Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: Mr. Scientist on 08/01/2009 21:53:22
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Consider an infinite plain, from a to b, without recourse or stop, we can say according to phsyical math as:
∞=[∞-(-∞)]²
Which has the coordinates:
(-∞,∞) x (-∞,∞)
Which then suggests it must have an infinite plain - by simplifying ∞=[∞-(-∞)]² we get;
∞=(2∞)²
And so
∞=4∞²
Which must give since according to prove ∞=[∞-(-∞)]²
b
∫ =f(t) dt=∞
a
Easily states that f(t) does not have a boundary Ω between a and b, so it must imply some infinite range. We should now consider a second infinite plane, which would allow us to calculate one infinite plain which is incoming, and a second which is outgoing. We could express this as:
x<z<z
x>y>z
(If we simply replace the x and z coordinates here with infinities, we find one infinity ascending in the positive direction, we find another descending in the opposite.
Thus instead of just saying, (1) ∞→-∞, we also have (2) -∞→∞, and so now I suppose each plain, call (1) the conjugate of plain (2). This assigns each plain a special property which equally relates the two. If plain (1) acts as a boundary to plain (2) upon mathematical cobjugation, we actually find the following:
b
∫ =f(t) dt=∞
a
and
-b
∫ =f(t) dt=-∞
-a
So that the boundary of |∞|² given as the absolute square of the infinities in question, which is analogous to -∞•∞ where one can consider some renormalization, since being conjugates, one could have the value 1. The only problem one would consider here, is that the mathematical expression -∞•∞ yields normally undefined, because -∞•∞ should yield simply -∞. But perhaps infinite qualities (but only certain kinds as acting like conjugates of each other), could break the mathematical dogma?
My thoughts remind me of Cantors proof of infinities showing that one infinity can be larger than another infinity which initially suggests some countable difference between the two elements. If two infinities as quickly shown:
Lim of (n →∞)^n =∞
And
Lim of (n →∞)^n² =∞
(1)
Where Lim of n →∞^n² =∞ implies a greater infinity than Lim of n →∞^n =∞, then we may be able to have two infinities, one moving in the positive direction and the other moving in the negative direction to both have values which together yield a single finite value may be possible. It could help maybe by mathematical notation that ∞=(a+bi) and ∞=(a-bi), where the two of them multipled would yield some real positive answer.
(1)- This is like saying that:
(∞∑ of n=1)^n≠(∞∑ of n=1)^n²
My Question
I have studied the fallacies of mathematics for a few years now, and I always here you cannot subtract an infinity from another, unless you do not define one not being a little larger than the other in some arbitrary way, likewise, you can’t divide them unless you specify the qualities, as Cantor once proved. So now I ask, why can one not define a real finite value, unless this value was calculated by multiplying two infinities acting as conjugates of each other, where there is some boundary upon contact, much like how one might envision the infinite wave function collapsing upon some measurement? So in another set of words, why can’t we define two infinities by multiplication if they have properties which can cancel, but not maybe completely, their own internal properties?
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Hold on, i am missing a part.. i will modify first post
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I'm afraid that doing sums with infinity involved will often be a case of
garbage in = garbage out
unless you really know what you're doing.
I think you have proved that.
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Well, the limits i had shown is a very lame example of cantors proof. With that knowledge aside, you must treat the limits i have shown as just that, but have a bit of descretion about yourself.
Can you answer my question though...
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If you want a precise answer then you ask a precise question. Alternatively, take a course in higher Maths.
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Don't patronize me please. I am currently studying for my Certificate in mathematics. But i do believe i understand enough about this to ask directly:
Why can we not specificy two infinities as conjugates of each other, (with each infinity) having exactly the properties to deflate their infinities to some countable real value? If it can be done in renormalization theory by (arbitrarily subtracting infinite values), surely there must be some way we can multiply them in some complexified way to yield a real value, by again, treating them as conjugates.
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In reflection of my question, we can deal with two infinite waves in quantum wave dynamics which uses two infinite wave functions as conjugates which yield
∫_Ω |Ψ|²=1
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Actually, in this case the integral is finite, although the limits are infinite. That is OK because the function is an appropriate one.
I think you are trying to run before you can walk.
I wish you well in your studies. Perhaps, if this is a field that you are actually studying, you ask your tutors about this. If not, are you really qualified?
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But this is my point, silly.
We can as you said, make a limit on the integration. If you integrate two ifninities, the property of them having a boundary should be able to analyze under the processes i gave. I am not trying to run. I think i am taking this under a serious analysis, in which you keep reverted to slagging me off.
