Question

Insert 3 rational numbers between

Answer 1

Answer:

$\frac{ \sqrt{20} }{20} \\ \sqrt{ \frac{1}{4} \times \frac{ \sqrt{20} }{20} } \\ \sqrt{ \frac{1}{5} \times \frac{ \sqrt{20} }{20} }$

Step-by-step explanation:

If you have two numbers x and y, their geometric mean is defined as

$\sqrt{xy}$

Let that number be m, and let x be the smallest of the two numbers.

You know for sure that

$x \leqslant m \leqslant y$

Since

$m \geqslant \sqrt{ {x}^{2} }$

And

$m \leqslant \sqrt{ {y}^{2} }$

With equality holding if and only if x=y.

So, we can use the geometric mean to generate numbers which are for sure between two desired values. For example:

$\frac{1}{4} < \sqrt{ \frac{1}{4} \times \frac{1}{5} } < \frac{1}{5}$

The middle term is of course irrational, since it's equal to

$\frac{1}{ \sqrt{20} } = \frac{ \sqrt{20} }{20}$

Now we can do the same thing with our newfound value which we know to be between 1/4 and 1/5:

$\frac{1}{4} < \sqrt{ \frac{1}{4} \times \frac{ \sqrt{20} }{20} } < \frac{1}{5}$

And

$\frac{1}{4} < \sqrt{ \frac{1}{5} \times \frac{ \sqrt{20} }{20} } < \frac{1}{5}$