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tommyk
1085316.  Fri Jul 18, 2014 11:56 pm Reply with quote

Above all things, maths is interesting. Engineers may claim it as useful, mathematicians as beautiful (which it surely is) and schoolchildren may decry it as pointless (if they have a bad teacher), but maths remains one of our most stimulating and accessible pursuits. I emphasise accessible - one does not always need to work through all the abstract calculations to appreciate the concept. Well explained maths can be understood by children, almost regardless of the difficulty of the intermediate calculations.

A clever child could easily understand Gabriel’s Horn. This is a mathematically defined 3D shape that looks a bit like a horn where the end you would blow in extends infinitely into the distance, becoming vanishingly smaller. It can be shown that the surface area of this horn is infinite - you would never have enough paint to paint all of its surface - yet its volume is finite, so that the horn could be filled with a finite amount of paint!

I'm sure most people on these forums will have heard of a Mobius strip - a shape with only one side such that an ant (or an elf) walking along the strip will eventually return back to where they started. Straying further into the abstract, two Mobius loops can be combined to form a Klein Bottle - a bottle with one side and no boundary! Unfortunately this could not be imagined by schoolchildren, yet that is not to their discredit as it cannot be imagined by anyone! A Klein Bottle does not live in 3D space like the Mobius Strip, but in 4D space, more than our evolved, savannah-plane inclined brains can handle.

Returning to something more tangible, it is a counterintuitive but true result that 1=0.999999…. Most people have a bias that 0.9999… is very close to 1, but not actually 1, it just seems to be common sense. Lemons and limes. This is wrong. The algebra to show this is very easy:
1/9=0.1111....
9 x 1/9 = 9 x 0.1111...
1=0.99999... !
Maths often doesn’t care about what ‘common-sense’ would tell us.

Advancing into some basic calculus, I have always found the Gaussian integral especially interesting. The Gaussian function gives a distribution of certain continuous properties of a population, for example, height. An integral is just the area under a curve. It can be shown that the integral of the Gaussian function is equal to the square root of pi, which to the casual child's eye would look fairly uninteresting. But as pointed out by Eugene Wigner, what does the area under a curve of height distributions have to do with the circumference of a circle?!

Pushing further into the abstract, we mine the Banach-Tarski paradox, or ‘a pea can be chopped up and reassembled into the Sun’ paradox. This is a proven mathematical theorem that states that a ball in 3 dimensions can be cut up into a finite number of pieces and then reassembled to make two balls, the same size as the original ball! This seems to violate the conservation of volume!

Our word for 100, ‘one hundred’, actually derives from the Old Norse hundrath which was actually used to mean 120, under a base 12 number system, commonly used in European countries of the time until the expansion of Christianity brought with it the decimal system. With this in mind I will assume that when the elves recommended a 500 word piece, they meant 5 hundrath, thus giving me 600 words to play with, and hope they don't look down on me too severely.

 
gruff5
1089794.  Fri Aug 15, 2014 11:42 am Reply with quote

tommyk wrote:
Above all things, maths is interesting... A Klein Bottle does not live in 3D space like the Mobius Strip, but in 4D space, more than our evolved, savannah-plane inclined brains can handle.

Returning to something more tangible, it is a counterintuitive but true result that 1=0.999999…. Most people have a bias that 0.9999… is very close to 1, but not actually 1, it just seems to be common sense. Lemons and limes. This is wrong. The algebra to show this is very easy:
1/9=0.1111....
9 x 1/9 = 9 x 0.1111...
1=0.99999... !

...

You point to some of the most interesting parts of maths. They actually have a Klein Bottle (made from transparent glass) in the Science Museum, London - or it is labelled as such!

What about the Gauss integral relationship to sq. root of pi - any intuitive explanation for that?

Asserting 1/9=0.1111.... can be regarded as equally debatable as asserting 1=0.999999… *, so your algebraic demo doesn't show anything. In my opinion, that '=' sign should be regarded in this case as a shorthand symbol for: "in the limit equals".

*Before anyone pipes up to say it, I do fully realise that this subject has been discussed before on QI forum.

 
tommyk
1090174.  Mon Aug 18, 2014 9:07 am Reply with quote

Im afraid I can't give an 'intuitive' explanation without a little maths. Here's my best, yet woeful, effort:

There is a short story at the start of Eugene Wigner's essay 'The Unreasonable Effectiveness of Mathematics in the Natural Sciences' (found here: https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html) on unlikely mathematical relationships, of which Gaussian integral surely is.
The Gaussian integral is written
I=∫e^(-x˛) dx = √π, where the integral takes place between ±∞.
Once we see the pi symbol, we suspect that circles must be involved somehow, though it is not immediately clear how.

