To determine which number, when added to 0.4, produces an irrational number, we need to examine each of the given options:
A. [tex]\(\frac{2}{7}\)[/tex]
B. [tex]\(0.444 \ldots\)[/tex]
C. [tex]\(3 \pi\)[/tex]
D. [tex]\(\sqrt{9}\)[/tex]
Let's analyze each option:
1. Option A: [tex]\(\frac{2}{7}\)[/tex]
- [tex]\(\frac{2}{7}\)[/tex] is a rational number because it can be expressed as a fraction of two integers.
- Adding 0.4 (a rational number) to [tex]\(\frac{2}{7}\)[/tex] (also a rational number) will result in a rational number.
2. Option B: [tex]\(0.444 \ldots\)[/tex]
- [tex]\(0.444 \ldots\)[/tex] is also a rational number because it is a repeating decimal that can be expressed as a fraction.
- Adding 0.4 (a rational number) to [tex]\(0.444 \ldots\)[/tex] (also a rational number) will result in a rational number.
3. Option C: [tex]\(3 \pi\)[/tex]
- [tex]\(3 \pi\)[/tex] is an irrational number because it is a product of 3 (a rational number) and [tex]\(\pi\)[/tex] (an irrational number).
- Adding 0.4 (a rational number) to an irrational number ([tex]\(3 \pi\)[/tex]) results in an irrational number. This is because the sum of a rational number and an irrational number is always irrational.
4. Option D: [tex]\(\sqrt{9}\)[/tex]
- [tex]\(\sqrt{9}\)[/tex] equals 3, which is a rational number.
- Adding 0.4 (a rational number) to 3 (also a rational number) will result in a rational number.
From the analysis, the only option that produces an irrational number when added to 0.4 is option C: [tex]\(3 \pi\)[/tex].
The irrational sum when 0.4 is added to [tex]\(3 \pi\)[/tex] is approximately [tex]\(9.82477796076938\)[/tex].
Therefore, the correct answer is:
C. [tex]\(3 \pi\)[/tex]
