The Escapist's Math Corner

[kurokotetsu]

Screw mathematics and calculus. My life has been miserable ever since they got involved.(and substantially improved after they were gone) Math is EVIL.

Which is why I switched my major to the computer programming.

Funny, I started studying programming and Discrete Mathematics was a requirement, which is a mash of different parts used in computer science and includes a little bit of Calculus.

I have to ask, which part of mathematics - calculus in particular - did you use? When I say calculus, I mean advanced mathematics. Not binary - decimal conversion, not simple multiplication, etc.

I'm pretty sure that Knuth's concrete math had some limits in there. His treatment in the early stages of the book of combinatorial. It also includes derivatives ( http://alg.csie.ncnu.edu.tw/~ykshieh/b2.pdf ). Of course, DIscrete Math may not be a requirement of all curricula, but if you get more into the nit and gritty of CS it is a form of Math. That is why the complexity argumetn of P vs. NP-complete is one of the Millenium Porblems. It would depend on your professor how deep you go, but some pretty advanced math can be found in computing, and it can be a lot of fun. Buy I know most people don't share my love for Math.

Hmm, well I didn't knew that. Thankfully it is only limited to computer science, and only to certain algorithms with approximate value.

As for the *fun* part, I will never under stand that. I was in engineering major before(wasn't a very good student though) and it was mostly derivatives, integrals and 3000 page book of CRC m
anual.

Complexity is a very important branch of larger algorithms, since it determines the time taken by the algorithm. If you program games younwon't run into it (unless you go into advanced AI stuff) but if you want to create a new search algorithm you need tl see how efficient it is. As for Calculus, well most CS is Discrete Math and Knuth only talks about dericatives becuade his Concrete Math is both continuos and discrete, and as such thibgs like limita make sense in the continuos branch.

As to why is fun... well very persona? but somving theorems and thibking of proofs and exploring things ao bizarre there are literally no waya tl express them but with math is fun for me. Math ia not resolving an integral. It is defining it, finding properties and sre what it does and even how it can be applied.

Well, I had in my homework a question about a variable (w) which is composed by either the positive absolute value of a Gaussian variable or minus the absolute value decided by the sign of multiplying 2 other Gaussian varaibles (x and y). It comes off as another Gaussian vector.
A question later was about whether it's dependant on x or y invidividually which is false UNLESS one of them is zero, but apparently we should disregard that scenario, my best guess why was it's because the chance of zero is zero. Am I right?

Ok, so if I get it you have a function of a random variable w, which is basically w=xy (it could have extra term, but this is the variabel that interests us in the begning), where xy random variable in a normal distribution. Th function goes like this:

The positive branch of the Gaussian if xy are greater than one and the negative branch of the Gaussian if y are less. A nice thing to this definition is that it is continuos in 0. And it has the orm of a standar Gaussian for this random variable w, and I would need to do the math, but probably it has a form related to the mulitplication of the standard deviation of the two other variables (if w is indeed xy).

If one of the random variables is 0, it doesn't mean that the probaility is 0 (the normalized normal distribution can show an example of a probability density function where 0 doesn't equal 0 probability). The variables seem independant form one another, but w is clearly dependant on both of them (changing one will alter the result). It is false on every point excpet the zero for one of them, because when one is zero, the mulitplication is zero and as such it doens't matter the value fo the otehr variable, as it will always be zero so it is independant form the non-zero variable.

Acutally it was W=|Z| if x*y>0 and W=-|Z| else, and the answer was that the odds of either X or Y be zero is zero, so it doesn't count. Technically it is zero or at least ininitely small as you pick a point out of a line, thanks!

Yes I undersrand it is just that W may be a combinationnof x amd y. But if the problem.is having real variablws then yes, the probability of either one of them.bein 0 is 0, per Kolmogorov' theorem. Glad I could try to help a bit.

I personally have wondered if there is some base system in which the various constants turn out to be nice round numbers. I don't expect there to be, but IMHO, that would serve as compelling proof for God.

Also, if not, if the universe could have been set up differently so that there was.

This three were discussed in the previous thread and are completely correct. I would like to add one of my own conoction (meaning that I thoght of it in my way home, although it is already an existing proof probably):

Spoiler: Properties of the reals

One of the properties of the reals is that they are dense. What does dense mean? It is that if a<b then there exists a number c such that a<c<b.

Lets say 0.9...=/=1

So, 0.9...<1 as such there exists a number x such that 0.9...<x<1.

