In the beginning ...

[Andrew D]

If I owe the bank money, and I don't have the money to repay it, my bank balance has a negative number in it.

Exactly. If you owe the bank money, and you do not have the money to repay your debt, the bank uses the useful fiction of stating your bank balance as a negative number. In reality, of course, the amount of money which you have in the bank is zero, and the amount of money which you owe the bank is a positive number. But for accounting purposes, it is convenient to state the amount of money which you have in the bank as a negative number. What about this is eluding you?

[Andrew D]

Your posting which begins "Hardly an interesting point" runs off the right edge when I view it; some words are missing. Would you please repost it so that (I hope) I can read it as you wrote it? Thanks.

[Scooter]

The use of limits is not restricted to calculus. They are, for example, used in connection with asymptotes.

And if you are going to use Wikipedia to back up your assertions, then you shouldn't be quoting from it so selectively, because that very same article goes on to state that division by zero is an operation that is illegitimate, i.e. produces a value that is undefined. It further goes on to say that, "it is sometimes useful to think of a/0, where a ≠ 0" as being infinity, which again is not the same as saying that a/0 equals infinity .

[Andrew D]

First hypothesis: I have a basket which contains zero apples.

In the western half of the basket, there are zero apples; and in the eastern half of the basket, there are zero apples. (That is division: the whole of the space within the basket divided by two.) In the northern half of the basket, there are zero apples; and in the southern half of the basket, there are zero apples. (That is also division: the whole of the space within the basket divided by two.) In the northwestern quarter of the basket, there are zero apples; in the northeastern quarter of the basket, there are zero apples; in the southwestern quarter of the basket, there are zero apples; and in the southeastern quarter of the basket there are zero apples. (That is also division: the whole of the space within the basket divided by four.)

I can go on like this forever: In each eighth of the basket, there are zero apples; in each sixteenth of the basket, there are zero apples; in each thirty-second of the basket, there are zero apples; etc.

Why? Because zero is nothing.

Second hypothesis: I have a basket which contains one whole apple.

I divide that whole apple by zero. I do not have two halves of an apple; that would have been dividing one whole apple by two. I do not have three thirds of an apple; that would have been dividing one whole apple by three. Etc.

I can divide that whole apple by zero an infinite number of times, and I will still have one whole apple. Hence, 1÷0=∞.

Why? Because by dividing the whole apple by zero, I have done absolutely nothing to that whole apple. The proposition that (1) I can do absolutely nothing to an apple an infinite number of times and still end up with a whole apple makes perfect sense.

And that makes it entirely unlike the proposition that (2) although a minute is not equal to an hour, the number of minutes in an infinite number of hours is equal to that infinite number of hours. Proposition (1) is entirely sensible; proposition (2) is babbling nonsense.

[Andrew D]

And if you are going to use Wikipedia to back up your assertions, then you shouldn't be quoting from it so selectively ....

Quite the habit of his.

[thestoat]

The use of limits is not restricted to calculus. They are, for example, used in connection with asymptotes.

I did say "a good example being integration", not "the only example being integration"

And if you are going to use Wikipedia to back up your assertions, then you shouldn't be quoting from it so selectively, because that very same article goes on to state that division by zero is an operation that is illegitimate, i.e. produces a value that is undefined. It further goes on to say that, "it is sometimes useful to think of a/0, where a ≠ 0" as being infinity, which again is not the same as saying that a/0 equals infinity .

And if *you* read further on, the same article states

Also, the fraction 1/0 is left undefined in the extended real line

. It also mentioned that 1 / 0 = infinity three times. No "limits" are required - not applicable in that equation.

[thestoat]

Your posting which begins "Hardly an interesting point" runs off the right edge when I view it; some words are missing. Would you please repost it so that (I hope) I can read it as you wrote it? Thanks.

The post was this:

Andrew D wrote:
To his credit, thestoat concedes that he “can’t explain it.” One might well think that that should be the end of the matter.

Hardly an interesting point. There are vast tracts of maths and physics (probably other sciences too) that the finest minds on the planet cannot explain. In fact, Richard Feynman once said "If you think you understand quantum mechanics, you don't understand quantum mechanics". Conceptually I can understand infinity, when to use it and when to approximate. I think it is fairly obvious you do not. I have given you a mathematical definition of it (which you ignored).


Andrew D wrote:
Once it is acknowledged that the statement “the distinction is important in more complex cases relating to some sort of infinity” does apply to “the simple infinity we are discussing,” the contrary statement that “999999999999999999999999 (which we could approximate to an infinite time)” should be withdrawn.


Not at all, and I find it truly bizarre that you keep coming back to this point that has been explained so many times. Here I shall repeat it ...

Approximations are used all the time in many branches of physics and maths. They have to be since no measurement is perfect. Now, 9999999999999999999999999 millennia might not be a good approximation of infinity when dealing with the age of the universe, but it would be a good approximation when dealing with the lifespan of a ladybird.

Or consider rays of light from the sun. We consider them to be parallel. They are not parallel, but near as dammit when making calculations. They would only be parallel if the sun was infinitely far away, which of course it isn't.


