The numbers `7.478478….` and `1.101001000100001` ….. are

Options

A. Rational and irrational respectively
B. Both rationals
C. Both irrationals
D. None of these

Answer

To determine whether the numbers \(7.478478...\) and \(1.1010010001...\) are rational or irrational, we can follow these steps:

### Step 1: Analyze the first number \(7.478478...\)

1. **Identify the nature of the decimal**: The decimal \(7.478478...\) has a repeating part, which is \(478\).
2. **Definition of rational numbers**: A number is rational if it can be expressed in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\).
3. **Conclusion for the first number**: Since \(7.478478...\) is a non-terminating but repeating decimal, it can be expressed as a fraction. Therefore, \(7.478478...\) is a rational number.

### Step 2: Analyze the second number \(1.1010010001...\)

1. **Identify the nature of the decimal**: The decimal \(1.1010010001...\) does not have a repeating pattern. The number of zeros between the ones increases progressively (1 zero, then 2 zeros, then 3 zeros, etc.).
2. **Conclusion for the second number**: Since \(1.1010010001...\) is a non-terminating and non-repeating decimal, it cannot be expressed as a fraction. Therefore, \(1.1010010001...\) is an irrational number.

### Final Conclusion

- The first number \(7.478478...\) is a rational number.
- The second number \(1.1010010001...\) is an irrational number.

Thus, the answer is that the first number is rational and the second number is irrational.

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