Imagine then, in your own language, we have an integral that is finite, meaning upon intreraction of two infinities (given the postulates i gave), can you give a reason why
∫_Ω |∞|²=1
Without implying a physical field could not work? This is a serious question. Treat is as being serious, instead of some cheap way to slag me off.
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If you integrate two infinities
What does that mean?
Do you mean integrate two functions over infinite limits?
How can one integrate an infinity?
You could talk of 'adding two infinities', possibly, but that would be fraught.
Why do you take a technical comment so personally? I am seriously suggesting that you need to know more about this Maths before you start trying to tell others about it.
No one is forcing to post here.
What sort of response do you get from your tutors if and when you get so objectionable?
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One could, if one had two infinities which cancelled each other out. If you take two infinite wave functions, they can upon multiplication cancel out, as this is what defines a boundary on the integration.
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two infinities which canceled each other out
And what might they be?
Give me a rigorous definition of what you mean.
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I will need time. I haven't mathematically defined to myself what two infinite conjugates could be otherwise calling them positive and negative directions.
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Take your time then. Come back when your ideas are a bit more complete and checked.
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A mathematician once told me, that if you think something can be done in math, it usually can within 80% of the time be represented. I do believe this can be respresented, and i will sit within my spare time to show this one way or another.
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At the risk of showing my complete lack of understanding of maths.
How can something be bigger or smaller than infinity.
To my poor understanding something is either infinitely large(everything) or infinitely small(nothing)am I missing something here [???].
Cheers
justaskin
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The idea of infinity is a sophisticated one and it is very often bandied about with a great lack of understanding.
But, as to having infinities of 'different sizes':
Imagine adding a series of numbers together
1+2+3+4+5 +++++ and keep going as long as you wanted, and a bit longer - 'for ever'. That would give you what we would call 'infinity'.
Then you added up the squares of the numbers:
1+4+9+16+25 ++++++ and keep going for the same length of time.
The second total would always be bigger than the first total.
Both totals would be 'infinity', if you keep on for ever but the second total would always be a lot more than the first total. So there you have the idea of infinities of different sizes.
There are many cases where you can 'compare infinities' which are, on their own, incomprehensible, but the ratio of the two can have a meaning.
Infinity is definitely not for the squeamish or the unwary but lots of my whacky Mathematician friends get very excited about it.
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The idea of infinity is a sophisticated one and it is very often bandied about with a great lack of understanding.
It seems to me quite simple.
Infinite as in not finite
the same as indestructable is not destructable.
Or infertile as in not fertile.
So to my mind once something is finite it can't be not finite.
But, as to having infinities of 'different sizes':
Imagine adding a series of numbers together
1+2+3+4+5 +++++ and keep going as long as you wanted, and a bit longer - 'for ever'. That would give you what we would call 'infinity'.
Then you added up the squares of the numbers:
1+4+9+16+25 ++++++ and keep going for the same length of time.
The second total would always be bigger than the first total.
That is as it maybe but notice the word total there.Surely once you stop and presumably thats what you would do to get a total(or totals)you have finite number do you not.A number that could be added to doubled or raised to what ever power you like and still it would not be infinite.
Both totals would be 'infinity', if you keep on for ever but the second total would always be a lot more than the first total.
That seems to be a contradiction in terms.How could you go on for ever(infinite) and have totals(finite)
So there you have the idea of infinities of different sizes.
There are many cases where you can 'compare infinities' which are, on their own, incomprehensible, but the ratio of the two can have a meaning.
Infinity is definitely not for the squeamish or the unwary but lots of my whacky Mathematician friends get very excited about it.
Well it might be what the mathematicans call it but it is not the English description.
Cheers
justaskin
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There is not a coherent 'English' description of Infinity - in fact dictionaries usually have a shamelessly circular definition which uses examples of its use rather than its meaning in other terms.
You have to go to Maths for a safe way to define it and even then it may not be satisfactory.
Your objections just go to show how difficult it is to define.
My two series of numbers each increase as you go further. One increases faster than the other so, however far you go, it is always bigger.
There are times, however, when you can use infinity more safely. For instance, if you add up the series of numbers
1+1/2+1/4 + 1/8 +++ you will never get a total more than 2 - even for an infinite number of terms. We can use the word 'infinite' safely because the final result is finite. That is a 'convergent' series, as opposed to a 'divergent' series like the first two examples.
Studying what happens 'on the way' to infinity can give you a clue about what would happen if you just carried on and on. . . .
It's just too hard for a simple definition and you can't expect one any more than you can put Relativity or Quantum Mechanics in simple terms.
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I know it's just a typo, but
ifninities
really amused me.
I think we should devise a definition of it and promote it's use.
