Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: Saganist on 05/03/2008 07:38:02
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Hi all,
I have been wondering about infinity in mathematics. How can mathematical operations involving real, finite numbers give a sum of infinity?
Even if you start with a very large number and multiply it or assign huge exponents to it, this should still generate a finite, albeit huge number. But such a number is still not even a small fraction of ...infinity.
So how is infintity reached in physics equations?
Cheers or as we say in Quebec; Thank you infinitely.
Saganist
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Hi all,
I have been wondering about infinity in mathematics. How can mathematical operations involving real, finite numbers give a sum of infinity?
Even if you start with a very large number and multiply it or assign huge exponents to it, this should still generate a finite, albeit huge number. But such a number is still not even a small fraction of ...infinity.
Did you try with division? 1/0 = ?
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Infinity is an awkward mathematical term that gives rise to a lot of problems and obsessions it also has obvious problems when dealing with physical situations as you say.
When you are summing a series of numbers you ususally talk about the sum to infinity and the fact that an infinite series of numbers can sum to infinity is not very interesting. it is much more interesting if an infinite series of numbers adds up to a finite number.
There is no difference between a mathematical expression and a physical expression it's all mathematics The fact that we might be talking about predicting the result of a real experiment has nothing to do with it.
Many equations involving physical quantities require the integration of a product of two equations over the range from minus infinity to plus infinity in all three dimensions. in most circumstances it is possible to prove mathematically that the bit from some particular value of the dimensions to infinity is less than some value ie the error of the bit outside a sphere is small and gets smaller as the sphere gets bigger so depending on how accurately you need the answer ho know how far to go so the awkward infinities can be ignored.
There are some cases where this is not possible and you cannot prove that there are not any unexpected turns that could make the answer very wrong. At the moment the quantum theory of gravity is one of these.
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The truth of the matter is not to worry about it too much it usually means just carry on until the accuracy is good enough for you.
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There are some cases where this is not possible and you cannot prove that there are not any unexpected turns that could make the answer very wrong. At the moment the quantum theory of gravity is one of these.
Which "quantum theory of gravity"? I thought it doesn't exist, yet.
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The solution of quantum theory equations using the techniques that have been used for electromagnetig weak and strong interections requires the calculaton of arrays of product integrals. if you use the same techniques ti try to develop a quantum theory of gravity the integrals all gallop off to infinity so you can't get a sensible answer. That's why it's so difficult
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I have always found it fascinating that you can have a series with an infinite number of steps such as that which you use to calculate 'e' and end up with a finite result.
When I was at school I tried to convince my teachers that the product of 1/0 and 0/1 should be unity by the cancellation of the zeros i.e 1/1 with no luck!