Infinite numbers of decimals for a finite point in a line

Question

Recently I started studying real analysis. In the beginning itself I was introduced to numbers which can't be represented as ratios of other natural numbers. But before studying them I had doubts about numbers which can be represented as ratios. Why is that numbers such as $\frac{1}{3}$, which represents a finite point in a line, needs infinitely many decimal numbers to represent itself?

I know how by computation we can keep on adding more and more numbers after the decimal point but never find a number $n$ such that $3n=1$. But intuitively it feels wrong that to locate a finite point in a number line you need to divide the space between $0.3$ and $0.4$ infinitely many times

Answer 1

Not a complete answer, but a kind of hint:

First, do you know about different "bases", like base 2 and base 10 numbers?

If so, you might want to consider how to express the number we (in base 10) write as $1/3$ ... in base $3$.

Second, let me argue in the other direction.

Suppose I agree with you and say "all finite points between 0 and 1 (to make it concrete) should be representable by just a finite number of digits". (It's not at all clear what the first "finite" in that sentence means, by the way).

How many digits are enough? If you take, say, 2 digits, then you can only express the numbers $.00, .01, .01, \ldots, .99$. There are only a hundred of those, and if you drew a picture, you'd see lots of gaps between them, so the number "line" wouldn't really be a number line.

If you took 20 digits, you'd only get $10^20$ numbers...and there'd still be gaps between them. But you COULD write them down in order, just as I did for the 2-digit numbers above.

With a little more work, you could imagine making a table like this:

1   .0
2   .1
3   .2
4   .3
5   .4
6   .5
7   .6
8   .7
9   .8
10  .9
11  .00
12  .01
13  .02
...

and every finite-decimal representation would appear on the right hand side. (Some numbers, like 0, appear multiple times, in position 0 and position 11, but that's OK). Anyhow that'd be a complete list of all the numbers you want to allow.

Now I need one more idea:

If you have an increasing sequence of numbers, and it never gets greater than $1$, then it approaches some limiting number. For instance, the numbers

$$ 1/2 - 1/2\\ 1/2 - 1/3\\ 1/2 - 1/4\\ 1/2 - 1/5\\ $$ are all less than one, and they approach the number $1/2$ as a limit.

(The "all less than 1" avoids problems like the sequence $1, 2, 3, \ldots$, which does not approach any single number as a limit.)

OK. Let's look at the numbers $$ 1/10\\ 1/10 + 1/100 \\ 1/10 + 1/100 + 1/1000 \\ 1/10 + 1/100 + 1/1000 + 1/10000 \\ \ldots $$ They're all less than $1$; they're an increasing sequence...so they have to approach some single number. What's the decimal representation of that number?

Hint: the decimal representations of the numbers in the sequence are $0.1, 0.11, 0.111, 0.1111, \ldots$.

Answer 2

Why is that numbers such as 1/3 which represents a finite point in a line needs infinitely many decimal numbers to represent itself ?

This is the wrong question to ask.

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The question you should be asking is:

Why should we expect a finite point to have a finite decimal representation?

And the answer is we shouldn't. Any point with a finite decimal representation can be written as $$\frac{m}{10^k}$$ for some value of $k$. There is no reason to expect every number to be written in that form, and $\frac13$ is one such point that cannot be written as "something divided by 10 to some power".