To determine which number produces a rational number when multiplied by 0.5, we need to analyze each option one by one.
Option A: [tex]\( \sqrt{3} \)[/tex]
- [tex]\( \sqrt{3} \)[/tex] is an irrational number because it cannot be expressed as a fraction of two integers.
- Multiplying an irrational number by 0.5 (which is a simple fraction 1/2) will still produce an irrational number.
- Therefore, [tex]\( \sqrt{3} \times 0.5 \)[/tex] is irrational.
Option B: [tex]\( 0.54732814 \ldots \)[/tex]
- The number [tex]\( 0.54732814 \ldots \)[/tex] appears to be a non-repeating, non-terminating decimal, which means it is an irrational number.
- Multiplying an irrational number by a rational number (0.5) will still produce an irrational number.
- Therefore, [tex]\( 0.54732814 \ldots \times 0.5 \)[/tex] is irrational.
Option C: [tex]\( -1.73205089 \ldots \)[/tex]
- The number [tex]\( -1.73205089 \ldots \)[/tex] appears to be a non-repeating, non-terminating decimal, which indicates it is an irrational number.
- Multiplying an irrational number by a rational number (0.5) will still result in an irrational number.
- Therefore, [tex]\( -1.73205089 \ldots \times 0.5 \)[/tex] is irrational.
Option D: [tex]\( \frac{1}{3} \)[/tex]
- The number [tex]\( \frac{1}{3} \)[/tex] is a rational number because it can be expressed as a fraction of two integers.
- Multiplying a rational number by another rational number (0.5 or [tex]\( \frac{1}{2} \)[/tex]) will result in a rational number.
- Hence, [tex]\( \frac{1}{3} \times 0.5 = \frac{1}{3} \times \frac{1}{2} = \frac{1}{6} \)[/tex], which is a rational number.
Among the given options, the only number that results in a rational number when multiplied by 0.5 is [tex]\( \frac{1}{3} \)[/tex].
The answer is:
[tex]\[ \boxed{4} \][/tex]
