The decimal representation 0.393993999399993999993... is an interesting number. Let's analyze whether it is rational or irrational:
1. Rational Numbers:
- A number is rational if its decimal representation either **terminates** (ends) or repeats.
- For instance, 0.75 is a rational number because its decimal representation ends.
- Similarly, 0.3333... (repeating 3s) is also rational because it repeats³.
2. Irrational Numbers:
- An irrational number is one whose decimal representation neither terminates nor repeats.
- The number 0.393993999399993999993... falls into this category.
- It has a non-terminating and non-repeating decimal pattern.
- As a result, it cannot be expressed as a ratio of two integers.
- In other words, it cannot be written in the form of p/q, where p and q are integers and q ≠ 0.
- Therefore, 0.393993999399993999993... is an irrational number.
In summary, the number 0.393993999399993999993... is irrational due to its non-repeating and non-terminating decimal expansion.
