Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: thedoc on 04/12/2016 09:23:02
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Will Harris asked the Naked Scientists:
I would like to ask the Naked Scientists the following question concerning the concept if infinity, which i would like to divide into 3 sub-questions if i may to illustrate my queries.
Question 1. If you take the number 10, which is essentially a whole number, and divide it by 3 you get three equal figures of 3.3333 recurring. How is this the case in that a completely whole number becomes 3 infinite numbers? I.e. 3.333 recurring never reaches an end therefore cannot form a whole.
Question 2 and 3 are kind of similar. I cannot comrehend that space is infinite. Everything has to have an end to it. If so what is the other side of that end point? Am i right in saying the universe is made up of galaxies eyc (i may be wrong!). Assuming this what is outside of that universe, and if there is something, what is outside of that? An so on!
Question 3 How small can you get in terms of space, and does space have a measurable size? Considering the atomic size and the sub-atomic size as a measurable quantity, what is there that is smaller, and what smaller than that etc? Plus if space were to contain no particles of matter whatsoever, can you measure size in that space?
Apologies for the lengthy question (no pun intended), can you shed any light on this?
Many thanks
What do you think?
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Hi Will, I'm not a scientist, so question everything. Just to start the ball rolling, i'll have a go. I'm in hospital and struggling to master my daughter's tablet, so I'll take it a bit at a time.
1. Are you confusing mathematical infinities with anything that might be considered. as a physical infinity?
Cantor defined a range of infinities, and showed how they could be manipulated mathimatically, but even he found there was an "absolute" infinity which was resistant to maths.
Also scientists tend to use "infinite" as though it were synonymous with "unbounded". Your recurring decimal is "infinite" only in that sense. You can go on adding 3s for ever, but you'll never reach a point where you can say: "this number is now infinite"
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You say you can't comprehend infinite space. Join the club :).
Our "reality" is a 3+1D Univrse which we have evolved to understand (sort of), infinity is outside our comfort zone. Consider this though: There can never have been nothing, otherwise there would still be nothing now. (cue alan for a critical quality), so something must be eternal/infinite. There are plenty of experts who argue that the Universe came from nothing, but nothing is a bit like infinite, it has so many definitions that it has become almost necessary to use "absolutely nothing" when what you really mean is "nothing". That's as tautologus as "absolutely infinite".
The concept of "outside the Universe" is an interesting one that I'd like to come back to, and hope someone better qualified than I might comment on.
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What is the smallest you can get? The Planck length, c 10e-35m, is generally considered the smallest length that has any meaning. No one Knows if space is quantized, but I believe recent observations indicate that if it is, the quanta will ccertainly not be larger than the Planck length.
If space contained no mater, what would there be to measure?
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I would say that space is a continuum and the objects that occupy the space are quantised. However we then end up with the conundrum of zeno's paradox. If you chop distance in half over and over you would end up with an infinite number of slices. So at some point we must say that there is a limit to the scale we can measure. This is the Planck length.
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I'm OK with that, Jeffrey, but it still leaves the question: is Planck's length the quantum of space?
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I'm OK with that, Jeffrey, but it still leaves the question: is Planck's length the quantum of space?
You can start your measurement at an infinite number of points along the continuum. If space were really quantised it would indicate a fixed lattice which also assumes a fixed background. How does this gel with relativity?
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How would You know that you could start at an infinite number of points? If points were 10e-35m apart, we have no instruments capable of making such small measurements.
Would quantum foam equate to a fixed lattice?
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Question 1. If you take the number 10, which is essentially a whole number, and divide it by 3 you get three equal figures of 3.3333 recurring. How is this the case in that a completely whole number becomes 3 infinite numbers?
This is probably the easiest of these questions to address.
- The number 10 is finite, and a bit less than 11.
- The number 10/3 is finite, and a bit less than 4.
- The fact that a decimal representation of 10/3=3.33333.... never ends is just an accident of the fact that we have 10 fingers, and use the mathematical notation called "Base 10".
- If we were chickens with 3 visible toes, we might choose to use "Base 3", in which case the answer comes out as a very concise 10.13 = 3.33333....10 (where the little 10 means "Base 10").