On another forum that I used to visit we defined 'snese' as being the description of something that at first seems sensible but actually isn't, once you've thought about it. For example; "a lot of what is said about infinity makes complete snese"
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It's this keyboard, you see. :-)
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There is not a coherent 'English' description of Infinity -
boundless,limitless.never ending,going on for ever, beyond imagination.
You have to go to Maths for a safe way to define it
And that definition is?.
[quote
My two series of numbers each increase as you go further.
[/quote]
Yes
One increases faster than the other
Yes
so, however far you go, it is always bigger.
Yes and this defines infinity how?.
There are times, however, when you can use infinity more safely. For instance, if you add up the series of numbers
1+1/2+1/4 + 1/8 +++ you will never get a total more than 2 - even for an infinite number of terms. We can use the word 'infinite' safely because the final result is finite. That is a 'convergent' series, as opposed to a 'divergent' series like the first two examples.
Are you not just using infinity here to describe how large the series could be.could you not use some of the other descriptions such as limitless or boundless.Infinity has no roll in the actual calculation other than to describe its boundary yes?
Studying what happens 'on the way' to infinity can give you a clue about what would happen if you just carried on and on. . . .
And hence there is no need to and you can pack up and go down the pub for a beer. [;D]
It's just too hard for a simple definition and you can't expect one any more than you can put Relativity or Quantum Mechanics in simple terms.
I think I have heard this used before to make people think they are inferior.
So what is the mathematical value of infinity?.
We have a value for Hubble's constant.
We have a value for the speed of light.
So what is the value for infinity.
Cheers
justaskin
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Infinity only makes sense to me as a dynamic value, which doesn't work as a static term.
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With a bit of simple thought it is quite obvious that infinity cannot be treated just like any number in an equation
For example infinity plus one is clearly also infinity as is infinity plus any number up to infinity. So it is clearly not possible to treat it in equations as the originator of this topic suggest so any conclusions are likely to be fallacious or at best partial.
As to the different orders of infinities these follow from the equations that control them as a mathematicion I can show how if I have an equation that "blows up" and becomes infinite at some value. for example 1/x as x tends to zero I can easily create an equation that blows up faster by multiplying it by another number that blows up say (1/x)^(1/x) as x tends to zero.
I can always increase this by rasing it to a power of 1/x and so on without limit.
The answer when x=zero is always infinity but each of the infinite series of equations is always infinitely bigger bigger than the previous one.
This may be obscure mathematics but there are physical situations where this kind of thing happens.
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I think I have heard this used before to make people think they are inferior.
No. To make everyone realise that they ARE inferior.
The only way to rise above this state is to realise our limitations and make the best of it.
There is no 'value' for infinity. How could there be if it is the result of a process which has no limit?
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There is no 'value' for infinity. How could there be if it is the result of a process which has no limit?
Hey I agree with that.Does that make me a mathematician. [;D]
Cheers
justaskin
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a concept of infinity that I always thought of on this subject (ever since questioning the math teacher, and him being unable to tell me much of infinite geometric sequences [;D]) was that Infinite is any number than cannot really have an end value pasted on it... something like a destination, rather than a number. just like the horizon, you know it's there, but you will never reach it.
say, perhaps, you have point a, and point b. point a is 25 centimeters from point b. but it can also be measured in inches, half inches, millimeters, feet, tens of feet, in infinite number of ways, because the list never ends.
but then you have the size of infinity.
the 1+2+3+4 example is one way, but I thought of another definite way to put it. 1/0 is infinite, because of the process used in division. in division, you essentially subtract the denominator from the numerator, increase the end result by 1, and then repeat until you cannot subtract anymore.
for 8/2, it would go like this:
8-2 = 6, result = 1
6-2 = 4, result = 2
4-2 = 2, result = 3
2-2 = 0, result = 4. can no longer subtract, therefore 8/2 = 4.
but for 1/0, it would go like this:
1-0 = 1, end result = 1
1-0 = 1, end result = 2
1-0 = 1, end result = 3
...
that cycle will never end, so therefore, the end result is infinite.
but say you turn it into 2/0, it would be a infinite that is larger, since, if you could put it another way, and you divide them, (2/0)/(1/0), then you divide the denominators, assign that as a value of 1, because any identical values divided will equal one, then the numerator becomes 2. Therefore, theoretically, 2/0 as an infinity is twice as large as 1/0.
however, you can also say that 0/0 is an infinite, invalidating the previous paragraph.
but given the fact that I don't really yet know calculus, and haven't worked with infinity much, I don't know if my thoughts are really valid.
anybody care to correct me?
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As a school boy I tried to argue that zero times infinity was one from the arithmetical statement 1/0 equals infinity and 0/1 equals zero therefor 1/0 * 0/1 the zeros cancel out and one is left with 1/1 which equals one.
