Limits - Mathematical certainties?

Hey guys.

I was really bored at work and i started thinking about something that has bugged me ever since year ten. Limits.

Just in case people don't know what i am on about:

Limits in mathematics is the process by which the 'space' between one set of numbers and another set of numbers is so infinitely small that it is in fact, for all intents and purposes, 0.

The perfect example is 9/9. 9/9 = 1, but it also equals 0.99999999999999999999999999 repeated infinitely. However, 0.9 repeater is the same as one because, at some far distant nine in the future the difference between 1 and 0.9 repeater becomes so bloody small that the difference is zero.

I reckon that is a crock of shit. What do you guys reckon?

Tom

BTW Andy I have d/lded the audio book "The universe in a nutshell" if you want a copy. Its a lot easier than reading it i can tell you that bloody much, even so it is still making my brain bleed.

Tom wrote:BTW Andy I have d/lded the audio book "The universe in a nutshell" if you want a copy.

You rule. Damn straight I want a copy! And I just recently downloaded the audiobook of the first Flashman novels, and you should get a copy of that. Funny stuff.

Also, great topic. Hope it sparks some discussion.

Yo A-dawg! Hit me with the flashman!

You got it!

I do (to a degree) share your sentiments about it being “a crock of shit,” but I have come across a couple of sources that assure me that, at least in the specific case you just mentioned, it is the case. So, while I don’t entirely understand it, and therefore can’t entirely back it up, I’ll try my best.

There might be a difference (there also mightn’t – I’m not sure) between limits of a function and these infinitely repeating decimals. But I’ll just focus on the repeating decimals because that’s all that these books dealt with, and it’s what you mentioned.

I think the reason that 0.9 repeater is equal to 1 is not because as 0.9 repeater grows it gets closer and closer to 1 (which seems natural enough, I agree) but because it actually is 1; it’s just another way of writing 1 – another symbol for it. This is much in the same way that we usually accept that 0.3 repeater is the same as a 1/3, it’s just that, the explanation goes, we don’t intuitively like the idea of a nice round integer, like 1, as being represented in that decimal way, whereas 1/3 doesn’t evoke such feelings.

So, I think it’s to do with the peculiar properties of infinity (obviously); in that, if we say “the amount between 0.9 repeater and 1 is tiny, sure, but it’s still there” this implies that we’ve stopped somewhere (I think), but the nature of infinity is that it doesn’t stop.

But doing a search on the net threw up one source (at least) that seemed to vaguely suggest that it might not be the case (not definitively). But I think the weight of the opinions is on the other side. What ever that’s worth.

Wikipedia’s got a whole page devoted to it! (Geez, overkill) http://en.wikipedia.org/wiki/0.9_%3D_1

But 1/3 is 0.3 repeater. Where as 0.9 repeater is 9/9 which is also 1. Not exactly the same because there is a two leap process to get you there as opposed to it being the same thing.

And an infinitely small number is still a small number.

But if A = B = C, then A = C...

C’mon, I know you know that, Tom (I can’t read if there’s a sarcastic tone in your comments).

trying to put my words into words

A = B = C. So, to use an old philosophical debating thing to show the possible flaws in impirical observation: european swans are white so all swans ever observed by europeans were white. They therefore assumed the following

Swans are white therefore if it is not white it is not a swan.

To take it further:

Oaks have green leaves, that tree has green leaves, therefore that tree is an oak

To take it further still

murder is killing another person, murder is wrong, therefore what he did was wrong

In each there is a logical leap, a juncture where the logic must reach from one conclusion or premise to the next. However, even though the statement holds true in some cases in others it is not logically valid.

Looking at the murder one. Murder is indeed killing. However there are many different shades of grey surrounding the killing of one person by another. There is self defence, there is killing in war time, there is manslaughter so we can say that, although murder is killing killing is not murder.

There is also varying degrees of wrong associated with killing. If you had been the man who killed, say, Hitler, you would not have been viewed in the same light as a man who kills children (as gruesome as that is). So we can also say that killing is not necessarily wrong for a given value of wrong.

In the case where you put A = B = C you can say that the same applies (i certainly do). 1 = 1 = 1 is correct.

1 = 9/9 is correct
9/9 = 0.9 repeater is correct
but to say that just because one of the variables in those two equations is constant is to say that all are equal i don't believe it.

We say that 9/9 is the same as 1 because a number divided by that same number is equal to one. In this case it is also 0.9 repeater.
But i would say that 0.9 repeater is not the same as one. As i said before, an infinitely small number is still a small number.

Screw you limits!

Hmm. I personally don’t think the philosophic examples you’ve given (as useful as they may be) have relevance in this case.

A = B = C
Therefore A = C

I believe that this relation, known as transitivity, is very fundamental to logic and math. I’m pretty sure it applies here. That is, only if 9/9 = 0.9 repeater and 1 = 9/9 — if those are both true then 1 = 0.9 repeater must be true, I believe. But you could always argue that 9/9 = 1 is false, or whatever.

But is an infinitely small number still a number? Perhaps that’s making the same mistake as saying that ‘infinity’ itself is a number? Maybe, maybe not?

When I said "you could argue 9/9 = 1 is false," what I was more meant to say was "you could argue 9/9 = 0.9 repeater is false," which is, perhaps, more debatable — I didn't mean to sound like a smart alec, since obviously 9/9 = 1, and there is no debate on that.