To determine which number produces an irrational number when added to [tex]\(\frac{1}{3}\)[/tex], let us analyze each option step-by-step:
### Option A: [tex]\(2\)[/tex]
1. Add [tex]\(2\)[/tex] to [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{6+1}{3} = \frac{7}{3}
\][/tex]
2. [tex]\(\frac{7}{3}\)[/tex] is a rational number because it can be expressed as a fraction of two integers. Therefore, adding [tex]\(2\)[/tex] to [tex]\(\frac{1}{3}\)[/tex] produces a rational number.
### Option B: [tex]\(2\pi\)[/tex]
1. Add [tex]\(2\pi\)[/tex] to [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
2\pi + \frac{1}{3}
\][/tex]
2. We know that [tex]\(\pi\)[/tex] is an irrational number, and any non-zero multiple of an irrational number is also irrational. Therefore, [tex]\(2\pi\)[/tex] is irrational.
3. Adding a rational number (such as [tex]\(\frac{1}{3}\)[/tex]) to an irrational number results in an irrational number. Hence, [tex]\(2\pi + \frac{1}{3}\)[/tex] is irrational.
### Option C: [tex]\(0.166\)[/tex]
1. Convert [tex]\(0.166\)[/tex] to a fraction:
[tex]\[
0.166 = \frac{166}{1000} = \frac{83}{500}
\][/tex]
2. Add [tex]\(\frac{83}{500}\)[/tex] to [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
\frac{83}{500} + \frac{1}{3}
\][/tex]
3. Finding a common denominator (1500):
[tex]\[
\frac{83 \times 3}{500 \times 3} + \frac{1 \times 500}{3 \times 500} = \frac{249}{1500} + \frac{500}{1500} = \frac{249 + 500}{1500} = \frac{749}{1500}
\][/tex]
4. [tex]\(\frac{749}{1500}\)[/tex] is a rational number. Therefore, adding [tex]\(0.166\)[/tex] to [tex]\(\frac{1}{3}\)[/tex] produces a rational number.
### Option D: [tex]\(\frac{2}{3}\)[/tex]
1. Add [tex]\(\frac{2}{3}\)[/tex] to [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
\frac{2}{3} + \frac{1}{3} = \frac{2 + 1}{3} = \frac{3}{3} = 1
\][/tex]
2. [tex]\(1\)[/tex] is a rational number. Therefore, adding [tex]\(\frac{2}{3}\)[/tex] to [tex]\(\frac{1}{3}\)[/tex] produces a rational number.
### Conclusion
After evaluating all the options, we find that the only number that produces an irrational number when added to [tex]\(\frac{1}{3}\)[/tex] is [tex]\(2\pi\)[/tex].
Thus, the correct answer is:
B. [tex]\(2\pi\)[/tex]
