September 18th, 2014, 05:54 AM  #21 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  
September 18th, 2014, 07:23 AM  #22 
Member Joined: Aug 2014 From: Somewhere between order and chaos Posts: 44 Thanks: 5 Math Focus: Set theory, abstract algebra, analysis, computer science 
I agree with v8archie. You are exhibiting a fundamental misunderstanding of the concept of infinity. You are trying to treat infinite and infinitesimal numbers like real numbers. There is no difference between 0, +0, and 0 for instance. Surreal numbers can not be expressed in the traditional way (don't ask me what the proper way of expressing them is, because I know very little about surreal numbers other than that they are a theoretical possibility). You also show a fundamental misunderstanding of limits. Your statement about the limit of zero being something other than zero is meaningless. If you want to truly understand limits, I suggest you look up the Cauchy convergence test. You might want to look up the Cauchy test for convergent sequences first, since it uses fewer variables and is somewhat easier to understand; it's a good intermediate step towards understanding limits of functions. 
September 18th, 2014, 09:17 AM  #23 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
Let me take this in a different direction. There are lots of systems other than the real numbers out there. For example, there is the free monoid on, say, ASCII characters. In that system "0" is different from "+0" since the latter is two letters and the former is only one, and "0" is different from both since the first letters differ. Or take sequences of real numbers relative to some fixed nonprincipal ultrafilter U. You could define $\alpha=(1,1/2,1/3,1/4,\ldots)$ and (as usual) identify the real number $x$ with the sequence $(x,x,x,\ldots)$. Then you have $\alpha<0<\alpha,$ just as you like (though you can't use the symbol +0 for what I have called $\alpha$ since +0 has its own meaning in that system), regardless of your choice of U. Note that for any real number $\varepsilon>0$ you have $0<\alpha<\varepsilon$ so this $\alpha$ is an actual infinitesimal. But it doesn't make sense to ask for its decimal expansion  it's a hyperreal, not a real. (You could ask for the decimal expansion of its shadow, which is the real number 0 = 0.000000000....) 
September 20th, 2014, 12:40 AM  #24  
Senior Member Joined: Nov 2013 Posts: 160 Thanks: 7  Quote:
My method has not been mainstream. I have my own way. I need to believe in what I do, because I don't have always proofs at my disposal. I have noticed that it is possible to arrive at right answers in two way: even a wrong method can lead to a correct answer, as well as a right method of course. All my "prophetic proofs" have been systematically denied in all forums where I have written them. When I drop them, and when I choose to write instead my "normal" proofs which I consider as facts, these are also the same way systematically denied. Might it be better not to write any proofs at all?  
September 20th, 2014, 03:49 AM  #25 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,649 Thanks: 2630 Math Focus: Mainly analysis and algebra 
If you want to have a proper discussion about your mathematical ideas, you need proofs to show why your ideas aren't just wishful thinking and handwaving that are based on a misunderstanding of theories that are wellknown, rigorouslesslly tested, formally proved and completely understood by the mathematical community. I don't recall reading any proof of yours in this thread. 
September 20th, 2014, 04:48 AM  #26 
Banned Camp Joined: Feb 2013 Posts: 224 Thanks: 6  Another Dimension
What lies beyond infinity is another dimension. 1/3 is done by infinite binary divisions in 1 dimensional space (a number line). But can easily be accomplished in 2 dimensions. Angle trisection can be done by infinite angle bisection. But can easily be done in 3 dimensional space. 
September 20th, 2014, 08:26 AM  #27  
Senior Member Joined: Nov 2013 Posts: 160 Thanks: 7  Quote:
I can also tell about my approach to how to divide by three, and what happens if the division is continued for ever. It is an example which is posted on sciforums, I avoid writing all again so I will write the link here It is also an example of how I try to prove what I say so that my ideas aren't just wishful thinking and handwaving. I am writing that dividing by three leaves a nonzero infinitesimal, but usually it is "swept under the rug" because it is so small that noone knows or sees it, but dividing for example by 2 does not leave an infinitesimal. The decimal number system allows an infinitely small decimal number, which is nonzero. The infinitesimal x has no exact value, therefore it is not equal to 0, it has a limit which is 0. Limit of x is 0. The mathematical community has not accepted my ideas because they think that the infinitesimals are always equal to zero, therefore they don't exist. My ideas are not accepted, but I don't see what is wrong with my ideas. I might as well stop writing on these forums if I realized what is wrong. This is after all just my hobby. Life would be much easier for me if I did not need face the flamewars into which my ideas lead. I am not trying to teach anyone to accept my ideas, I am not a teacher, if my ideas are not accepted, it does not matter. Last edited by TwoTwo; September 20th, 2014 at 08:33 AM.  
