Question

Determine if 0, point, 9, 4, 9, 4, 4, 9, 4, 4, 4, 9, 4, 4, 4, 4, 9, 4, 4, 4, 4, 4, 9, dot dot dot0.949449444944449444449... is rational or irrational and give a reason for your answer.

Answer 1

To determine if the number \(0.949449444944449444449...\) is rational or irrational, we can analyze the decimal expansion.

The number given can be expressed as:

\[
0.949449444944449444449\ldots
\]

We can observe the pattern in the decimal: it starts with "94", followed by a series of "4"s and the digit "9" interspersed. The sequence appears to be a repeating pattern, specifically:

- The digit "9" appears after each group of "4"s.
- After the initial "94", there is one "4" followed by "9", then two "4"s followed by "9", three "4"s followed by "9", and so on.

This suggests that the decimal does not terminate and is not purely periodic; however, it can be broken down in terms of formatting if we mark the segments of the pattern explicitly.

To formalize whether such decimals are rational, we observe that:

A number is rational if it can be represented as a fraction of two integers \( \frac{p}{q} \) (where \( q \neq 0 \)). This implies its decimal expansion must either terminate or eventually become periodic (repeat a block of digits indefinitely).

In this case, the sequence does not stabilize to a repeatable block but rather continues to grow in the number of "4"s before each repeating "9". Thus:

1. The number does not terminate (the sequence continues infinitely).
2. The decimal does not become periodic, as the segment of "4"s grows with increasing repetition.

Consequently, the number \(0.949449444944449444449...\) is not a quotient of two integers and therefore is classified as an **irrational number**.

Thus, the answer is:

\[
\text{The number } 0.949449444944449444449\ldots \text{ is irrational.}
\]