The square of the integral is pi:
I˛=∫e^-(x˛+y˛) dx dy = π.
We get some circle connection when we square the function. Squaring the function means means we gain a dimension, and so instead of giving the area under the graph, the integral of the function now gives the volume. Therefore the total volume between the graph of e^-(x˛+y˛) and the xy-plane is π. x˛+y˛ is the square of the distance from the origin (Pythagoras), and so the total volume is rotationally symmetric - hence the π.

If that was not very clear (I suspect it wasn't), it might be easier to consider some pictures: http://www.wolframalpha.com/input/?i=plot+e%5E-%28x%5E2+%2By%5E2%29
The Gaussian Curve in 2 dimensions (x and y) is the top image.
The circles can be seen on the contour plots.

 
tommyk
1090177.  Mon Aug 18, 2014 9:13 am Reply with quote

The Klein bottle in the Science museum, whilst being very impressive and lovely to look at, isn't a true Klein bottle - which can only exist in 4-D - but a 3D 'projection' of a 4D object.

You are quite right about the algebra for 'proving' that 1=0.9999... It was just a quick note as people usually don't have any problem with the idea that 1/9=0.1111111...

 
Zziggy
1090183.  Mon Aug 18, 2014 9:28 am Reply with quote

0.999... = [sum from n=1 to infinity] 9/10^n
= 9 * [sum from n=1 to infinity] 1/10^n
= 9 * (1/10)/(1 - 1/10) as the sum of a geometric series
= 9 * 1/9
= 1
QED.

 
gruff5
1090327.  Tue Aug 19, 2014 6:34 am Reply with quote

Zziggy wrote:
0.999... = [sum from n=1 to infinity] 9/10^n
= 9 * [sum from n=1 to infinity] 1/10^n
= 9 * (1/10)/(1 - 1/10) as the sum of a geometric series
= 9 * 1/9
= 1
QED.

Sorry, this just looks like another obtuse algebraic tautology and in the same vein as tommyk's.

I think if the concept of a limit is not understood and regarded as a core part of that deceptively simple phrase "sum to infinity" then the whole thing falls over. Infinity can only be reasonably used in classical mathematics if the limit is regarded as the actual thing you are working with, rather than infinity itself (whatever that is!)

 
Zziggy
1090329.  Tue Aug 19, 2014 6:57 am Reply with quote

Well we can consider it as the sum from n=0 to p, and then take the limit as p -> infinity if you like. The answer is exactly the same.

The concept of a limit is understood and defined, and

lim{p->infinity} sum{n=1 to p} 1/10^n = 1/9

can be proven by this definition.

 
gruff5
1090334.  Tue Aug 19, 2014 7:34 am Reply with quote

tommyk wrote:
Im afraid I can't give an 'intuitive' explanation without a little maths. Here's my best, yet woeful, effort ...

The square of the integral is pi:
I˛=∫e^-(x˛+y˛) dx dy = π.
We get some circle connection when we square the function. ....

Yes, thanks, that does give me some feel of where the circle is. I suppose regarding the bell curve, it does look a little like an "unwrapped" circle or something ...

 
gruff5
1090336.  Tue Aug 19, 2014 7:45 am Reply with quote

Zziggy wrote:
.... The concept of a limit is understood and defined, and
...

It may be for mathematicians, but it probably isn't for non-mathematicians when trying to explain why 0.999... is regarded as equal to 1. Without the limit concept, a non-mathematician might say: "But another 9 can always be added on to that long series of 9s" It's a perfectly reasonable thing to say and brings up the ancient objection as to whether absolute infinity, rather than potential infinity, is a legitimate concept to be using in a logical process. The objection can be avoided by explaining to the non-mathematician what a limit is and that we are working with limits and not infinity itself.

