Can TWO different points on the number line ever be represented by the SAME infinite decimal?

Question

So I am visualizing this problem, and on the surface it seems like there is no way two individual and unique values could have the same infinite decimal representation. Any thoughts on this? It is for my History of Mathematics course. Thanks, peeps.

Answer 1

No.

Proof. Suppose $x = d_{1}d_{2}d_{3}\ldots$ and $y = d_{1}d_{2}d_{3}\ldots$ are two distinct real numbers with the same decimal representations. Then $0 = d_{1}d_{2}d_{3}\ldots - d_{1}d_{2}d_{3}\ldots = x - y$. But $x$ and $y$ are distinct, contradiction.

Answer 2

For simplicity, let's just consider numbers between $0$ and $1$, to avoid the extra complexity of the whole part of numbers.

To answer your question, it's worth trying to understand a bit better what a decimal expansion is. When we say that a number $x \in (0,1)$ has a decimal representation $0.d_1 d_2 d_3...$, what we mean is that $x$ is equal to the infinite series: $$ x = \sum \limits_{n=1}^{\infty} \frac{d_n}{10^n} = \frac{d_1}{10} + \frac{d_2}{100} + \frac{d_3}{1000} + \cdots $$ Then we must again ask, what does it mean to be equal to an infinite series? Well, the sum of an infinite series is definited by the limit of its partial sums; so in fact we have: $$ x = \lim \limits_{N \rightarrow \infty} \sum \limits_{n=1}^N \frac{d_n}{10^n} $$ So, if $x$ and $y$ had the same decimal expansion, then they would both be limits of this sequence. Thus the question "can two distinct numbers have the same decimal expansion" in fact amounts to "can a sequence have two distinct limits". The answer is no: limits are unique! (This is a basic fact of real analysis.)