Скачать или смотреть Problem 1.1.2. | Chapter 1 I A Maron - Problems in Calculus of One Variable

1.1.2. Prove that any number, with zeros standing in all decimal places numbered 10^𝑛 and only in these places, is irrational.

Key Idea:

The decimal expansion of such a number has zeros only in positions 10,100,1000,…and all other decimal places are filled with non-zero digits. This pattern ensures that the decimal expansion is non-terminating and non-repeating, which implies the number is irrational.
Proof:

Assume the number is rational:
Suppose the number x is rational. Then, by definition, its decimal expansion must either terminate or eventually repeat.

Decimal expansion of xx:
The decimal expansion of xx has zeros only in positions 10^n (i.e., positions 10,100,1000,…), and all other positions are filled with non-zero digits. This means:

The zeros are spaced farther and farther apart as n increases.

The number of non-zero digits between consecutive zeros grows without bound.

Non-repeating nature:
For the decimal expansion to eventually repeat, there must exist a fixed block of digits that repeats indefinitely. However, in x, the spacing between zeros grows without bound (e.g., the first zero is at position 10, the next at position 100, the next at position 1000, etc.). This means there is no fixed block of digits that repeats, as the pattern of zeros becomes increasingly sparse.

Contradiction:
If x were rational, its decimal expansion would have to either terminate or eventually repeat. However, the decimal expansion of x neither terminates nor repeats, because the zeros are spaced farther and farther apart, and the non-zero digits between them do not follow a repeating pattern. This contradicts the assumption that xx is rational.

Conclusion:
Since assuming xx is rational leads to a contradiction, xx must be irrational.

Final Answer:

Any number with zeros standing in all decimal places numbered 10^n (for n=1,2,3,…) and only in these places is irrational, because its decimal expansion is non-terminating and non-repeating, which violates the definition of a rational number.

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