Infinite number of decimal places?

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.


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 $24.9999\ldots = 25$.


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.

Tags:

Real Numbers

Related

For $X$ contractible, deformation retract of $CX$ onto $X$. Why is the Pontraygin dual a locally compact group? Structure of groups of order $pq$, where $p,q$ are distinct primes. Is an arbitrary number of the form xyzxyz divisible by 7, 11, 13? How do I construct a function $\operatorname{sog}$ such that $\operatorname{sog}\circ\operatorname{sog} = \log$? How to solve this confusing permutation problem related to arrangement of books? Prove that $\frac{1}{^n\sqrt{1+m}}+\frac{1}{^m\sqrt{1+n}}\geq1$ Not every function on $[0,1]$ is a pointwise limit of continuous functions on $[0,1]$ Set builder notation: Colon or Vertical Line Show that $f'(x) = \lim\limits_{h \rightarrow 0} \dfrac{f(x+h)-f(x-h)}{2h}$ A disease spreading through a triangular population Definition of the ordered triple (a, b, c) according to Kuratowski's Set Theory.

Recent Posts