Let's think of 2 of these rays of light falling on a book you are reading (let's just consider 2 rays, one on one extremity of the book, the other on the other). The rays of light can be considered to form an isosceles triangle. One side of this triangle is about 6 inches long. The other 2 sides are 93 million miles long. Question - what are the internal angles of the triangle?

Any normal person with a reasonable understanding of maths would say "hmmm. Let's pretend that this 93 million miles is infinity." When you do that, the answer becomes 90 degrees, 90 degrees and 0 degrees. That answer is normally good enough. But if you insist that you can never approximate infinity they you will say

"Ah ha - wrong answer. The actual answer is ...".

I wonder - can you tell me? My calculator can't calculate such accurate numbers. It will be something like
0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 degrees and
2 lots of
89.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999998

Most people would, I suspect, be happy to say "well, call it two lots of 90 and 1 lot of zero degrees", which is what you will get if you assume, in this case, 93 million miles is infinity.

Perhaps you can see that approximating large numbers to infinity can actually be very useful, if you understand the maths. Or perhaps you still don't agree?

That is what I mean by a simple calculation. Therefore I most certainly do not concede I am in error. Everything I have said has a very firm basis in logic.


Andrew D wrote:
Speaking of weird, thestoat concedes that he “can’t explain” how the number of minutes in an infinite number of hours can be the same as that infinite number of hours without a minute’s being equal to an hour


Hang on - didn't you just say that at the start of the post?

Andrew D wrote:
How can one rationally agree with a proposition which one finds inexplicable?


Ah, I see. Then please explain to me the nature of quantum mechanics - or do you disagree with QM? Unless you refute this statement in your next sentence, in which case I wonder why you bring it up.

Andrew D wrote:
Perhaps he will be good enough to bring them to my attention.


See above.

[thestoat]

Andrew

How would you define 1 / 0? The answer is infinity. But how can you describe that?

Sorry - I should have been clearer. I meant how would you describe the definition? Giving an example is great - but that is not a definition.

[Sean]

First hypothesis: I have a basket which contains zero apples.

In the western half of the basket, there are zero apples; and in the eastern half of the basket, there are zero apples. (That is division: the whole of the space within the basket divided by two.) In the northern half of the basket, there are zero apples; and in the southern half of the basket, there are zero apples. (That is also division: the whole of the space within the basket divided by two.) In the northwestern quarter of the basket, there are zero apples; in the northeastern quarter of the basket, there are zero apples; in the southwestern quarter of the basket, there are zero apples; and in the southeastern quarter of the basket there are zero apples. (That is also division: the whole of the space within the basket divided by four.)

I can go on like this forever: In each eighth of the basket, there are zero apples; in each sixteenth of the basket, there are zero apples; in each thirty-second of the basket, there are zero apples; etc.

Why? Because zero is nothing.

Sorry Andrew but I don't think you get the concept. In your hypothesis you are trying to divide an undefined space (rather than a number) by zero which is not possible.

The whole question of minutes Vs hours is not too difficult. We shouldn't try to think of infinity as a number (as it, by definition, cannot be measured and labelled) but as a concept. Note: Not a fiction but a concept. There is a difference.

I'm not hunting for the infinity symbol so I'm just going to call it 'I'...

Minutes in infinity = I
Hours in infinity = I

This would seem to suggest that minutes = hours but:

Minutes in infinity = I x 60
Hours in infinity = I

is also true.

In fact:

Minutes in infinity = I x (any number)
Hours in infinity = I x (any number)

The point being that 'I' multiplied by anything is still 'I'.

[loCAtek]

ow it is that the number of minutes in an infinite number of hours is equal to that infinite number of hours.

A X ∞ = ∞
There can be no greater amount.

Same as:

A X 0 = 0

There can be no smaller amount.

[Sean]

Actually Lo in maths there a many amount <0.

They are negative numbers...

[loCAtek]

...yea, but even a negative times zero is Zero: Zero is Hero!

[Sean]

Still doesn't mean that 0 is the lowest number Lo...

[loCAtek]

'Kay, what is the lowest number? -∞?

[Sean]

Spot on Lo!

[thestoat]

Actually, there are different levels of infinity - we are currently just talking about "countable infinities".

A X ∞ = ∞
There can be no greater amount.

Spot on. I actually quoted that a while ago but the post was ignored by Andrew - presumably an inconvenient fact

[thestoat]

... or maybe the maths was too complicated, since he then started adding and subtracting apples ...

[rubato]

Mathematics does not include an operation which says "there exists an X such that x=" the output of a mathematical statement.

Which is why the number of minutes in an infinite number of hours is mathematically = 60 x the former. The mathematical operation of multiplication simply ignores the fact that "infinity" cannot be empirically defined (operationally defined) but however big that quantity really is you can still multiply it. The term 'infinity' is merely hypothetical.

yrs,
rubato

[thestoat]

Tumbleweed ...

[loCAtek]

...right, bring on the 72 virgins and bottomless flagons of mead!