Answer :
Let's examine each option in detail to see which one produces an irrational number when added to [tex]\(\frac{1}{3}\)[/tex].
### Option A: [tex]\(\sqrt{9}\)[/tex]
First, let's simplify [tex]\(\sqrt{9}\)[/tex]:
[tex]\[ \sqrt{9} = 3 \][/tex]
Next, we add [tex]\(\frac{1}{3}\)[/tex] to 3:
[tex]\[ 3 + \frac{1}{3} = \frac{9}{3} + \frac{1}{3} = \frac{10}{3} \][/tex]
The result [tex]\(\frac{10}{3}\)[/tex] is a rational number.
### Option B: [tex]\(-\frac{1}{7}\)[/tex]
Next, let's add [tex]\(\frac{1}{3}\)[/tex] to [tex]\(-\frac{1}{7}\)[/tex]:
[tex]\[ \frac{1}{3} - \frac{1}{7} \][/tex]
To add these fractions, we need a common denominator. The least common multiple of 3 and 7 is 21:
[tex]\[ \frac{1}{3} = \frac{7}{21} \quad \text{and} \quad \frac{1}{7} = \frac{3}{21} \][/tex]
Now, subtract the fractions:
[tex]\[ \frac{7}{21} - \frac{3}{21} = \frac{4}{21} \][/tex]
The result [tex]\(\frac{4}{21}\)[/tex] is a rational number.
### Option C: [tex]\(2.4494897 \ldots\)[/tex]
Now, let's add [tex]\(\frac{1}{3}\)[/tex] to [tex]\(2.4494897 \ldots\)[/tex]:
[tex]\[ \frac{1}{3} + 2.4494897 \ldots \][/tex]
Given the sum is approximately:
[tex]\[ 2.7828230333333335 \][/tex]
This result is an irrational number because an irrational number added to a rational number (like [tex]\(\frac{1}{3}\)[/tex]) results in an irrational number.
### Option D: [tex]\(0.464646 \ldots\)[/tex]
Finally, let's add [tex]\(\frac{1}{3}\)[/tex] to [tex]\(0.464646 \ldots\)[/tex]. The number [tex]\(0.464646 \ldots\)[/tex] is actually a repeating decimal and can be expressed as a fraction (a rational number):
[tex]\[ 0.464646 \ldots = \frac{46}{99} \][/tex]
Add [tex]\(\frac{1}{3}\)[/tex] to [tex]\(\frac{46}{99}\)[/tex]. First, convert [tex]\(\frac{1}{3}\)[/tex] to the same denominator:
[tex]\[ \frac{1}{3} = \frac{33}{99} \][/tex]
Now, add the fractions:
[tex]\[ \frac{33}{99} + \frac{46}{99} = \frac{79}{99} \][/tex]
The result [tex]\(\frac{79}{99}\)[/tex] is also a rational number.
### Conclusion:
From these calculations, only Option C ([tex]\(2.4494897 \ldots\)[/tex]) produces an irrational number when added to [tex]\(\frac{1}{3}\)[/tex]. Therefore, the correct answer is:
[tex]\[ \boxed{2.4494897 \ldots} \][/tex]
### Option A: [tex]\(\sqrt{9}\)[/tex]
First, let's simplify [tex]\(\sqrt{9}\)[/tex]:
[tex]\[ \sqrt{9} = 3 \][/tex]
Next, we add [tex]\(\frac{1}{3}\)[/tex] to 3:
[tex]\[ 3 + \frac{1}{3} = \frac{9}{3} + \frac{1}{3} = \frac{10}{3} \][/tex]
The result [tex]\(\frac{10}{3}\)[/tex] is a rational number.
### Option B: [tex]\(-\frac{1}{7}\)[/tex]
Next, let's add [tex]\(\frac{1}{3}\)[/tex] to [tex]\(-\frac{1}{7}\)[/tex]:
[tex]\[ \frac{1}{3} - \frac{1}{7} \][/tex]
To add these fractions, we need a common denominator. The least common multiple of 3 and 7 is 21:
[tex]\[ \frac{1}{3} = \frac{7}{21} \quad \text{and} \quad \frac{1}{7} = \frac{3}{21} \][/tex]
Now, subtract the fractions:
[tex]\[ \frac{7}{21} - \frac{3}{21} = \frac{4}{21} \][/tex]
The result [tex]\(\frac{4}{21}\)[/tex] is a rational number.
### Option C: [tex]\(2.4494897 \ldots\)[/tex]
Now, let's add [tex]\(\frac{1}{3}\)[/tex] to [tex]\(2.4494897 \ldots\)[/tex]:
[tex]\[ \frac{1}{3} + 2.4494897 \ldots \][/tex]
Given the sum is approximately:
[tex]\[ 2.7828230333333335 \][/tex]
This result is an irrational number because an irrational number added to a rational number (like [tex]\(\frac{1}{3}\)[/tex]) results in an irrational number.
### Option D: [tex]\(0.464646 \ldots\)[/tex]
Finally, let's add [tex]\(\frac{1}{3}\)[/tex] to [tex]\(0.464646 \ldots\)[/tex]. The number [tex]\(0.464646 \ldots\)[/tex] is actually a repeating decimal and can be expressed as a fraction (a rational number):
[tex]\[ 0.464646 \ldots = \frac{46}{99} \][/tex]
Add [tex]\(\frac{1}{3}\)[/tex] to [tex]\(\frac{46}{99}\)[/tex]. First, convert [tex]\(\frac{1}{3}\)[/tex] to the same denominator:
[tex]\[ \frac{1}{3} = \frac{33}{99} \][/tex]
Now, add the fractions:
[tex]\[ \frac{33}{99} + \frac{46}{99} = \frac{79}{99} \][/tex]
The result [tex]\(\frac{79}{99}\)[/tex] is also a rational number.
### Conclusion:
From these calculations, only Option C ([tex]\(2.4494897 \ldots\)[/tex]) produces an irrational number when added to [tex]\(\frac{1}{3}\)[/tex]. Therefore, the correct answer is:
[tex]\[ \boxed{2.4494897 \ldots} \][/tex]