Now we focus on 0.9...<x. Let's say the frst different digit is the one in the r position between 0.9... and x. So the first r-1 digits of both of those are 9. If the first different digit is less than 9 then x would be less than 0.9...

If it is 9 then it wouldn't be different.

Therefore it should be greater than 9. But there are no digits greater than 9 (in a base 10 system suhc as what we are suing). Therefore we have arrived at a contradiction. SO our hypothisis is wron and 0.9...=1

I think that is a better proof that the previous three.

Thabks for liking my reductio ad absurdum.proof.

As flr a base system all constanta are round numbers, there ia no such base system for things like 2^(1/2) or pi. Those are irratiinal numbers if you go from the standard construction of rationals (and I do not know of an alrernate constructiin) no matter what you do amd all round numbera are whole numbers which are rational. For physical constants a base system will still work in the same way and all irrational constants will be irrational (and a consequence of the Kolmogorov Theorem is that a phtsical measure has a probabilty of 1 of being irratinal) if you stay in the same units that we usually use. There are renormalisations of the units though which make the most used constanst a simple number, usually 1, and change the rest accorfing to that. So instead of having the speed of light in meters per second you define it as 1 and derive meters from it. Still every measurement tou take has a probability of 1 of being irrational.

A world like rhat though would be voring and with less interesting things I thinlk. You wpuld be stuck woth rationals only, not a field, some equationa would not have solutions... We would be back at acient Greexe in terms of our ubdersranding of numbwrs and we would be missing a lot of great stuff.

 

[Rosiv]

How would one suggest the learning of statistics, for a would-be biologist? I am interested in genomics and bioinformatics but lack a strong math or coding background.

Final year theoretical biology masters student here. I'd try to get a copy of "the analysis of biological data". It's the statistics book we used in the first year of my bachelor.

However, while it very well explains how statistics should be used on biological data, it's not the best book at explaining the statistics behind bioinformatics. However, that's a seriously complicated subject to begin with, to be honest. Furthermore: bioinformatics has become such a huge field that the word is slowly becoming a useless descriptor. What do you mean with bioinformatics?

To be honest I have no idea. I purchase these bioinformatics books a while back because I was interested in the field. I had previously taken an "introduction to genomics" class as well as a discussion based class on genomics. So I figured drilling through some books would be a good idea. I have not had a chance to touch them yet though since I have been busy with coursework.

I am unsure if I want to go to grad school and study the field further. My computer science skills are poor, as I flunked out of the major originally.

I just would like to develop my skills in my spare time to get to a respectable level and flourish in a graduate school program if I do decide to attend.

The alternative would be doing anything in the medical field, like a PA, PT, or Medical tech. I would enjoy the possible nice pay, but doubt I could handle the stress/loans/MCATS.

(For context in my lack of skill in biology, upon taking my Biology GRE exams I scored a 40 percentile, which I was told was poor)


As of now, I am planning to take 1 extra year to "figure things out", taking more coursework correlated to the medical side (Anatomy, Biochemistry, Microbiology...) while also hoping to work as a laboratory technician at my school, helping to prep student labs.

Any advice you could give would be appreciated.

 

[The Almighty Aardvark]

A topologist is someone who can't tell the difference between their ass and a hole in the ground. They can, however, tell the difference between their ass and two holes in the ground.

I'll be back, I like this idea of a math thread

 

[Alleged_Alec]

How would one suggest the learning of statistics, for a would-be biologist? I am interested in genomics and bioinformatics but lack a strong math or coding background.

Final year theoretical biology masters student here. I'd try to get a copy of "the analysis of biological data". It's the statistics book we used in the first year of my bachelor.

However, while it very well explains how statistics should be used on biological data, it's not the best book at explaining the statistics behind bioinformatics. However, that's a seriously complicated subject to begin with, to be honest. Furthermore: bioinformatics has become such a huge field that the word is slowly becoming a useless descriptor. What do you mean with bioinformatics?

To be honest I have no idea. I purchase these bioinformatics books a while back because I was interested in the field. I had previously taken an "introduction to genomics" class as well as a discussion based class on genomics. So I figured drilling through some books would be a good idea. I have not had a chance to touch them yet though since I have been busy with coursework.

There are basically three fields people can talk about when they mean bioinformatics. I use some slightly arbitrary naming below, mostly to categorize them in a way which I think makes the most sense. I'll use some jargon, so if you're not sure what something means, do ask.

