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!
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$\begingroup$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.
$\endgroup$ 5 $\begingroup$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 $24.9999\ldots = 25$.
$\endgroup$ 9 $\begingroup$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.
$\endgroup$ 3 $\begingroup$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.
$\endgroup$ $\begingroup$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!
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