To express $0.123\overline{3}$ as a rational number, let's follow these steps:

Let: $x = 0.123\overline{3}$ This means the decimal repeats indefinitely with 3s.

Step 1: Eliminate the repeating decimal

Multiply $x$ by 10 to shift the decimal point one place to the right: $10x = 1.23\overline{3}$

Now subtract the original $x$ from this equation to eliminate the repeating part: $10x - x = (1.23\overline{3}) - (0.123\overline{3})$

This simplifies to: $9x = 1.11$

Step 2: Solve for $x$

Now, divide both sides of the equation by 9: $x = \frac{1.11}{9}$

Step 3: Convert $1.11$ to a fraction

Notice that $1.11$ is a terminating decimal, so we can express it as a fraction: $1.11 = \frac{111}{100}$

Step 4: Substitute back into the equation

Substitute $\frac{111}{100}$ back into the equation for $x$: $x = \frac{\frac{111}{100}}{9} = \frac{111}{900}$

Step 5: Simplify the fraction

Now, simplify the fraction $\frac{111}{900}$. The greatest common divisor (GCD) of 111 and 900 is 3, so divide both numerator and denominator by 3: $x = \frac{111 \div 3}{900 \div 3} = \frac{37}{300}$

Thus, $0.123\overline{3}$ as a rational number is: $x = \frac{37}{300}$

Would you like more details on this solution or have any further questions?

Here are 5 related questions:

1. How do you convert other repeating decimals into fractions?
2. What is the significance of rational numbers in number theory?
3. Can non-repeating decimals be expressed as fractions?
4. How does the method differ for repeating patterns longer than one digit?
5. What are examples of repeating decimals that represent simple fractions?

Tip: When converting repeating decimals to fractions, it's useful to know the general formula $x = \frac{n}{d}$, where $n$ is the repeating part's sum and $d$ is the number of repeating digits times 9.