Q4:

Express $0.99999 ....$ in the form $\frac{p}{q}$. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.

Step 1: Variable Assignment and Equation Formulation

To express the non-terminating, repeating decimal $0.99999...$ (which can also be written as $0.\overline{9}$) in the rational form $\frac{p}{q}$, we begin by assigning it to a variable $x$.

Let $x = 0.99999... \quad \text{--- (Equation 1)}$

Step 2: Algebraic Multiplication

We observe the periodicity of the repeating decimal. Since exactly one digit (the digit $9$) repeats infinitely, we multiply both sides of Equation 1 by $10^1$ (or $10$). [Per the fundamental properties of equality and base-10 positional notation, multiplying by 10 shifts the decimal point one place to the right].

$10x = 9.99999... \quad \text{--- (Equation 2)}$

Step 3: Subtraction and Isolation of the Variable

To eliminate the infinitely repeating fractional part, we subtract Equation 1 from Equation 2. Because the infinite tail of $9$s is identical in both equations, their difference is exactly zero.

$10x = 9.99999...$
$- \quad x = 0.99999...$

Now, solve for $x$ by dividing both sides by $9$:

$x = \frac{9}{9}$

$x = 1$

Step 4: Expressing in $\frac{p}{q}$ Form

The integer $1$ can be expressed as a rational number in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$.

$x = \frac{1}{1}$

Step 5: Analytical Discussion & Theoretical Justification

The result $0.99999... = 1$ often causes initial surprise because they appear to be distinct numbers. However, in the real number system, they are two different representations of the exact same mathematical value. This makes perfect logical sense when analyzed through multiple mathematical frameworks:

• Argument 1: Fractional Equivalency
  Consider the fraction $\frac{1}{3}$. In decimal form, $\frac{1}{3} = 0.33333...$
  If we multiply both sides of this equation by $3$, we get:
  $3 \times \frac{1}{3} = 3 \times 0.33333...$
  $1 = 0.99999...$


• Argument 2: Infinite Geometric Series (Real Analysis)
  The decimal $0.999...$ can be expanded as a sum of fractions:
  $0.9 + 0.09 + 0.009 + \dots = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \dots$
  This is an infinite geometric series where the first term $a = 0.9$ and the common ratio $r = 0.1$.
  [Per the formula for the sum of an infinite geometric series, $S = \frac{a}{1 - r}$ for $|r| < 1$]:
  $S = \frac{0.9}{1 - 0.1} = \frac{0.9}{0.9} = 1$.


• Argument 3: The Archimedean Property
  In the real number system, if two numbers are strictly different, there must exist a non-zero distance between them. The difference between $1$ and $0.999...$ is $0.000...$ with an infinite string of zeros. Since there is no "final" position to place a $1$, the difference is exactly $0$. Therefore, the numbers are identical.

Final Solution: The repeating decimal $0.99999...$ expressed in the rational form $\frac{p}{q}$ is $\frac{1}{1}$. The value is exactly equal to $1$, which is mathematically sound and justified by the properties of infinite geometric series, fractional equivalency, and the definition of limits in the real number system.