1: Theoretical biology- in this field, mostly mathematical models (for example ordinary differential equations) are used to describe biological systems. Examples of these systems includes a wide range of subjects, such as virus-host and predator-prey interactions.

2: Computational biology- although this has a lot of overlap with 1, I do think it's important to separate them, since a different skillset is required. Here, it's not mathematical models which are important, but computer model. A lot of work here is done in spatial systems, such as for example Ten Tusscher's work on root topology. Additionally: this field also includes the wide class of evolutionary models.

3: Bioinformatics- this is the big data analysis. Making evolutionary trees, correlating transcriptomes from different species, that sort of stuff is all done by the bioinformatics group in the research group I've worked in. Their toolset often includes software such as BLAST and HAMMER. Research done by these people is often along the lines of finding out what was in the genome of common ancestors of species, or looking at co-evolutionary properties of genes, both in the genome as well as in how they are transcribed. Lastly, I'd also say the metagenomics would be a part of this category.
Their work I have least experience with, to be honest, but from what I've heard it's one part coding, two parts biological knowledge and three parts experience.

I am unsure if I want to go to grad school and study the field further. My computer science skills are poor, as I flunked out of the major originally.

Yeah. Coding can be pretty hard. Which language did they make you learn in? I found Python as my first language much better than the stuff they did in my CompSci minor. It is high-level enough that you don't have to worry about too much, and once you have learned programming logic in Python, then switching to other languages becomes much easier, since it becomes a matter of learning the syntax and quirks of that language, rather than also having to worry about the logic of programming in general.

I just would like to develop my skills in my spare time to get to a respectable level and flourish in a graduate school program if I do decide to attend.

The alternative would be doing anything in the medical field, like a PA, PT, or Medical tech. I would enjoy the possible nice pay, but doubt I could handle the stress/loans/MCATS.

Let me ask you this: is either something you would see yourself doing for more or less the rest of your life? A good pay is nice, sure. However, if your heart's not in it, you're eventually going to crash and burn. Believe me: I know.

(For context in my lack of skill in biology, upon taking my Biology GRE exams I scored a 40 percentile, which I was told was poor)

Note: I'm not too familiar with USA curricular terminology, but from my looking stuff up, it seems like GRE is a test you need to enter a masters degree, right?

As of now, I am planning to take 1 extra year to "figure things out", taking more coursework correlated to the medical side (Anatomy, Biochemistry, Microbiology...) while also hoping to work as a laboratory technician at my school, helping to prep student labs.

Given your earlier statements of doubt, I think this is a good thing to do. If you do want to continue your education, you will want to do something where you're genuinely interested in.

Hope if this helps in any manner. If you have any further questions, be sure to ask them, and I'll answer to the best of my abilities.

 

[Maze1125]

THe thing is, Leibniz did construct his calculus from a different idea, not about limits, but that the numbers can have a really really small "minimum number", which is called an infinitesimal. And in the sixties there was a non-standard analysis, which is denying the supremum property by thinking there exists infinitesimals (and even before that in the therties, non-standard arithmetic). As such you can basically construct reals (what are usually called hyperreals it seems) in a way where 0.9...=/=1, as there is an infinitesimal between 0.9... and 1, and as such are different numbers. It is a not often explored branch of mathematics, but it exists. For more information: https://en.wikipedia.org/wiki/Infinitesimal

So is 0.9...=1? It depends on what system you are using and how are you defining the reals, how you are working and what theoretical construction you are using. In what we mostly use, it is an equality. But it can be not. And that is exaclt why I want a thread about Math here in The Escapist.

Of course you can construct systems where 0.999... =/= 1. All you have to do is redefine 0.999... (and is also required if you try and use the hyper-reals, as the hyper-reals have something known as the "transfer property" where any number (it's a bit more general than that) that exists in the real numbers also exist in the hyper-real numbers and has the same value, hence if 0.999... = 1 in real numbers then 0.999... = 1 in the hyper-reals unless you redefine 0.999... itself)

So the question is, if you're claiming that 0.999... doesn't have to equal 1, then what is your definition of 0.999...? And is this new definition consistent with the other properties we wish 0.999... to have?

 

[Rosiv]

Spoiler

How would one suggest the learning of statistics, for a would-be biologist? I am interested in genomics and bioinformatics but lack a strong math or coding background.

Final year theoretical biology masters student here. I'd try to get a copy of "the analysis of biological data". It's the statistics book we used in the first year of my bachelor.

