To determine which number produces a rational number when added to \(\frac{1}{2}\), let's consider the definitions and properties of rational and irrational numbers.
Step 1: Understanding Rational and Irrational Numbers
- A rational number is a number that can be expressed as the quotient of two integers, i.e., \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\).
- An irrational number is a number that cannot be expressed as a simple fraction, meaning it cannot be written as \(\frac{p}{q}\). Common examples include \(\pi\) and \(\sqrt{12}\).
Step 2: Evaluating Each Option
Option A: \(\pi\)
\(\pi\) is a well-known irrational number. Adding an irrational number to a rational number yields an irrational result:
[tex]\[ \pi + \frac{1}{2} \text{ is irrational.} \][/tex]
Option B: \(\sqrt{12}\)
\(\sqrt{12}\) can be simplified to \(2\sqrt{3}\) and is also irrational. Hence:
[tex]\[ \sqrt{12} + \frac{1}{2} \text{ is irrational.} \][/tex]
Option C: 0.314
This number is written in decimal form and terminates. Any terminating decimal or repeating decimal is a rational number, so:
[tex]\[ 0.314 + \frac{1}{2} \text{ is rational.} \][/tex]
Option D: \(4.35889894 \ldots\)
If this number represents a non-repeating, non-terminating decimal, it is irrational. Therefore:
[tex]\[ 4.35889894 \ldots + \frac{1}{2} \text{ is irrational.} \][/tex]
Step 3: Conclusion
Among the given options, the only number that, when added to \(\frac{1}{2}\), produces a rational number is:
[tex]\[ \boxed{0.314} \][/tex]
Therefore, the number that produces a rational number when added to [tex]\(\frac{1}{2}\)[/tex] is [tex]\(0.314\)[/tex].
