Question

find two irrational numbers between 2/3 and 8/9

Answer 1

Therefore, we can choose two decimals between 22 and 33; if we give them nonrepeating patterns, they certainly cannot be expressed as fractions. A few examples:

2.10110011100011110000...2.10110011100011110000...

2.9898998998999888999888...2.9898998998999888999888...

2.322332222333222222...

Answer 2

Answer:

$\bf\textbf{The required number = } \frac{4}{3\sqrt{3}}$

Step-by-step explanation:

$\text{We are given two numbers = }\frac{2}{3}\text{ and }\frac{8}{9}\\\\\text{Now, let }a=\frac{2}{3}\text{ and b = }\frac{8}{9}$

Since, both the given numbers are rational, so to find an irrational number between two rational numbers, we take the square root of the product of both the numbers and check whether the resultant is irrational or not.

Let the irrational number which is to be inserted = r

$\implies r = \sqrt{a\cdot b}\\\\\implies r = \sqrt{\frac{2}{3}\times \frac{8}{9}}\\\\\bf\implies r = \frac{4}{3\sqrt{3}}\\\\So, \sqrt{3}\text{ is an irrational number and }\frac{4}{3}\text{ is an non-zero rational number}$

Now, product of non-zero rational no. and irrational no. is irrational no.

$\bf\textbf{The inserted number r = } \frac{4}{3\sqrt{3}}\textbf{is irrational.}$