- 10/3 is described as a "rational number", in that it can be represented as the ratio of two whole numbers. If the two numbers are finite (and the lower one is not zero), the rational number is finite.
- You may remember in school being told that "to work out the circumference of a circle from the diameter of the circle, multiply the diameter by 22/7"=3.14285..., which is a rational number.
- While this is close enough for many practical purposes, unfortunately, this is not quite right. This ratio is called "pi", and is an example of an "irrational number", in that it cannot be written as the ratio of two finite whole numbers (and not because it is insane). But pi itself is quite finite, being slightly less than 4, at roughly 3.14159265358... etc etc, never repeating.
Another odd fact: If you add up 3.3333... + 3.33333... + 3.33333... = 9.999999...., which looks quite different from the 10 that you started with. But it is possible to show mathematically that 9.999999... is infinitesimally different from 10, and in fact that 9.99999.... = 10 (exactly).
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If it is possible to prove, mathimatically that 9,999...equals 10, does that mean that maths confirms my assertion that you can go on adding 3s for ever, but never reach a point where you can say this is now infinite, because it will always be finite, or is that too much to expect?
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I have a book on real and complex analysis which goes into the 0.999 recurring issue. I will try to find the relevant section.
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Question 1
Take one cubical object made of ten atoms, for simplification. This object might be subdivided into 3 equal pieces only if matter would be made of something continuous (often associated with "classical" in Physics). But matter is made of atoms, so you can't divided it into 3 equal pieces.
Question 2 is simply an unknown, the universe might be finite or not. Most people think it is finite. I think it is finite because the universe contains finite structures in it.
Question 3
The Planck length represents the smallest length beyond which the actual known mathematics of physics make sense. At smaller scale, it is a total unknown. Some physicists think it is a very special and important dimension and some physicists don't think it is that important.
Concerning space, according to Einstein's equation of general relativity, it should exist without matter. The problem is that Einstein didn't want this result and he didn't know about this when he published his theory. Was Einstein right or his equation right? I am convinced he was right at the time of writing and space is a product of matter or an ingredient of matter which cannot exist by itself (like a tree without leaves). The nature of space is very different in the theories of Relativity and Quantum physics, so it is still an unknown. A single theory explaining both is needed.
If there were no matter and there was space, there would be no way to measure it. It doesn't mean space wouldn't be made of pieces (or be continuous).
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If it is possible to prove, mathematically that 9.999...equals 10...
You can show it this way:
- 9.9999... can be represented as 9+9/10+9/100+9/1000....=9(1+1/10+1/100+1/1000....)
- Mathematically, this is called an "infinite series".
- There is a high-school maths formula that lets you sum this series: a(1+r+r2+r3+r4....) = a/(1-r)
- This sum is finite if "a" is finite, and -1<r<1
- In this case, a=9, and r=0.1
- The sum becomes 9.9999... = 9/(1-0.1) = 9/(0.9) = 10
does that mean that maths confirms my assertion that you can go on adding 3s for ever, but never reach a point where you can say this is now infinite, because it will always be finite, or is that too much to expect?
You can start at 3.0, and keep adding 3s after the decimal point:
- 3.33333... can be represented as 3+3/10+3/100+3/1000....=3(1+1/10+1/100+1/1000....)
- In this case, a=3, and r=0.1
- Since -1<r<1, the sum is finite
- The sum becomes 3.33333... = 3/(1-0.1) = 3/(0.9) = 10/3, which was the observation in the OP
- So, no matter how many 3's you add, the answer is always greater than 3.0, and less than 4.0
- As you add more 3's, the answer converges closer and closer to 10/3, which is finite
This is a defect in some of Zeno's paradoxes; the ancient Greek mathematicians did not understand that you could have an infinite sum with a finite total (provided -1<r<1)
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I guess I won't be bothering with the book then.
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Sometimes I wish I had a good background in maths, and sometimes I'm glad I don't. I think this is one of those latter times. :)
Thanks, anyway.
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The word “infinite” or "infinity" is not a number, but a relation/condition whereby an entity cannot be quantified. Consequently there is no assigned value and the entity cannot be measured (contrary to G. Cantors work). Eg. try to measure a stick with only one end!
