] How many Irrational numbers
irrational-numbers
Best Answer
There are uncountably infinite number of irrational numbers and countably infinite number of rational numbers.
Suppose we have a irrational numbers $q_0<q_1$ then we can construct a bijective mapping from $\mathbb R$ to $(q_0,q_1)$ that preserves rationality. This would show that the number of rational numbers and irrational numbers respectively in the interval is the same as the number of rational and irrational numbers overall.
We construct these by using a bijective mapping $\theta:\mathbb R\to(0,1)$ with the same properties and then use $f(x) = q_0 + (q_1-q_0)\theta(x)$. The mapping $\theta$ can be given as:
$$\theta(x) = \begin{cases} {1\over 2-x} & \text{ if } x<0 \\ {1+x\over 2+x} & \text{ if } x\ge 0 \end{cases}$$