Is 0.999... Really 1?
Is an infinite decimal "infinitely close" or "exactly the same"?
Unraveling intuitive discomfort with algebraic proofs.
1. The Paradox
When told that "$0.999\dots = 1$", many people feel an intuitive rejection.
"I understand that as you add $0.9$, $0.99$, $0.999\dots$, it gets infinitely close to $1$. But it's only 'approaching' it; doesn't it technically never reach $1$?"
This sensation stems from our tendency to perceive numbers not as "static points on a number line" but as a "dynamic process of approaching." However, in mathematical definition, these two values represent exactly the same point.
2. Proof 1: Algebraic Approach
The most famous algebraic proof. We derive the answer by cleverly eliminating the infinite recurring part.
Let $x$ be the number in question:
$$ x = 0.999\dots $$
Multiply both sides by $10$. The decimal point shifts one place to the right.
$$ 10x = 9.999\dots $$
Subtract the original $x$ from this equation.
$$ \begin{align} 10x &= 9.999\dots \\ -)\; x &= 0.999\dots \\ \hline 9x &= 9 \end{align} $$
Thus, we conclude that $x = 1$.
3. Proof 2: Fraction Approach
A more intuitive approach using fractions.
It is widely accepted that $1 \div 3$ equals $0.333\dots$.
$$ \frac{1}{3} = 0.333\dots $$
Multiply both sides of this equation by $3$.
$$ \frac{1}{3} \times 3 = 0.333\dots \times 3 $$
The left side becomes $1$, and on the right side, each digit $3$ becomes $9$.
$$ 1 = 0.999\dots $$
4. Dr.WataWata's Insight
Why does this equality hold? The essence lies in the lack of "uniqueness of representation."
We tend to assume that "one number has only one way to be written." However, just as "$\frac{1}{2}$", "$\frac{2}{4}$", and "$0.5$" all represent the same number, "$1$" and "$0.999\dots$" are merely two different names pointing to the exact same spot on the number line.
If $0.999\dots$ and $1$ were different numbers, there must be a "gap" between them, and another number must exist in that gap. However, the difference $1 - 0.999\dots$ is $0.000\dots$, and no non-zero digit ever appears no matter how far you go.
It does not "approach"; it is "there" from the beginning. This is the conclusion brought by the continuity of real numbers.