[14th century. Via Old French from Latin infinitus , literally “not bounded,” from finitus “finished, finite.”]
With integers:
n=da+r. The remainder r will always be <d the divisor, and therefore not divisible by d.
The example simply shows 10 is not a multiple of 3, and 1 cannot be divided by 3. For integers with no common factors, pursuing division using the continuum, there are only (d-1) possible remainders, after which they must repeat.
Zeno's Paradox - Achilles
Zeno argues that motion is impossible.
Using a geometric progression, he divides a distance into a series of smaller distances.
This progression is a continuing process or iteration loop and therefore by it's definition never ends. To move a fraction of a distance requires the same fraction of time needed to move the whole distance. This is inferred or explicitly stated such as, "while Achilles moves to...", i.e. the motions are simultaneous.
The key is, if both distance and time are divided in the same ratio, the speed is still constant and the time required to complete the motion is a simple division of (initial separation)/(difference in speed). This solution makes no use of an infinite series.
By definition an infinite (unending) series that converges to a limit can never equal the limit without contradicting itself. (This applies to statement 2 below.)
Zeno showed that the concept of continuous space and time led to illogical results.
1. u=1
2. lim (9/10^n) for n=1 to inf = 1.
Statement 1 is an equality, statement 2 is a limit. They are NOT the same. If they were one would be redundant. Statement 2 is rounded up to 1 since the difference is insignificant relative to the application.
Show me the magic 'n' where the 9 in the fraction in statement 2 becomes 10.
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They are NOT the same. If they were one would be redundant.
Philosophically speaking:
1. 1 - 2 =-1
2. ei x pi = -1
Is one of these statements redundant?
I would suggest that there are many ways to represent a mathematical value or statement.
Some will be more useful in certain contexts.
Just because they give the same answer does not make them redundant.
Similarly, I would argue that mathematically, 0.999999.... = 1.00000.....
They are not redundant, just different ways of representing the same thing.
Just like some computers have different binary representations for +0 and -0, but they both represent the same number.
Correction: Sometimes I can't do simple subtraction, let alone trigonometry with complex arguments....
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Phyti, whilst I agree with most of what you say, I have to put in a word in defence of Cantor. I struggled for a long time with the way in which "infinite" is used in scientific circles, but it was Cantor's work that brought me to realise that when scientists/mathematics talk of infinities, they are talking about mathematical infinities, which are quite distinct from anything that might be considered as a physical infinity. He also established that absolute infinity had to be a "reality".
I find I can work comfortably with that, without getting into constant wrangling about the meaning of infinity.
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evan_au #16:
1. and 2. are both equalities.
Mathematically speaking, the point is the difference between a limit and an equality.
Why would mathematicians define a limit if an equality would serve the same purpose?
Limits are a necessary part of calculus, and used in the analysis of functions. The idea is to prove that a function can be made as close to a specific value as desired.
A simple example: y=f(n)=1/n, lim(1/n) (for n=1 to inf) = 0. So y can never be <0 and since inf is not a number, and there is no largest integer, y can never equal 0.
Another example:
y1=sqrt(x^2+1), and y2=x, an asymptote to y1.
Lim (y1) (for n=1 to inf) = x, but the two plots never meet.
If y=sum(9/10^n), then 1-y=1/10^n, or y=1-1/10^n.=1-r. Remainder r never =0, and it doesn't just disappear. The representation of numbers with infinite decimals are incomplete, and truncated out of necessity. No one uses the actual value of pi, but an approximation, since that is sufficient for any purpose.
Excluding magic, how can a sequence of nines become a sequence of zeroes?
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since inf is not a number
Most modern computers use IEEE Floating point (https://en.wikipedia.org/wiki/IEEE_floating_point#Exception_handling), and have an explicit representation for infinity, which is returned if you do 1/0.
But it is a dangerous number to use, because it loses a lot of information; there are literally an infinite number of ways to get there!
Some symbolic maths packages can solve problems with limits, and produce exact answers. Because you can deal with (some types of) infinities exactly, if you know how you got there.
