To understand why [tex]\( 3 \frac{1}{9} \)[/tex] must be a repeating decimal, let's break it down step-by-step.
### Step 1: Understand the Mixed Number
[tex]\( 3 \frac{1}{9} \)[/tex] is a mixed number, where 3 is the whole part and [tex]\( \frac{1}{9} \)[/tex] is the fractional part.
### Step 2: Convert the Fraction to Decimal
Let's focus on the fractional part [tex]\( \frac{1}{9} \)[/tex].
[tex]\[ \frac{1}{9} = 0.\overline{1} \][/tex]
This means that when you divide 1 by 9, the result is a repeating decimal: [tex]\( 0.1111... \)[/tex], where the digit '1' repeats infinitely.
### Step 3: Combine the Whole Part and the Fractional Part
Now, add this repeating decimal to the whole part of the mixed number.
[tex]\[ 3 + 0.1111... = 3.1111... \][/tex]
### Step 4: Express the Combined Number as a Decimal
The resulting decimal number [tex]\( 3.1111... \)[/tex] includes the integer part 3 and the repeating decimal part [tex]\( 0.\overline{1} \)[/tex].
### Conclusion
Because the fractional part [tex]\( \frac{1}{9} \)[/tex] converts to a repeating decimal [tex]\( 0.\overline{1} \)[/tex], the entire mixed number [tex]\( 3 \frac{1}{9} \)[/tex] results in a repeating decimal:
[tex]\[ 3.1111... = 3.111111111111111 \][/tex]
Thus, we can conclude that [tex]\( 3 \frac{1}{9} \)[/tex] is a repeating decimal due to the repeating nature of its fractional part [tex]\( \frac{1}{9} \)[/tex].
