The number in question is:

$0.010010001000010000010000001000000010000000010000000001\ldots$

This sequence consists of alternating 1s and an increasing number of zeros between them. The pattern of zeros grows by one additional zero between each successive 1.

To prove this number is irrational, let's break down the reasoning:

Key Points:

1. Non-Repeating, Non-Terminating Decimal:

   • A rational number, when written as a decimal, either terminates (e.g., $0.25$) or repeats periodically (e.g., $0.3333\ldots$).
   • The given number does not terminate, and it does not form a repeating pattern. The number of zeros between the 1s continually increases without repeating any fixed block.

2. Increasing Gaps Between 1s:

   • The distance between consecutive 1s grows indefinitely. Specifically, the sequence starts with 1 zero between two 1s, then 2 zeros, then 3, and so on. This makes it impossible to express the number as a ratio of two integers, as no repeating block of digits exists.

3. Contradiction with Rationality:

   • If this number were rational, its decimal expansion would eventually settle into a repeating pattern. However, the pattern here is ever-expanding, which violates the property of rational numbers. Hence, the number cannot be written as a fraction of two integers.

Therefore, the number is irrational because its decimal expansion is non-repeating and non-terminating.

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:

1. What is the definition of a rational number in terms of decimals?
2. How can a number’s decimal expansion help classify it as rational or irrational?
3. What other examples of non-repeating, non-terminating decimals are there?
4. How can we prove the irrationality of numbers like $\pi$ or $e$?
5. What is the difference between periodic and non-periodic decimal expansions?

Tip: When determining whether a number is rational or irrational, always check if the decimal either terminates or eventually repeats a pattern.