Answer
See explanation
Work Step by Step
A standard way to show \(0.1999\ldots\) (repeating 9s) equals \(0.2\) is to use a simple algebraic manipulation:
1. **Let** \(x = 0.1999\ldots\).
2. **Multiply both sides by 10** to shift the decimal point:
\[
10x = 1.9999\ldots
\]
3. **Subtract** the original \(x\) from this equation:
\[
10x - x = 1.9999\ldots - 0.1999\ldots
\quad\Longrightarrow\quad
9x = 1.8.
\]
4. **Solve for \(x\)**:
\[
x = \frac{1.8}{9} = 0.2.
\]
Hence \(0.1999\ldots = 0.2\).
Another perspective:
- \(0.1999\ldots\) can be viewed as \(0.2 - 0.0000\ldots1\), but that “\(...1\)” is an infinitely small quantity in decimal representation and thus equals 0 in the real‐number system. Either way, the rigorous conclusion is that \(0.1999\ldots\) and \(0.2\) represent **the same real number**.
