Three Irrational Numbers between 5/7 and 9/11

Introduction:
Irrational numbers are the numbers that cannot be expressed in the form of p/q, where p and q are integers. They are non-repeating and non-terminating decimals. In this question, we have to find three different irrational numbers between the rational numbers 5/7 and 9/11.

Explanation:
To find the irrational numbers between 5/7 and 9/11, we need to find a number that lies between them and is irrational. One way to do this is to convert the rational numbers into decimals and then find a number that lies between them.

Step 1: Convert 5/7 and 9/11 into decimals.
5/7 = 0.714285714...
9/11 = 0.818181818...

Step 2: Find a number between 0.714285714... and 0.818181818... that is irrational.
We can take √2 which is an irrational number. Its decimal value is 1.41421356...

Step 3: Convert 1.41421356... into a fraction.
Let's take the first 6 decimal places of √2 which are 1.414213. Now, multiply this number by 10^6 to get rid of the decimal point. We get 1414213. Now, this number can be written as a fraction by putting it over 10^6. So, √2 = 1414213/1000000.

Step 4: Repeat Steps 2 and 3 to find two more irrational numbers between 5/7 and 9/11.
Another irrational number that lies between 0.714285714... and 0.818181818... is e which is the base of natural logarithm. Its decimal value is 2.718281828...

To convert e into a fraction, we can use the continued fraction expansion of e which is [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, ...]. We can take the first few terms of this expansion to get a good approximation of e. Let's take the first 5 terms which are [2; 1, 2, 1, 1]. Now, we can convert this continued fraction into a fraction by using the following formula:
[2; 1, 2, 1, 1] = 2 + 1/(1 + 2 + 1/(1 + 1)) = 2 + 1/5 = 11/5.

Therefore, e can be written as 11/5.

The third irrational number between 5/7 and 9/11 can be π which is the ratio of the circumference of a circle to its diameter. Its decimal value is 3.141592653...

To convert π into a fraction, we can use the following formula:
π = 4 arctan(1) = 4 (1 - 1/3 + 1/5 - 1/7 + 1/9 - ...)

We can take the first few terms of this series to get a good approximation of π. Let's take the first 5 terms which are 4(1 - 1/3 + 1/5 -

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