To determine which number, when added to [tex]\(\frac{1}{3}\)[/tex], produces an irrational number, let's analyze each option one by one.
Consider the definition of irrational numbers: An irrational number is a number that cannot be expressed as a simple fraction or a ratio of two integers. Its decimal representation goes on forever without repeating.
### Option A: [tex]\(\frac{2}{3}\)[/tex]
Adding [tex]\(\frac{1}{3}\)[/tex] and [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[
\frac{1}{3} + \frac{2}{3} = \frac{3}{3} = 1
\][/tex]
The result is 1, which is a rational number.
### Option B: 2
Adding [tex]\(\frac{1}{3}\)[/tex] and 2:
[tex]\[
\frac{1}{3} + 2 = \frac{1}{3} + \frac{6}{3} = \frac{1 + 6}{3} = \frac{7}{3}
\][/tex]
The result [tex]\(\frac{7}{3}\)[/tex] is a rational number.
### Option C: 0.166
Adding [tex]\(\frac{1}{3}\)[/tex] and 0.166:
[tex]\[
\frac{1}{3} \approx 0.333\ldots
\][/tex]
[tex]\[
0.333 + 0.166 = 0.499
\][/tex]
The number [tex]\( 0.499 \)[/tex] terminates and can be expressed as [tex]\(\frac{499}{1000}\)[/tex], which is a rational number.
### Option D: [tex]\(2\pi\)[/tex]
Adding [tex]\(\frac{1}{3}\)[/tex] and [tex]\(2\pi\)[/tex]:
[tex]\[
\frac{1}{3} + 2\pi
\][/tex]
- Given [tex]\(2\pi \approx 6.283185307179586\)[/tex].
Adding a rational number, [tex]\(\frac{1}{3}\)[/tex], to an irrational number, [tex]\(2\pi\)[/tex], results in an irrational number. Since [tex]\(2\pi\)[/tex] is irrational, adding any rational number to it will still produce an irrational number.
Thus, the number that produces an irrational number when added to [tex]\(\frac{1}{3}\)[/tex] is:
[tex]\[ \boxed{2\pi} \][/tex]