However, while it very well explains how statistics should be used on biological data, it's not the best book at explaining the statistics behind bioinformatics. However, that's a seriously complicated subject to begin with, to be honest. Furthermore: bioinformatics has become such a huge field that the word is slowly becoming a useless descriptor. What do you mean with bioinformatics?

To be honest I have no idea. I purchase these bioinformatics books a while back because I was interested in the field. I had previously taken an "introduction to genomics" class as well as a discussion based class on genomics. So I figured drilling through some books would be a good idea. I have not had a chance to touch them yet though since I have been busy with coursework.

There are basically three fields people can talk about when they mean bioinformatics. I use some slightly arbitrary naming below, mostly to categorize them in a way which I think makes the most sense. I'll use some jargon, so if you're not sure what something means, do ask.

1: Theoretical biology- in this field, mostly mathematical models (for example ordinary differential equations) are used to describe biological systems. Examples of these systems includes a wide range of subjects, such as virus-host and predator-prey interactions.

2: Computational biology- although this has a lot of overlap with 1, I do think it's important to separate them, since a different skillset is required. Here, it's not mathematical models which are important, but computer model. A lot of work here is done in spatial systems, such as for example Ten Tusscher's work on root topology. Additionally: this field also includes the wide class of evolutionary models.

3: Bioinformatics- this is the big data analysis. Making evolutionary trees, correlating transcriptomes from different species, that sort of stuff is all done by the bioinformatics group in the research group I've worked in. Their toolset often includes software such as BLAST and HAMMER. Research done by these people is often along the lines of finding out what was in the genome of common ancestors of species, or looking at co-evolutionary properties of genes, both in the genome as well as in how they are transcribed. Lastly, I'd also say the metagenomics would be a part of this category.
Their work I have least experience with, to be honest, but from what I've heard it's one part coding, two parts biological knowledge and three parts experience

I am unsure if I want to go to grad school and study the field further. My computer science skills are poor, as I flunked out of the major originally.

Yeah. Coding can be pretty hard. Which language did they make you learn in? I found Python as my first language much better than the stuff they did in my CompSci minor. It is high-level enough that you don't have to worry about too much, and once you have learned programming logic in Python, then switching to other languages becomes much easier, since it becomes a matter of learning the syntax and quirks of that language, rather than also having to worry about the logic of programming in general.

I learned Java in school. Although outside of school i tried to teach myself various languages like C++ and python. In our genomics class we also did some very basic stuff in python. Id say im pretty much beginner level though given my lack of continued practice.

I just would like to develop my skills in my spare time to get to a respectable level and flourish in a graduate school program if I do decide to attend.

The alternative would be doing anything in the medical field, like a PA, PT, or Medical tech. I would enjoy the possible nice pay, but doubt I could handle the stress/loans/MCATS.

Let me ask you this: is either something you would see yourself doing for more or less the rest of your life? A good pay is nice, sure. However, if your heart's not in it, you're eventually going to crash and burn. Believe me: I know.

Well the good pay would be so I could live independently. In the US college is expensive, so I had to take out loans just for undergrad. Im closing in on 50k of loans so far and I do not know how I could pay that off in a reasonable amount of time while working a 30k a year job.

(For context in my lack of skill in biology, upon taking my Biology GRE exams I scored a 40 percentile, which I was told was poor)

Note: I'm not too familiar with USA curricular terminology, but from my looking stuff up, it seems like GRE is a test you need to enter a masters degree, right?

If you want to get to the "top 10 schools" in the US, the Biology GRE is recommended. I was just curious as to how much skill I had in my major, and that is why I took it.

As of now, I am planning to take 1 extra year to "figure things out", taking more coursework correlated to the medical side (Anatomy, Biochemistry, Microbiology...) while also hoping to work as a laboratory technician at my school, helping to prep student labs.

Given your earlier statements of doubt, I think this is a good thing to do. If you do want to continue your education, you will want to do something where you're genuinely interested in.

Hope if this helps in any manner. If you have any further questions, be sure to ask them, and I'll answer to the best of my abilities.

Thanks for your input.

 

[kurokotetsu]

A topologist is someone who can't tell the difference between their ass and a hole in the ground. They can, however, tell the difference between their ass and two holes in the ground.