The representation of numbers with infinite decimals are incomplete, and truncated out of necessity
This is talking about finite pieces of paper, or computers with a finite number of bits.
But "1" is just a shorthand for the infinite decimal 1=1.00000000.... and we don't consider this to be inexact or truncated!
Mathematically, you don't need to truncate them in all cases, and mathematicians often talk about results "at infinity" (where they are defined).
The examples you give are all "well-behaved", and it makes sense to discuss their values "at infinity".
For example, Lim(n->∞) 1/n = 0. There are no problems here; the limit is defined and converges to a single number, 0.
In fact, the answer is the same as Lim(x->0) x = 0, if you just substitute x=1/n. There are no infinities here!
and Lim(x->∞) sqrt(x2+1) = x is perfectly well-behaved, even if your calculator can't reliably enter infinity in the square root function.
Another well-behaved example is: Lim(x->0) sin(x)/x = 1.
If you try to work out x=0 on a calculator, it will get upset with dividing by zero (you have lost too much information), but symbolic maths packages handle this just fine (L'Hopital's rule works fine, and I will show another way of doing it below).
Lim(x->0) e-1/x2=0 is perfectly well behaved, as is Lim(x->∞) e-1/x2=1
Now, let's look at a poorly-behaving limit:
Lim(x->0) sin(1/x) = (somewhere between -1 and +1)
This does not converge to a specific number, so it does not have a specific limit, although you can place bounds on it.
Mathematically speaking, the point is the difference between a limit and an equality.
I know they make a big fuss about this distinction in high school, but the fact that calculus works at all is because in many useful cases, the there is no difference between the limit and the exact value (chaotic and fractal systems fail this test, which is one reason they are so hard to analyze).
Many people have seen the equality sin(π/2)=1 (where the angles are measured in radians). This is an equality.
But you can also show that sin(x) = x - x3/3! + x5/5! - x7/7! + x9/9! .....
This is called an "infinite series (https://en.wikipedia.org/wiki/Sine#Series_definition)" or Taylor series; I think you would call this a limit?
If you plug in x=π/2, you get sin(π/2) = π/2 - (π/2)3/3! + (π/2)5/5! - (π/2)7/7! + (π/2)9/9! ..... =1
It seems that you are arguing that in one case, sin(π/2)=1 is an equality, but in the other case, sin(π/2) is a limit that can never equal 1!
I argue that you can talk about limits "at infinity", if they are well-defined.
Back to the earlier example of Lim(x->0) sin(x)/x:
Using the series expansion of sin(x)=x - x3/3! + x5/5! - x7/7! + x9/9! .....
You can see that sin(x)/x = (x - x3/3! + x5/5! - x7/7! + x9/9! .....)/x
= x/x - x3/3!x + x5/5!x - x7/7!x + x9/9!x .....
= 1 - x2/3! + x4/5! - x6/7! + x8/9! .....
As x->0, sin(x)/x is perfectly well-defined, and equals 1.
In fact, with the series expansion, you can insert the value x=0, and it is perfectly well-defined on any calculator (but the buttons might wear out first!).
Conclusion (skip here to avoid the maths)
So I suggest that in many real-world cases, you can talk about the results "at infinity" (eg Achilles overtaking the tortoise); while in some frustrating cases, you can't (eg black hole event horizons and the weather).
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Question 1. If you take the number 10, which is essentially a whole number, and divide it by 3 you get three equal figures of 3.3333 recurring. How is this the case in that a completely whole number becomes 3 infinite numbers? I.e. 3.333 recurring never reaches an end therefore cannot form a whole.
As explained above 3.333... is an infinite series whose sum is 10/3. What this means is that 10/3 is the number which the sum gets closer to as you add more digits.
Question 2 and 3 are kind of similar. I cannot comrehend that space is infinite.
If space is actually infinite then join the club. :) But it's also possible that space is finite.
Everything has to have an end to it.
Not at all.
Am i right in saying the universe is made up of galaxies eyc (i may be wrong!).
Yes.
Assuming this what is outside of that universe, and if there is something, what is outside of that? An so on!
There is no outside of the universe.