Thread: Do infinite integers exist?

Consider the number <<<####ad infinitum ###373737373737. This number, whose digits proceed to the left ad infinitum, would appear to be of infite magnitude. But now consider the following.

Let x = <<<####ad infinitum ###373737373737 . Then 100x = <<<####ad infinitum ###37373737373700 .
Subtracting, we get, for x - 100x:

<<<####ad infinitum ###373737373737
- <<<####ad infinitum ###373737373700
------------------------------------------------
37

So, 37 = x - 100x = -99x. With the result that x = -37/99 , a very finite, albeit negative, number.

Another example. Consider <<<#######ad infinitum#####999999999999 .
This is about as infinite as an integer could be. However, let us add 1 to it.

<<<#######ad infinitum#####999999999999
+1
----------------------------------------------------
<<<########ad infinitum######00000000000

If y + 1 = 0, then y = -1. So, <<<#######ad infinitum#####999999999999 = -1.

These findings suggest that infinite strings of digits to the right do not form infinite quantities, but rather negative numbers.

To address the question asked in the subject line, infinite integers do not exist, at least not if you use the usual definition of "integer".

Originally Posted by Atomic-S

These findings suggest that infinite strings of digits to the right do not form infinite quantities, but rather negative numbers.

Well, I guess you could take that tack. The more usual interpretation would be that infinite strings of non-zero digits on the left-hand side of the decimal point do not represent defined numbers. You can define them as "infinity" if you like, but infinity does not (because it cannot!) follow the same familiar rules of arithmetic as the conventional numbers. As a result, if you assume all of these laws of arithmetic are true, and apply them to infinite quantities, you can prove things that are not true.

The familiar infinite series has the value on the range . (See the 0.999... thread for some people who might argue it is undefined, or defined and not equal to this quantity, or undefined and yet somehow equal, despite being undefined, to something really really close but not equal to .) A standard error is to declare that therefore , so an infinite sum of increasing positive numbers is actually negative. The intepretation I would place on this result is, some of the steps in the derivation of the formula are not valid outside the range .

Originally Posted by Atomic-S

Another example. Consider <<<#######ad infinitum#####999999999999 .
This is about as infinite as an integer could be. However, let us add 1 to it.

<<<#######ad infinitum#####999999999999
+1
----------------------------------------------------
<<<########ad infinitum######00000000000

If y + 1 = 0, then y = -1. So, <<<#######ad infinitum#####999999999999 = -1.

These findings suggest that infinite strings of digits to the right do not form infinite quantities, but rather negative numbers.

Wait, why not do the same trick you did with the 37s? Multiply by 100 and then subtract that?

x - 100x = 99, so x=-1

As Coelacanth points out, those findings suggest that numbers with an infinite string of digits to the left do not really fit into our number system.

Wait, why not do the same trick you did with the 37s? Multiply by 100 and then subtract that?

x - 100x = 99, so x=-1

It is interesting that, using this different arithmetic, you get the same answer that I did.

Even more interesting is that you start with a positive number and deduce that it's negative, I'd think. That's gotta say something right there.

What do you get when you divide it by 3?

Originally Posted by grapes

Even more interesting is that you start with a positive number and deduce that it's negative, I'd think. That's gotta say something right there.

Never thought I'd find myself at a board where 0.999... does not equal one, but !

(Preemption - grapes, that is an exclamation point, not a factorial symbol )

It does have me wondering if you can define some kind of self-consistent arithmetic by defining, for example, to be . Well, of course you can, just define the addition and multiplication operations to be consistent with the definition just stated, but then the question is whether the intuitive, algorithmic "definitions" of these operations work.

Then would need to work out some kind of limit theorem for non-repeating infinite digit strings left of the decimal point.

Which is bigger, <<<####ad infinitum ###373737373737 or <<<####ad infinitum ###373737373700 ?

Did I take the first one and subtract 37 to get the second one, or did I take the first one and multiply by 100? Does it affect the answer when I subtract the two?

Why?

Originally Posted by grapes

Which is bigger, <<<####ad infinitum ###373737373737 or <<<####ad infinitum ###373737373700 ?

Did I take the first one and subtract 37 to get the second one, or did I take the first one and multiply by 100? Does it affect the answer when I subtract the two?

Why?

Well, let's see. By the proposed method of definition, the first one would be equal to . So if you subtract , you get . So I can't tell whether you subtracted or multiplied by , since both operations get you to the same place

Which is bigger? By absolute magnitude, the second is "bigger", but since they're both negative, the first one is > the second one.

Does it affect the answer when I subtract them? What does the answer look like when I subtract them? is it 37, or is it -37? Or neither?

Originally Posted by grapes

Does it affect the answer when I subtract them? What does the answer look like when I subtract them? is it 37, or is it -37? Or neither?

I think we have to go with ...3737373737-...373737373700=37. Any other answer is inconsistent with the algorithmic "definitions" of addition and multiplication. (I'm not yet totally convinced that any definition at all is consistent with the intution of the OP, but if there is one, then it has to be the one above, doesn't it?)