Q8:

Find three different irrational numbers between the rational numbers $\frac{5}{7}$ and $\frac{9}{11}$.

Step 1: Decimal Expansion of the Given Rational Numbers

To find irrational numbers between two rational numbers, we must first determine the decimal expansions of the given boundaries. This allows us to establish a clear numerical range.

• First Rational Number ($\frac{5}{7}$):
  Performing long division of $5$ by $7$:
  $5 \div 7 = 0.714285714285\dots$
  Since the block of digits "$714285$" repeats infinitely, we express this as $0.\overline{714285}$.


• Second Rational Number ($\frac{9}{11}$):
  Performing long division of $9$ by $11$:
  $9 \div 11 = 0.818181\dots$
  Since the block of digits "$81$" repeats infinitely, we express this as $0.\overline{81}$.

Step 2: Establishing the Range and Criteria

[Per the fundamental properties of the Real Number System], an irrational number is defined as a number whose decimal expansion is strictly non-terminating and non-repeating. Therefore, our objective is to construct three distinct decimal numbers, say $x$, $y$, and $z$, that satisfy two conditions:

1. Range Condition: $0.\overline{714285} < x, y, z < 0.\overline{81}$
2. Irrationality Condition: The decimals must not terminate and must not settle into a repeating periodic block.

Step 3: Construction of the Irrational Numbers

We can systematically construct non-terminating, non-repeating decimals by choosing starting digits that fall strictly between $0.71$ and $0.81$, and then appending a pattern of digits separated by an increasing number of zeros. This guarantees the sequence never repeats a fixed block.

• First Irrational Number ($x$):
  Choose a prefix greater than $0.714\dots$, such as $0.72$.
  Construct the non-repeating pattern: $x = 0.720720072000720000\dots$


• Second Irrational Number ($y$):
  Choose a prefix greater than $0.72$ but less than $0.81$, such as $0.75$.
  Construct the non-repeating pattern: $y = 0.750750075000750000\dots$


• Third Irrational Number ($z$):
  Choose a prefix greater than $0.75$ but less than $0.818\dots$, such as $0.80$.
  Construct the non-repeating pattern: $z = 0.808008000800008\dots$

Step 4: Visualizing the Real Number Line

The following high-precision number line maps the exact spatial relationship between the rational boundaries and the newly constructed irrational numbers.

Note: There are infinitely many irrational numbers between any two rational numbers. The three numbers constructed above are just one valid set of examples.

Final Solution: Three different irrational numbers between $\frac{5}{7}$ and $\frac{9}{11}$ are $0.720720072000\dots$, $0.750750075000\dots$, and $0.80800800080000\dots$.