September 20th, 2014, 09:37 AM  #28 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,649 Thanks: 2630 Math Focus: Mainly analysis and algebra 
The most important thing I can say about that post is that 0.333333... (and the whole of the decimal system) is no more than a representation of numbers. For your argument to hold (that division by three leaves some some small remainder) you will have to explain $\frac{1}{3}$. Where is the remainder now? How is this different to $\frac{1}{2}$. I rather suspect that the only difference is that 3 is coprime to 10, which means that your infinitesimals depend entirely on the arbitrary choice of base. If we chose to use base 3, we'd have $\frac12 = 0.111\cdots$. Does that mean that division by 2 both leaves an infinitesimal and that it doesn't? Your idea seems to be either inconsistent or to introduce a quantity that can be made to disappear whenever we want by changing our notation. And since changing the notation doesn't change the object, I'm forced to conclude that it never existed anyway. As a mathematician, I have always disliked using decimals, because of the loss of accuracy that comes from using finite representations. I always prefer to use fractions and symbols that represent irrationals such as $\pi$ and $\mathbb e$. 
September 20th, 2014, 10:44 AM  #29  
Senior Member Joined: Nov 2013 Posts: 160 Thanks: 7  Quote:
Quote:
$\frac{1}{3}$, because the question assumes a priori that $\frac{1}{3}$ has a decimal representation. The right question is "Does $\frac{1}{3}$ have a decimal representation?" $\frac{1}{2}$ is not different than $\frac{1}{3}$ if they both had no decimal representation. Quote:
I did not think what happens to infinitesimals if we chose to use a different base. It is possible that they disappear, their existence depends on the base. The base10 system that we use, seems to allow the existence of infinitely small decimal numbers. They seem to exist only because of the decimal system that we have chosen to use, the base10 with its decimals. Quote:
Don't ask what is the decimal represention of $\frac{1}{2}$, don't ask what is the decimal represention of $\frac{1}{2}$ in base 3. Ask only "does $\frac{1}{2}$ have a decimal representation?" Quote:
was infinitesimally small. Unlikely is also that matter is infinitely divisible. It is unthinkable that matter disappears if it is divided into infinitely small pieces. Therefore there must be a smallest size of everything, even though the smallest size was not infinitesimally small.The number system that we have chosen to use just seems to allow infinitesimals. They are probably just products of our choice of number system. If we used only fractions like $\frac{1}{3}$ we would not need to deal with infinitesimals. Quote:
Physicists need usually deal with approximations and decimal numbers suit for writing approximations, for example value of $\pi$ . There are fractional representations for $\pi$ but maybe noone uses them anymore. I have read that they were used in antiquity. Maybe we should abandon the decimals.  
September 20th, 2014, 10:58 AM  #30 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
To add to v8archie's point: one of the most important metaskills for a mathematician to learn is the ability to switch between different representations of a problem. For example, when working on my paper on odd perfect numbers, I often had to switch between thinking of numbers in an 'Archimedian' sense (real numbers which can be compared with < and >) and what I now call a 'supernatural' sense (integers are a particular case of possibly infinite products of primes raised to possibly infinite exponents). In the first case the natural order is total while in the second case the natural order (divisibility) is partial. For real numbers there are many forms. Binary (or decimal) expansion, continued fraction, the Engel or Pierce expansion, etc. If the number is rational you can also write it as a fraction of integers; if it's an integer the Zekendorff expansion; and so forth. There's nothing special about the decimal expansion. In fact, it's poorly suited to most purposes. Use it when it works, but please don't mistake it for anything more than it is. 

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