I'll be back, I like this idea of a math thread

Lol. I knew a form of that joke, but a lot less crass. And I remember a topology class starting with a topologist bitterly remarking about that joke... But they deserve it... Hahaha

THe thing is, Leibniz did construct his calculus from a different idea, not about limits, but that the numbers can have a really really small "minimum number", which is called an infinitesimal. And in the sixties there was a non-standard analysis, which is denying the supremum property by thinking there exists infinitesimals (and even before that in the therties, non-standard arithmetic). As such you can basically construct reals (what are usually called hyperreals it seems) in a way where 0.9...=/=1, as there is an infinitesimal between 0.9... and 1, and as such are different numbers. It is a not often explored branch of mathematics, but it exists. For more information: https://en.wikipedia.org/wiki/Infinitesimal

So is 0.9...=1? It depends on what system you are using and how are you defining the reals, how you are working and what theoretical construction you are using. In what we mostly use, it is an equality. But it can be not. And that is exaclt why I want a thread about Math here in The Escapist.

Of course you can construct systems where 0.999... =/= 1. All you have to do is redefine 0.999... (and is also required if you try and use the hyper-reals, as the hyper-reals have something known as the "transfer property" where any number (it's a bit more general than that) that exists in the real numbers also exist in the hyper-real numbers and has the same value, hence if 0.999... = 1 in real numbers then 0.999... = 1 in the hyper-reals unless you redefine 0.999... itself)

So the question is, if you're claiming that 0.999... doesn't have to equal 1, then what is your definition of 0.999...? And is this new definition consistent with the other properties we wish 0.999... to have?

Well, I have to look deeper into the transfer principle of the hypeereals and how it works. It has excpetions (for example teh existance of the infninitesiamals is a true statement in R* but not true in R) and I need to verify why is that (I beleive it is the infnite number of terms, but I need to reread the thing). ALso I would like to find my Real Analysis notes, as we did the proof teh the supremum property rules out the infninitesimals and check it. And since we have an example of a statement that breaks the far too simplified if and only if between reals and hyperrreals, it has contrictions. I haven't read before of the transfer principles in general and I would need to infomr myself of the property, because some properties vary between the two fields, and whcihc and why needs more reasearch form my part.

 

[Maze1125]

You didn't answer my question: What is your definition of 0.999...?

 

[kurokotetsu]

You didn't answer my question: What is your definition of 0.999...?

The definition of a single number is quite irrelevant. 0.9... is just a rational number represented by a zro ad ecimal oint and and infnite number of nines. It may have some properties, but that depends on the field and the construction of it rather than the definition of the nuber itself (most numbers aren't really defined in their owne terms, in the contruction of a field,the identities if I recall correctly, and then it is finding "special" numbers and such but you don't define usually a single number). That is not the important question, it is ho we cosntruct the fiedl we are working on.

The proof about the infinitesimal, was based on the fact that the reals are desne. Lets us have a number x, and an infinitesimal &#969;, which is the smallest number there can be different than zero, smaller thatn all the otehr numbers in the reals.

SO since &#969;=/=0 then x < x+&#969;. Because the reals are dense tehre exists a number between this two numbers y, and what is more it can be constructed as the middle point betweeen both of them so y=x+(x+&#969;-x)/2= x+ &#969;/2.

We know that y<x+ &#969; so x+ &#969;/2 < x+ &#969; which implies that &#969;/2 < &#969;. But the infinitesimal is the smallest number and we have reached a contradiction. SO if the reals are dense (and they are dense because they have a least upper bound) then there are no infnitisimals in the reals.

THe proof went like that. The thing is that the definition of infinitesimals is different form that form what I'm seeing. the thing is, this little proof either proofs there are no infinitesimals, or that there are an ifnity of them, all smaller than the next and that the hyperrelas by extension have an infinite numbers of "infinite" numbers that can be multiplied by real numbers and still be larger than one anotehr (that doens't happen with ordianls or cardinals). It is interesting conclusion from a very quick roof.

THe thing is, I had the definition of infinitesimal as a single number that was smaller, as such if the reals have an infinitesimal like that, they are not dense and the proofs related to 1=0.9... are incorrect because they are usually based in the density of the reals (Dedekind's cuts, proof by density, the existance of a standard limit of a series) and you could argue that 0.9...+ &#969;=1 if there was a limit.

