My daughter wondered whether series like 12345… or 10000… could represent integers and I told her ‘no’ because at some base intuitive level it doesn’t seem like they could be, but I couldn’t give her a good reason, especially since we’d just talked about repeating decimals and irrationals.
So why isn’t 12345… a number? Is it? If not, what if anything could such a series represent then anyway?
I was thinking of that. There’s no reason you can’t have one that goes infinitely in the other direction take.
lim x->infinity (sum from n=0 to x ( 10^n ) ) = …1111111111
Addition is well defined here. and …111112 and …1111110 obviously have ordering. It’s just the infinite going towards the decimal place that breaks things. (Not that that doesn’t mean it’s not a number, just not an integer).
Edit: Actually, nevermind, integers are defined to have a finite number of digits*. It’s still “really close” to an integer, though.
Actually, one important property of integers is that any number can be expressed by a finite sequence of adding 1s and -1s. e.g. 2 = 1+1 or 1+1+1+(-1)
Yep. It’s also good proof that two’s-complement representation is the numeric system of the universe. …1111 + 1 (base 2) = 0.
Yeah–to be clear for the OP, the p-adic numbers are not integers. They behave like integers in many ways, but as Asympotically fat said, all integers have a finite number of digits, and so p-adic numbers cannot be integers.
Though, to try and put this in terms a kid learning about irrationals may understand:
Those can be numbers, but we don’t talk for them because for most practical purposes, they’re not particularly useful. They don’t have a lot of the nice properties that the numbers we usually use have – you can’t easily add 1 or 2 to them, for instance. There are numbers without decimal places that can repeat or have an infinite number of digits such as …12345, and they’re probably more useful than 12345…, but they’re not as useful as irrational numbers so we don’t talk about them as much.
There’s no reason you can’t have 12345… (or …12345…[point]…54321…, if you want to get crazy), but they’re the kinds of numbers that most people don’t worry about. If you make a wooden square with a beam down the middle with sides of length 1, then you have a real, actual beam with length sqrt(2). What does it mean to have …12345 apples? You can come up with times it would be interesting or useful to talk about, but the scenarios aren’t common or intuitive like pi being the ratio of a circle’s circumference to its diameter, or having 2/3 of a cake left, or a diagonal board in a square.
That’s trivial: The answer is that the sum is 12345…+1
I’m serious. The next one in the series is 12345…+2. After that you have 12345…+3. I can even do arithmetic with them. Here’s what happens when I subtract the first from the third:
Got a problem with that?
Some posters have said that an integer must have a finite number of digits, but I don’t know where they get that. All the integer needs is no fractions and nothing to the right of the decimal point. Nothing stops us from arbitrarily saying that “12345…” has an indeterminate number of digits, but all of them on the left side.
My opinion is that the proper label to put on this thing is that it is a variable, much like “x” or “i”. You can use it pretty much the same as x and i, subject to the constraint that it is, by definition and/or convention, an integer.
You can call it a number if you want, but it is a useless one, except for the things that infinity is useful for, I suppose.
The same would be true of Aleph-1. I can “define” its successor as Aleph-1 plus one, but that’s just a trick of labelling. You aren’t actually performing an operation, only substituting variables in an outline of an operation.
Only in an extended schema of numbers, going beyond the integers as we understand them, could a construct having “infinite” number of digits be considered a number. In the standard field model of integers, such an object cannot be categorized.
(Is it even…or odd? Is it prime…or composite? You can’t answer those questions.)
From a different direction: does 12345… - 12345… = 0? Integers must have an additive inverse (that is, for all numbers “a” there is a number called “-a” that when added to a the sum is zero). Given that infinity-infinity is an indeterminate form, I don’t think so.
The reason that these numbers aren’t integers is very simple:
[li]Every integer can be written as the difference of natural numbers.[/li][li]Every natural number is either zero, or the result of adding one to zero finitely many times.[/li][/ol]
Natural numbers don’t have infinitely many digits in any base, so neither do the integers.
You could make a class of numbers like this, although you’d have to give an explicit construction so that we know exactly what they are, but the trick is convincing people that they’re useful.
I always learned the equivalent (but different) construction that all integers are zero or can be expressed as a finite series of invocations of the successor (+1) and predecessor (+(-1)) functions starting at zero.
Or the other equivalent one that all integers are zero and can be expressed as a finite sum of the natural and negative natural numbers.
Really, there’s a lot of equivalent definitions, but “finite” always comes up somewhere in there.
Let’s think about repeating decimals (rational or irrational) for a minute. What do they mean? Take a relatively simple one, like 0.6363636363… (the decimal representation of 7/11). The value of this repeating decimal is the sum of the infinite series 610[sup]-1[/sup] + 310[sup]-2[/sup] + 610[sup]-3[/sup] + 310[sup]-4[/sup] + 610[sup]-5[/sup] + 310[sup]-6[/sup] + …
In other words, you can specify exactly what each individual digit represents: the first 6 after the decimal point represents 6 tenths; the next digit represents 3 hundredths; the next digit represents 6 thousandths; and so on.
In a “number” like 12345… or 10000…, what does each individual digit represent? That 2 in 12345… means 2 of what? Ask your daughter that.
I’d give the same answer as I gave in the .99999… = 1 thread. Such peculiar “numbers” are prohibited by the Axiom of Archimedes.
Very crudely put, that Axiom asserts that any two non-zero numbers must have a finite ratio.
Various definitions or “axioms” could be mentioned to prohibit the peculiar numbers. I like Archimedes’ Axiom (which Archimedes himself credited to Eudoxus) because the very name and antiquity should carry great dignity and weight. If it was good enough for the man often called “the greatest mathematician ever” …
Suppose we can somehow get to the highest digit… but the highest digit is actually infinitely .
infinity + (anything) = infinity.
the infinite number of highest digits are all infinity …
So infinity + infinity + infinity … doesn’t go anywhere
Or if you work from the lower digits and go up, as we keep working on a value, the value just keeps getting larger and larger, growing faster with more progress. Its not bounded.
recurring decimals can be evaluated , eg
0.12121212121212… is 12/99
0.123123123123123 is 123/999
Irrationals - we can only state the approximate value, eg we can state two rationals that it is between… so they are bounded. (the formula used to derive the approximation can be shown to head to toward one value - or at least a range with a high and low value. )
If the “infinite integers” are instead of the form …54321, then then it’s clear that the 2 is in the tens place.
No one has pointed it out yet, but there are uncountably many of these. Every real number 0.ABCDE… corresponds to a number …EDCBA. (Modulo handling numbers where the ellipses end in all trailing 9s, but there’s only a countably infinite number of those.)
I was never happy with the way p-adics were handled.
…9999 + 1 = 0? Not to this Computer Scientist. There’s an overflow digit that has to be accounted for. Let’s use “:” to account for this. I.e., …9999 +1 = 1:0. The number before the “:” is overflow from operations on the “lower” digits. Addition and subtraction can be defined more reasonably, etc.
Multiplication has to be able to cascade these overflows. E.g., 1:0 * 1:0 is 1:0:0.
But at least some key properties of primes and such are preserved.
Division gets trickier. 1/1:0 is some type of infinitesimal. Which I am a big fan of but not too many others.
BTW: What’s the name for the opposite of an infinitesimal? “Infinitesimax”?
Another problem with p-adics is years ago back on the sci.math newsgroup there was a problematic individual with an obsession with them, so they got tainted as a crank topic.