Decimal expansion of a fraction

Question

I was reading something which I found really special. It goes like this : Imagine we have a line with unity division (0,1,2,etc.) Now, we have a point on this line. The point can be on a point of division or it can be contained between two points of division. If it isn't on a point of division, then we can continue cutting the line into 10 equal pieces. (The first piece being 0, the second 1, the third 2,...) So, if my point was between 1 and 2, I'll have 1.0,1.1,etc. We continue this cutting until our point arrives at a point of division.If this arrives, we have two choices : We can pick the interval on the left or the interval on the right. And we can continue this way at infinity.

We also have the following formulae to express our numbers : g+a/10+a2/100+a3/1000+... g being an integer and a being the number of the piece which contains our point.

Now, my question concerns this :

https://i.stack.imgur.com/rszUH.jpg

How can it be possible that we can have 1/4=0.2499... and 1/4=0.25000... Also, for this idea to be valid, must it necessarily be an infinite decimal expansion or I could have the following 1/4=0.249 and 1/4=0.250 ?

Thank you !

Answer 1

The point here is that the two real numbers $$0.24999... \text{ and } 0.25000...$$ are the same. There are a couple of ways to see this. Let's just consider the number $$r=0.9999...$$ we can demonstrate that this is actually equal to 1: $$10r-r = 9$$ $$9r=9$$ $$r=1$$

If that isn't convincing, you can view this in terms of a geometric series: $$\sum_{n=1}^\infty 9 \cdot 10^{-n} = 9 \frac{10^{-1}}{1-10^{-1}} = 9 \cdot \frac{1}{9} = 1$$

Or finally from the definition of real numbers as equivalence classes of Cauchy sequences of rational numbers, we see that the two sequences $$\{0.9, 0.99, 0.999, ...\} \text{ and } \{1, 1, 1, 1,... \}$$ are in the same equivalence class. Thus the two real numbers are the same.

Answer 2

Real numbers are usually defined by a process or algorithm. This process produces a series of numbers $(a_n)_{n=1}^{\infty}$ (usually rational, or else the definition of real numbers is circular).

This series is said to converge if (using Cauchy's definition) for any $c>0$ there is a $N(c)$ such that for all $n, m > N(c)$, $\mid a_n-a_m \mid < c $.

Two of these processes, say $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$, are said to represent the same real number if for any $c>0$ there is a $N(c)$ such that for all $n > N(c)$, $\mid a_n-b_n \mid < c $.

This is why the sequences $(1, 1, 1, ...)$ and $(.9, .99, .999, ...)$ represent the same real number: because $a_n-b_n = \frac1{10^n} \to 0$.

You can do the same thing using Dedekind cuts (as in Landau's page-turner "Foundations of Analysis") but I find that Cauchy is easier to explain (and also has fewer letters).