# Infinite number of decimal places?

here's my question:

Let's say you have two numbers. For this example, it can be 24 and 25. Now, as I understand it, there can be decimal intervals between them, such as 24.2, 24.34 and even 24.788843.

Now, what I'm wondering is:

Is there the potential for an infinite number of decimal places? e.g. would it be possible to continue adding more decimal places indefinitely, such as in the pattern below?

24.8877544

24.88775443

24.887754438

24.8877544386

And so on...

If there was, that would imply an infinite quantity of numbers could be generated, without ever reaching 25. And if it that is the case, what sort of operation could be applied to create such a pattern?

Thank you very much!

• Every number $\gamma$ in base 10 can be represented as: $\gamma = \sum_{n=-\infty}^{+\infty}a_n 10^n, a_n\in \{0,...,9\}$, so yes, there are infinite numbers of decimal places. Oct 29, 2015 at 16:05

Is there the potential for an infinite number of decimal places?

No potential about it. There are an infinite number of decimals.

Consider the simple function $f(n)=25\times\frac{n}{n-1}$ as $n$ goes to infinity. It will approach $25$ but never get there.

• That answers my question quite nicely! and so n is used to represent infinity? Thanks a lot Oct 29, 2015 at 7:01
• No $n$ is usually used to represent an integer, but it could be a real in this case too. For example 123.567 works just fine.
– MaxW
Oct 29, 2015 at 7:05
• If you really want to start a flurry of posts ask if 24.999... = 25 :-)
– MaxW
Oct 29, 2015 at 7:06
• Ah, thank you for the clarification! I'll keep that in mind ;) Oct 29, 2015 at 7:10
• @MaxW I wouldn't expect very much of a debate about that issue on MSE... Oct 29, 2015 at 17:40

Did you know $1/3 = 0.33333\ldots$ with a $3$ recurring infinitely?

Actually you can append any sequence of digits to $24.$ to obtain numbers between $24$ and $25$. And if you append an infinite sequence of nines, you'll get $24.9999\ldots = 25$.

• I actually did not know that. As you can probably tell, I have very much to learn. Thanks for answering! Oct 29, 2015 at 7:07
• A number like 1/7 = 0.14285714285714285714285714285714... has a repeating pattern (142857) that repeats and repeats forever. All fractions n/m were n and m are integers, will have a repeating fraction or they will "terminate" and not go on forever. These are called the "rational" numbers. However there are more more numbers that go on forever without repeating patterns. These can not be written as fractions. These are called irrational numbers. There are MORE irrational numbers then rationals and any decimal expansion even if it is infinite is a number. Oct 29, 2015 at 7:16
• @fleablood I didn't go into that distinction deliberately. It's enough surprising for now to the OP that there is actualy infinite set of numbers betwee any two real numbers. Let them adopt that fact, then go into distinction between infinities. :) Oct 29, 2015 at 7:21
• Oh. I wasn't trying to get into the distinction of between infinities. You're right about that being overwhelming at first. But I figure once one realizes that numbers are a continuum and numbers one should realize all possible representative values must therefore exist, but then one shouldn't fall into the trap of thinking they can all generated finitely via rationals. Students are told that some numbers go on forever (especially pi) but I think few truly realize the significance of it. Oct 29, 2015 at 12:54
• @fleablood Yep. They leave school with this idea that pi is special BECAUSE of it's irrationality. Which simply is ridiculous. It would be much more special had it been rational. Oct 29, 2015 at 17:42

More numbers have infinite decimals expansions than do not. You may have heard that the expansion of $\pi = 3.1415....$ goes on forever and never repeats. Despite what you hear on pi day, that is one of the least interesting things about pi as nearly all numbers have expansions that go on forever without repeating.

And there's no need to find a "pattern". Any possible sequence of numbers will make a decimal number. So I could take the number 24.429385.... and just start typing digits at random forever and it will be a number.

The important thing to note, is that between any two numbers, say 24.2348605 and 24.2348606, we can always find a number between them, 24.23486055, and we can always find a different number as close to it as we possible want. If I have 24.2348605749372859437295748932789547329532...., can find also have 24.2348605749372859437295748932789547329533 which is only one 10000000000000000000000000th away. If we wanted to find a number one googolth away we could.

There's a lot more to it than that. But that's enough for now.

• Wow, this is quite a lengthy response... which I very much appreciate. Seriously, all of you guys are awesome! Oct 29, 2015 at 10:58
• "nearly all numbers" :-D Oct 29, 2015 at 15:13
• Yeah... well.... Oct 29, 2015 at 15:20

You don't need to consider decimals to see that there are infinitely many numbers between 24 and 25.

Start with 24 and a half, 24 and a quarter, 24 and an eighth, and keep on halving. You can clearly do this for ever and they are all between 24 and 25.

Maybe even simpler: 24 and a half, 24 and a third, 24 and a quarter, 24 and a fifth, 24 and a sixth, etc.

Look up Zeno's paradox. People have struggled with this subject for a long time.

Is there the potential for an infinite number of decimal places?

I assume we are talking about at least rational numbers here. If so, then indeed all the numbers in the interval $(24,25)$ have infinitely many digits in their decimal expansion. For rational numbers, they all recur after some point on. Examples are $24.8999\ldots, 24.375000\ldots, 24.245678678678\ldots.$ However, and much more so, the irrational numbers abound, for example $24.5055055505555055555\ldots$, which has a pattern, and $24.86382649403204847293746620846291646490163038\ldots,$ which I hope doesn't.

If there was, that would imply an infinite quantity of numbers could be generated, without ever reaching 25.

Yes, indeed. We call this the density property of the rational numbers and, in general, the real numbers (which include the irrationals). That is, between any two real numbers there is always one real number, which means there are infinitely many of them. For example, if $r$ and $s$ are real numbers such that $r<s$, then we have the real number $$x={r+s\over2}$$ satisfying $r<x<s$. So the reals are much different from the integers, yeah? ☺

And if it that is the case, what sort of operation could be applied to create such a pattern?

I don't know what you mean here. But if you simply mean a way of generating any real number between any two distinct ones, then the recipe in paragraph $2$ above is one; there are others. But if you mean the construction of real numbers in decimal form, then you can make only few of them, since infinitely many of them show no pattern in their digits, as far as we know. For example, the decimal expansion of $π$ has shown no pattern hitherto.

PS. The study of the real (and complex) numbers is the foundational focus of the mathematical field known as analysis. There are many more properties that $\mathbb R$, say, has that makes it more interesting than $\mathbb Q$. If you study this subject later on, you'd learn more. Good luck!