 
Zziggy
1090338.  Tue Aug 19, 2014 8:00 am Reply with quote

Are you suggesting non-mathematicians don't understand the concept of 'infinity'? I don't know, maybe I grew up in a strange family but I knew about it as a child. In fact, I would say the idea of infinity might be easier than the idea of a limit. In any case, right at the beginning tommyk described this as 'counterintuitive' (personally I'm not 100% certain I agree with the assertion that "Well explained maths can be understood by children, almost regardless of the difficulty of the intermediate calculations").

 
gruff5
1090369.  Tue Aug 19, 2014 12:39 pm Reply with quote

More than that, I would suggest no mortal can properly comprehend 'infinity'! I think non-mathematicians generally think of 'infinity' in the 'potential infinity' sense - something without end, going on and on, a process. In that sense, the series of 9s in 0.9999... just keeps going on and never completes to a value of 1.0

This is a perfectly reasonable assumption and trying to convince someone otherwise with a bit of circular algebra won't help and is not doing justice to the issue or the person asking the initial question.

Mathematics was in a real quandary after the introduction of calculus. The method worked and worked well, but it's logical foundations were extremely shaky and contradictory - mainly due to its reliance on the use of 'actual infinity' (from which came infinitesimals) , which could not be defined. The way out of the dilemma finally came about by basing the logic on the notion of a limit, so side-stepping the incorporation of 'actual infinity'.

You're right, the idea of a limit is non-trivial to understand (hence it took the mathematical world so long to arrive at it). But it is a tangible concept and one that can be grasped with the human imagination, unlike 'actual infinity'. One could use Zeno's paradox as a way in to the idea of limits.

 
Zziggy
1090402.  Tue Aug 19, 2014 5:09 pm Reply with quote

gruff5 wrote:
More than that, I would suggest no mortal can properly comprehend 'infinity'! I think non-mathematicians generally think of 'infinity' in the 'potential infinity' sense - something without end, going on and on, a process. In that sense, the series of 9s in 0.9999... just keeps going on and never completes to a value of 1.0.

But that is exactly what the proof relies on, so I don't see what the problem is. I'm not asking anyone to describe me Hilbert's Hotel or anything. And as has been said, yes, it is counter-intuitive.

And it is a proof, thanks. Not circular algebra or an obtuse algebraic tautology. The worst I committed was a bit of mathematical laziness.

 
gruff5
1090462.  Wed Aug 20, 2014 7:16 am Reply with quote

Apologies if I sounded rude about your proof, but if it relies on "potential infinity", then 0.999... does not equal 1.0, but is always a "little" bit less.

There are two types of infinity* - actual (or completed) infinity and potential infinity. The latter is easy to grasp, the former can no more be visualized by the human imagination than a 4D Klein bottle - hence the difficulty for many people with 0.9999... = 1.0

More about this distinction here:

http://en.wikipedia.org/wiki/Actual_infinity

wiki wrote:
Mathematicians generally accept actual infinities... The philosophical problem of actual infinity concerns whether the notion is coherent and epistemically sound.

Probably the greatest mathematician whoever lived, Gauss, said:
Quote:
I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction.

*Ignoring Cantor's different sized infinities

 
Zziggy
1090485.  Wed Aug 20, 2014 9:03 am Reply with quote

I don't know what more I can say, save that my proof is correct, and there are many other proofs in existence too. What you are calling 'actual' infinity'* appears to just be the concept of infinite sets? A wealth of research has been done on this subject in set theory. Gauss was clever and all but he did die like 15 years before set theory arrived.

I will grant that the discussion we are currently having might not be easily understood by someone without training in mathematics, but I still dispute that the proof or idea is too hard for non-mathematicians in general to understand.

*I'm not disputing that this may be the correct accepted term, only that I have not heard it before.

 
ConorOberstIsGo
1090514.  Wed Aug 20, 2014 3:16 pm Reply with quote

gruff5 wrote:
Apologies if I sounded rude about your proof, but if it relies on "potential infinity", then 0.999... does not equal 1.0, but is always a "little" bit less.

There are two types of infinity* ......


*Ignoring Cantor's different sized infinities


Sorry but I trained as a maths teacher so would like to point out that while the idea of 0.9(recurring) being equal to 1 may not have been explained well, it is most definitely true in a non-tautological fashion. If anyone is still unconvinced after a cursory youtube search of this, then I am happy to take them through it.

More importantly, while you may understand several types of infinity, there are actually about 20 different names given to different types of infinity. You might say that two or three are qualitatively the same but it's definitely not two different types plus all of Cantor's.


Last edited by ConorOberstIsGo on Thu Aug 21, 2014 12:52 pm; edited 1 time in total

 

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