To determine if the number 0.27322370013414... is rational or irrational, let's review the definitions and characteristics of these types of numbers.
### Rational Numbers
1. Definition: A rational number is any number that can be expressed as a fraction [tex]\(\frac{a}{b}\)[/tex] where both [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers and [tex]\(b \neq 0\)[/tex].
2. Decimal Representation: A rational number's decimal form either terminates (comes to an end) or repeats a pattern indefinitely.
### Irrational Numbers
1. Definition: An irrational number cannot be expressed as a simple fraction [tex]\(\frac{a}{b}\)[/tex].
2. Decimal Representation: An irrational number's decimal form neither terminates nor has a repeating pattern. It goes on infinitely without repetition.
Given Number Analysis:
- The number in question is 0.27322370013414..., which we need to examine closely.
- Decimal Expansion: The given number has a non-terminating and non-repeating decimal expansion, evident from the extended digits and the absence of a repeating sequence.
### Conclusion:
Since the number 0.27322370013414... has an infinite decimal expansion that does not repeat, it cannot be expressed as a simple fraction. Therefore, we conclude that the number is irrational.
This means:
The number 0.27322370013414... is irrational because its decimal representation is infinite and non-repeating.
