To classify the number [tex]\(1.01011011101111...\)[/tex], we need to carefully examine its decimal representation. Let's walk through the determination step-by-step:
1. Definition of Rational and Irrational Numbers:
- Rational Numbers: A number is considered rational if it can be expressed as the quotient [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers and [tex]\( q \neq 0 \)[/tex]. Additionally, the decimal representation of a rational number either terminates after a finite number of digits or repeats a fixed pattern indefinitely.
- Irrational Numbers: A number is considered irrational if it cannot be expressed as a simple fraction. In other words, its decimal representation is both non-terminating (goes on forever) and non-repeating (no repeating pattern).
2. Analyzing the Given Number [tex]\( 1.01011011101111... \)[/tex]:
- Let's carefully observe the digits after the decimal point in the number [tex]\( 1.01011011101111... \)[/tex].
- The sequence starts with [tex]\([0101]\)[/tex], then [tex]\([10]\)[/tex], then [tex]\([1110]\)[/tex], and keeps going on with increasingly complicated and non-repeating patterns.
3. Non-Terminating:
- The decimal expansion [tex]\( 1.01011011101111... \)[/tex] does not end. It goes on indefinitely, which means it is non-terminating.
4. Non-Repeating:
- Even more importantly, the sequence of digits after the decimal point does not exhibit a repeating pattern. Each segment grows in length without any fixed period that repeats. For example, we see longer and different sequences occurring rather than a uniform repeating segment such as [tex]\(0.3333...\)[/tex] (which is rational and repeating).
Based on these observations:
- Since the decimal expansion is non-terminating and non-repeating, it cannot be represented as a simple fraction [tex]\( \frac{p}{q} \)[/tex].
Therefore, we conclude that:
- The number [tex]\(1.01011011101111...\)[/tex] is irrational.
This means it cannot be expressed as a ratio of two integers, meeting the criteria for an irrational number.