 

[Pseudonym]

As for your last point... No mathematicain will just switch in the middle of an argument. THey might say it depends on your cinstrcut (for example the sum of the internal angles of a triangle) but if you are using the supremum or ZFC or any other set of axioms, they are not going to switch them to proof something (or at least they shouldn't and it is an incorrect proof). But those kind of "ridiculous" contructions have their place and can eb fascinating. Non-Euclidian geometries could be seen as such a mathematical "nonsense" but it has been fruitful to the study of how to reduce airplane travels and relativity. Making "2+2=5" if it had a ramification in math could be interesting and after all, there is already one that goes "2+2=0" with good results and applications, and studying such a ring might give us some understanding of rings in general (and later of Galois Theory) or the such, if it gives us an insight into the binary operations and the structure of things it can be useful (and fun). Just because a notion is "nonsense" in "real wordl" it doens't mean it shouldn't be explred. FOr example I would find it mightly interesting i the idea of the inverse of an infnitesimal (inifnitesimal are defined by giving the reals a number that is larger to any sum of ones and that has an inverse) could be related to how some of Feynman's manipulations could be done. Or if ti changes Complex Analysis and the extended complex numbers have a more simple or different results in non-standard analysis.Or many many possibilities I don't know of. You can pick a system and run with it. There is noting (apart form funding) that stops you to do it. You might find interesting results. Just stick to it and see what you find. And that is awsesome. Try a world of abanans. It might be usefu one day and you jsut don't know it.

To be clear, I don't believe that non-euclidean geometry, or rings with weird properties should not be studied nor that riciculous constructions don't have their place. My point is more one about language and what people say when they ask whether .9...=1. You may very well construct some ring and associated set of symbols that make '2+2=5' true in some specific language. That does not mean that 2+2=5. This is because when I say 2+2=4 I am speaking normal English and in normal English 2+2=4 refers to the natural numbers and regular addition and regular identity. Sure, any symbol could refer to any number of other things than what it actually does refer to, but when somebody asks me 'is .9...=1?' I would not tell them that it depends on how you construct the real numbers because the real numbers are ussually constructed in one specific way and most people refer to that way unless they specified otherwise. The way I understand what we refer to as the reals (and what we refer to by most standard mathematical notation), is a specific structure (specified up to isomorfism, that is) that I mentioned earlier. This is the structure that most people will work in, when they do math homework, or calculate something they need to know.

We can of course define another structure wherein the real numbers are not archimedean (like you did, if I'm not mistaken in your previous post to maze1125) but those simply aren't the reals and it is silly to take anyone to be referring to hyperreals when they clearly aren't.

As for the dedekind cuts not being intuitive. Without any mathematical training they aren't, admittedly. You are kind of right about that. I did actually come across something remarkably similar to them in Euclids Elements: (book 5, definition 5)

"Magnitudes are said to be in the same ratio, the first to the second, and the third to the fourth, when equal multiples of the first and the third either both exceed, are both equal to, or are both less than, equal multiples of the second and the fourth, respectively, being taken in corresponding order, according to any kind of multiplication whatever."

So the idea is at least old as Euclid which means it has some kind of lasting appeal. It at least seems to be an intuitive solution to do mathematics with quantities that have no rational ratio. (like square root of 2 and 1)

A topologist is someone who can't tell the difference between their ass and a hole in the ground. They can, however, tell the difference between their ass and two holes in the ground.

I'll be back, I like this idea of a math thread

I like that joke.

 

[kurokotetsu]

As for your last point... No mathematicain will just switch in the middle of an argument. THey might say it depends on your cinstrcut (for example the sum of the internal angles of a triangle) but if you are using the supremum or ZFC or any other set of axioms, they are not going to switch them to proof something (or at least they shouldn't and it is an incorrect proof). But those kind of "ridiculous" contructions have their place and can eb fascinating. Non-Euclidian geometries could be seen as such a mathematical "nonsense" but it has been fruitful to the study of how to reduce airplane travels and relativity. Making "2+2=5" if it had a ramification in math could be interesting and after all, there is already one that goes "2+2=0" with good results and applications, and studying such a ring might give us some understanding of rings in general (and later of Galois Theory) or the such, if it gives us an insight into the binary operations and the structure of things it can be useful (and fun). Just because a notion is "nonsense" in "real wordl" it doens't mean it shouldn't be explred. FOr example I would find it mightly interesting i the idea of the inverse of an infnitesimal (inifnitesimal are defined by giving the reals a number that is larger to any sum of ones and that has an inverse) could be related to how some of Feynman's manipulations could be done. Or if ti changes Complex Analysis and the extended complex numbers have a more simple or different results in non-standard analysis.Or many many possibilities I don't know of. You can pick a system and run with it. There is noting (apart form funding) that stops you to do it. You might find interesting results. Just stick to it and see what you find. And that is awsesome. Try a world of abanans. It might be usefu one day and you jsut don't know it.