My argument fell on deaf ears !
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it is fascinating that a convergent series taken to an infinite number of steps (Zeno's paradox ?)can have a finite value ones first impression would be that an infinite number of steps how ever small would add up to infinity.
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I had also thought that, but what if you took differing "values" of 0 and infinity? the only requirement for that kind of argument is that for the 0, you must have 0 in the numerator, and for infinity, must have 0 in the denominator. All of this, however, is more theoretical... I would think a lot of this is may be preconceived notions.
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Mathematically, chaps, I think we are all out of our depth.
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As a school boy I tried to argue that zero times infinity was one from the arithmetical statement 1/0 equals infinity and 0/1 equals zero therefor 1/0 * 0/1 the zeros cancel out and one is left with 1/1 which equals one.
My argument fell on deaf ears !
300,000/0 is also infinity, so 300,000/0 * 0/1 should be equal to 1?
0*∞ is undefined.
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yet there is another absolute of mathematics that that breaks. 0 times anything should be 0, so why wouldn't 0 times infinity be 0?
*calling all mathematicians*
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Infinity isn't a number in the usual sense of talking about numbers. Multiplication is defined as something you can do when you have two numbers. Therefore you can't multiply something by infinity. What's very useful in dealing with the usual real numbers is that when you add, subtract, multiply or divide (except by zero) two real numbers, you end up with a real number as a result. This means you can do any number of operations and be assured that you end up with a real number. (In mathematical lingo, the real numbers are a field under these operations).
What you're trying to do is to include ∞ as a real number so that you can do things like multiply by it. You can do this, but then addition, subtraction, multiplication and division (except by 0) have a lot of exceptions that aren't defined as real numbers anymore. Here's Wikipedia's list of what is usually defined and undefined: http://en.wikipedia.org/wiki/Real_projective_line#Arithmetic_operations_which_are_defined
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Thank you for clearing up that point it has perplexed me on and off for 70 years.
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What is infinity? … Half infinity doubled.
Is infinity plus one even greater?
Is half infinity smaller than infinity?
Is infinity just a concept?
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Infinity is a very strange subject.
Yet it seems to me that we are surrounded by it.
Infinity, isn't that when we lose count?
So, the possible amount of stars, molecules, atoms, quarks.
We're looking up through the eye of infinity doing our best to count our way out:)
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So, the possible amount of stars, molecules, atoms, quarks.
We're looking up through the eye of infinity doing our best to count our way out:)
If something is quantifiable, no matter how large a number, then it falls far short of being infinite.
Current thinking tends to speculate that even the universe is finite. Therefore the numbers of stars, molecules, atoms and quarks are also finite, as are the grains of sand in an hour glass.
If however, it turns out that the universe is unbounded and infinite then everything within it will also be infinite along with every permutation and possibility. This would also make the appearance of ‘life’ an inevitable certainty as every possible outcome would exist, regardless of how improbable. (Another topic?)
Infinity is more than very large - infinitly more!
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True:)
But who will count it down if finite.
An approximation will have to do.
But that's not the number.
So what we reach for, even if finite, will never be counted.
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True:)
But who will count it down if finite.
An approximation will have to do.
But that's not the number.
So what we reach for, even if finite, will never be counted.
Why? If something is finite, I call it "n" and that's all! [:)]
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yet there is another absolute of mathematics that that breaks. 0 times anything should be 0, so why wouldn't 0 times infinity be 0?
*calling all mathematicians*
Because what you say is true only if "something" = number.
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Ah! Infinity, I could go on about it forever....
There is no 'value' for infinity.
Quite so, if you try to put a value to infinity, you are giving it borders, you make it finite, don't you?
Infinity cannot be explained by figures and cannot be conceived by man.
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Well, it has been conceived by man:)
Or at least, found out::))
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This may sound like a put-down but there is no way you can get a handle on 'infinity' if you aren't prepared to go through all the formal steps of Mathematical Analysis. It's hard and it's rigorous and it doesn't involve a single poetic superlative. Whatever fancy words you use to describe infinity, there is no way to relate it to the infinity of Mathematicians. I'm afraid that many of the contributors to this particular thread are way out of their depth. I include myself in that. Don't expect to get anywhere unless you are prepared to take advice and go through it from scratch.
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SC, you can't mean that?
'and in wells of her eyes, infinity unfulfilled'
I once saw a little book. 'Mathematics for poets (and dreamers)' (?)
A very nice one, I almost understood it.
:)
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I wouldn't trust the Author to predict the outcome of a bit of Engineering if that was their only claim to fame.
As with many words, "infinity" has two separate meanings. They apply differently to different cases.
You can't make a mobile phone work with poetry, can you?
We need both left and right brain activity.