To express the fraction [tex]\(\frac{3}{13}\)[/tex] as a decimal, follow these steps:
1. Division: To convert a fraction to a decimal, divide the numerator by the denominator. Here, we need to divide 3 by 13.
2. Performing the Division: When you carry out the division, you find that [tex]\(\frac{3}{13} = 0.23076923076923078\)[/tex].
3. Rounding or Truncating: Depending on the level of precision required, you might need to round or truncate the decimal. Without rounding it, and just looking at the value at a further decimal point, we often present it as:
- [tex]\(0.230769\)[/tex] (exactly to 6 decimal places)
- If you were to expand more digits, you might say [tex]\(0.23076923076923078\)[/tex], as derived from the division.
4. Repetition or Not: Notice that [tex]\(0.23076923076923078\)[/tex] can be rounded to determine significant repetition in the sequence [tex]\(230769\)[/tex], showing it could be expressed with a repeating decimal notation. However, if you present results clearly, we can use:
- [tex]\(0 . \overline{230769}\)[/tex] which shows 230769 repeats indefinitely, but it is not a finite series.
- [tex]\(0.2307 \overline{69}\)[/tex] might appear less appropriate since it only partially shows repetition after [tex]\(2307\)[/tex].
Therefore, the correct closest representation with high precision and less ambiguous format in this context is [tex]\(\boxed{0.230769}\)[/tex]. This answer captures the essence of the decimal value derived from division.
