Answer :
To determine which number produces a rational number when added to \(\frac{1}{3}\), we need to analyze the nature of rational and irrational numbers.
1. Rational Numbers: These are numbers that can be expressed as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers, and \(b \neq 0\).
2. Irrational Numbers: These are numbers that cannot be expressed as a simple fraction. Irrational numbers have non-terminating, non-repeating decimal expansions.
Given the options, let's analyze each one:
- Option A: \(\pi\)
- \(\pi\) is well-known to be an irrational number.
- Adding \(\pi\) to \(\frac{1}{3}\) will result in an irrational number. Hence, \(\pi + \frac{1}{3}\) is irrational.
- Option B: \(\sqrt{10}\)
- \(\sqrt{10}\) is an irrational number because the square root of a non-perfect square is always irrational.
- Adding \(\sqrt{10}\) to \(\frac{1}{3}\) will result in an irrational number. Hence, \(\sqrt{10} + \frac{1}{3}\) is irrational.
- Option C: 0.22
- 0.22 is a terminating decimal, and all terminating decimals are rational numbers. It can be expressed as the fraction \(\frac{22}{100}\) which simplifies to \(\frac{11}{50}\).
- Adding \(\frac{11}{50}\) (0.22) to \(\frac{1}{3}\) gives a sum that can be verified as rational, but we need to check the result as well.
- Option D: \(5.38516480 \ldots\)
- The number 5.38516480... is represented with an ellipsis indicating that the number continues without repeating. These are characteristics of an irrational number.
- Adding \(5.38516480 \ldots\) to \(\frac{1}{3}\) will result in an irrational number.
Given that only rational numbers added to a rational number (\(\frac{1}{3}\)) will yield a rational result, we identify:
- Option C: 0.22 is rational.
Therefore, none of the given options produce a rational number when added to \(\frac{1}{3}\). The correct answer is:
```
None
```
1. Rational Numbers: These are numbers that can be expressed as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers, and \(b \neq 0\).
2. Irrational Numbers: These are numbers that cannot be expressed as a simple fraction. Irrational numbers have non-terminating, non-repeating decimal expansions.
Given the options, let's analyze each one:
- Option A: \(\pi\)
- \(\pi\) is well-known to be an irrational number.
- Adding \(\pi\) to \(\frac{1}{3}\) will result in an irrational number. Hence, \(\pi + \frac{1}{3}\) is irrational.
- Option B: \(\sqrt{10}\)
- \(\sqrt{10}\) is an irrational number because the square root of a non-perfect square is always irrational.
- Adding \(\sqrt{10}\) to \(\frac{1}{3}\) will result in an irrational number. Hence, \(\sqrt{10} + \frac{1}{3}\) is irrational.
- Option C: 0.22
- 0.22 is a terminating decimal, and all terminating decimals are rational numbers. It can be expressed as the fraction \(\frac{22}{100}\) which simplifies to \(\frac{11}{50}\).
- Adding \(\frac{11}{50}\) (0.22) to \(\frac{1}{3}\) gives a sum that can be verified as rational, but we need to check the result as well.
- Option D: \(5.38516480 \ldots\)
- The number 5.38516480... is represented with an ellipsis indicating that the number continues without repeating. These are characteristics of an irrational number.
- Adding \(5.38516480 \ldots\) to \(\frac{1}{3}\) will result in an irrational number.
Given that only rational numbers added to a rational number (\(\frac{1}{3}\)) will yield a rational result, we identify:
- Option C: 0.22 is rational.
Therefore, none of the given options produce a rational number when added to \(\frac{1}{3}\). The correct answer is:
```
None
```
