To convert the rational number \( \frac{4}{9} \) to its equivalent decimal form using long division, we can follow these steps:
-
Set up the division: We want to divide 4 by 9. Since 4 is less than 9, we can write this as \( 4.000000 \div 9 \).
-
Perform the long division:
- Step 1: 9 goes into 40 (the first two digits of 4.000000) 4 times, since \( 9 \times 4 = 36 \). Write 4 above the line.
- Step 2: Subtract 36 from 40, which gives us 4. Bring down the next 0 to make it 40 again.
- Step 3: Repeat the process. 9 goes into 40, again, 4 times. Subtract 36 from 40 to get 4, then bring down the next 0.
-
Recognizing the pattern: You will notice that the remainder is 4 every time you bring down a 0. This means that the process will repeat indefinitely.
The division looks like this:
0.44444...
__________
9 | 4.000000...
-36
----
40
-36
----
40
-36
----
40
...
- Conclusion: The decimal representation of \( \frac{4}{9} \) is \( 0.4\overline{4} \), where the \( 4 \) repeats indefinitely.
So, the equivalent decimal form of \( \frac{4}{9} \) is:
\[ 0.44444\ldots \text{ or } 0.4\overline{4} \]