To determine which number, when added to another number, produces an irrational number, we first need to understand the properties of rational and irrational numbers:
1. Rational Numbers: These are numbers that can be expressed as the ratio of two integers. Examples include fractions like [tex]\(\frac{5}{6}\)[/tex], repeating or terminating decimals like -0.625 and 0.777...
2. Irrational Numbers: These numbers cannot be expressed as a simple ratio of two integers. They have non-repeating, non-terminating decimal expansions. An example is the decimal [tex]\(0.13241536\ldots\)[/tex], where the digits continue without repeating in a predictable pattern.
Here's how we can decide which option, when added to another number, results in an irrational number:
- Adding a rational number to another rational number always yields a rational number. This is due to the closure property of addition for the set of rational numbers.
- Adding an irrational number to a rational number always yields an irrational number. This is because an irrational number cannot be represented as the ratio of two integers, which makes the sum irrational as well.
Given the choices:
- A. [tex]\(\frac{5}{6}\)[/tex] is a rational number.
- B. -0.625 is a rational number (it can be expressed as [tex]\(-\frac{625}{1000}\)[/tex] or [tex]\(-\frac{5}{8}\)[/tex]).
- C. [tex]\(0.13241536\ldots\)[/tex] is an irrational number (it has non-repeating, non-terminating decimal expansion).
- D. [tex]\(0.777\ldots\)[/tex] is a rational number (it repeats as [tex]\(0.\overline{7}\)[/tex] and can be expressed as [tex]\(\frac{7}{9}\)[/tex]).
Therefore, option C, [tex]\(0.13241536\ldots\)[/tex], when added to any number, will produce an irrational number because it is itself an irrational number. This ensures the sum cannot be expressed as a ratio of two integers, preserving its irrationality.
Thus, the answer is:
[tex]\[ \boxed{C} \][/tex]
