Q9(v):

Classify the following numbers as rational or irrational : (v) $1.101001000100001...$

Initial Setup & Theoretical Foundation

We are tasked with classifying the following real number:

$x = 1.101001000100001...$

To classify a real number as either rational ($\mathbb{Q}$) or irrational ($\mathbb{I}$) based on its decimal representation, we rely on the fundamental properties of real numbers:

• Rational Numbers ($\mathbb{Q}$): Possess decimal expansions that are either terminating (e.g., $0.25$) or non-terminating but repeating/recurring (e.g., $0.333...$). They can be expressed in the form $\frac{p}{q}$, where $p, q \in \mathbb{Z}$ and $q \neq 0$.
• Irrational Numbers ($\mathbb{I}$): Possess decimal expansions that are strictly non-terminating and non-repeating. They cannot be expressed as a ratio of two integers.

Step 1: Analyzing the Termination of the Decimal

Observe the given number $x = 1.101001000100001...$. The presence of the ellipsis ($...$) at the end of the number explicitly indicates that the decimal digits continue infinitely. [Per the standard mathematical notation for infinite sequences].

Therefore, the decimal expansion is definitively non-terminating.

Step 2: Analyzing the Recurrence (Repeating Pattern)

Next, we examine the sequence of digits after the decimal point to determine if there is a fixed, finite block of digits that repeats infinitely.

The sequence of fractional digits is: $101001000100001...$

Let us break down the structural pattern of zeros and ones:

• First segment: $10$ (contains exactly one zero)
• Second segment: $100$ (contains exactly two zeros)
• Third segment: $1000$ (contains exactly three zeros)
• Fourth segment: $10000$ (contains exactly four zeros)

Because the number of zeros between consecutive ones increases by exactly one each time, the sequence constantly changes its structure. [By the mathematical definition of periodicity, a sequence is periodic if $a_{n+p} = a_n$ for some constant period $p$. The strictly increasing run-length of zeros mathematically prevents any constant period $p$ from existing].

Therefore, the decimal expansion is non-repeating (non-recurring).

Step 3: Visualizing the Classification Framework

The following flowchart illustrates the logical path used to classify the given number based on its decimal expansion.

Step 4: Final Classification

Since the decimal expansion of $1.101001000100001...$ is proven to be both non-terminating and non-repeating, it cannot be expressed as a simple fraction $\frac{p}{q}$.

[Per the fundamental theorem of rational numbers, only numbers with terminating or repeating decimals belong to the set of rational numbers $\mathbb{Q}$. All other real numbers belong to the set of irrational numbers $\mathbb{I}$].

Final Solution: The number $1.101001000100001...$ is an irrational number.