To solve the problem of determining which number, when added to 0.53, produces a rational number, consider the properties of rational and irrational numbers. A rational number can be expressed as a ratio of two integers, whereas an irrational number cannot be expressed as such a ratio and has a non-repeating, non-terminating decimal expansion.
Given the options to consider:
- Option A: 0.2645751311...
- This number appears to be irrational because it has a non-repeating, non-terminating decimal expansion. Adding an irrational number to 0.53 will result in an irrational number.
- Therefore, this option is not suitable if we want to produce a rational result.
- Option B: 57
- This is a whole number and clearly rational. Adding a rational number to 0.53, which itself is a rational number, will always result in a rational number.
- Therefore, this option is suitable if we want to produce a rational result.
- Option C: √5
- The square root of 5 is an irrational number because it cannot be expressed as a ratio of two integers and its decimal expansion is non-repeating and non-terminating.
- Adding an irrational number to 0.53 will result in an irrational number.
- Therefore, this option is not suitable if we want to produce a rational result.
- Option D: π (pi)
- Pi is a well-known irrational number. Its decimal expansion never repeats and never terminates.
- Adding an irrational number to 0.53 will result in an irrational number.
- Therefore, this option is not suitable if we want to produce a rational result.
Based on this analysis, the number that produces a rational result when added to 0.53 is:
[tex]\[ \boxed{57} \][/tex]
