Sure! Let's analyze each option and see which one, when added to [tex]\(0.5\)[/tex], results in an irrational number.
#### Option A: [tex]\(\sqrt{3}\)[/tex]
First, let's consider [tex]\(\sqrt{3}\)[/tex]. The square root of 3 is an irrational number. When adding an irrational number to a rational number, the result is typically irrational. Let's see the result:
[tex]\[ 0.5 + \sqrt{3} \][/tex]
From the given data, we know:
[tex]\[ 0.5 + \sqrt{3} \approx 2.232050807568877 \][/tex]
This result is indeed an irrational number because it is a non-terminating, non-repeating decimal.
#### Option B: [tex]\(0.555 \ldots\)[/tex]
Next, let's consider the repeating decimal [tex]\(0.555 \ldots\)[/tex], which is equivalent to [tex]\(\frac{5}{9}\)[/tex]. This is a rational number. Adding two rational numbers results in a rational number:
[tex]\[ 0.5 + 0.5555555555555556 \][/tex]
From the given data, we know:
[tex]\[ 0.5 + 0.5555555555555556 = 1.0555555555555556 \][/tex]
This result is a rational number since it's a repeating decimal.
#### Option C: [tex]\(\frac{1}{3}\)[/tex]
Now, let's consider the fraction [tex]\(\frac{1}{3}\)[/tex]. This is a rational number. The sum of two rational numbers is also rational:
[tex]\[ 0.5 + \frac{1}{3} \][/tex]
From the given data, we know:
[tex]\[ 0.5 + \frac{1}{3} \approx 0.8333333333333333 \][/tex]
This result is also a rational number since it's a repeating decimal.
#### Option D: [tex]\(\sqrt{16}\)[/tex]
Finally, let's consider [tex]\(\sqrt{16}\)[/tex]. The square root of 16 is 4, which is a rational number. Adding two rational numbers results in a rational number:
[tex]\[ 0.5 + \sqrt{16} \][/tex]
From the given data, we know:
[tex]\[ 0.5 + \sqrt{16} = 4.5 \][/tex]
This result is a rational number because it is a finite decimal.
#### Conclusion:
The number that produces an irrational number when added to [tex]\(0.5\)[/tex] is [tex]\(\sqrt{3}\)[/tex]. Thus, the correct answer is:
A. [tex]\(\sqrt{3}\)[/tex]