To be clear, I don't believe that non-euclidean geometry, or rings with weird properties should not be studied nor that riciculous constructions don't have their place. My point is more one about language and what people say when they ask whether .9...=1. You may very well construct some ring and associated set of symbols that make '2+2=5' true in some specific language. That does not mean that 2+2=5. This is because when I say 2+2=4 I am speaking normal English and in normal English 2+2=4 refers to the natural numbers and regular addition and regular identity. Sure, any symbol could refer to any number of other things than what it actually does refer to, but when somebody asks me 'is .9...=1?' I would not tell them that it depends on how you construct the real numbers because the real numbers are ussually constructed in one specific way and most people refer to that way unless they specified otherwise. The way I understand what we refer to as the reals (and what we refer to by most standard mathematical notation), is a specific structure (specified up to isomorfism, that is) that I mentioned earlier. This is the structure that most people will work in, when they do math homework, or calculate something they need to know.

We can of course define another structure wherein the real numbers are not archimedean (like you did, if I'm not mistaken in your previous post to maze1125) but those simply aren't the reals and it is silly to take anyone to be referring to hyperreals when they clearly aren't.

As for the dedekind cuts not being intuitive. Without any mathematical training they aren't, admittedly. You are kind of right about that. I did actually come across something remarkably similar to them in Euclids Elements: (book 5, definition 5)

"Magnitudes are said to be in the same ratio, the first to the second, and the third to the fourth, when equal multiples of the first and the third either both exceed, are both equal to, or are both less than, equal multiples of the second and the fourth, respectively, being taken in corresponding order, according to any kind of multiplication whatever."

So the idea is at least old as Euclid which means it has some kind of lasting appeal. It at least seems to be an intuitive solution to do mathematics with quantities that have no rational ratio. (like square root of 2 and 1)

A topologist is someone who can't tell the difference between their ass and a hole in the ground. They can, however, tell the difference between their ass and two holes in the ground.

I'll be back, I like this idea of a math thread

I like that joke.

Well, yes. In plain English of course the equality is correct in the structure we use all the days. And indeed in most homeworks or calculations, in real with numbers, 1=0.9... as we always use them. But speaking about this wierd structures is what I like to do. I would usually anser with a "yes, but..." adn go on to rant bout the construction of the numbers. It is in part why I created this thread. TO discuss things like hyperreals and wierd things and challenge a lot of misconceptions about mathematics. And ineed you are not wring, the hyperreals are non-Archimedean ordered field. But bringing them up to a discussion about math, while a bit silly is the point of doing this thread.

And lovely Euclid quote. I didn't lnow of it. I usually read far more modern texts (especially since the calssics are so incliened towards Geometry, which is my least favorite branch of math) and it is really interesting that the idea of the least upper bound nad the sets is budding even then. Fascinating indeed. And very advanced for the Greeks, that didn't like the idea of irrationals.

 

[FalloutJack]

Well, in the strictest sense, 0.999999... can never be 1. Simply pushed towards that gray area in mathematics where the human says "Eh, close enough." and rounds up.

 

[kurokotetsu]

Well, in the strictest sense, 0.999999... can never be 1. Simply pushed towards that gray area in mathematics where the human says "Eh, close enough." and rounds up.

In the strictest sense, in the standard reals, there is no discussion. 0.9... is always exactly 1, there are many mathematical proofs of it. The human limitation comes more from the fact of denying it, not understanding in the full what infnite means and how it works, since it looks counter intuitive. There is no rounding up. It is in the reals always 1. If you disagree, please give a mathematical proof, suhc as the ones presented above. I would be delighted to discuss it.

 

[Zombie_Fish]

Would you rather a complete set of axioms or a sound set of axioms? Or in other words, would you rather a world where everything is consistent but some statements cannot be proven true or false, or a world where everything can be proven but some statements can be proven both true and false?

 

[kurokotetsu]

Would you rather a complete set of axioms or a sound set of axioms? Or in other words, would you rather a world where everything is consistent but some statements cannot be proven true or false, or a world where everything can be proven but some statements can be proven both true and false?

So where do I stand in G?del's Incompleteness theroem?

I would say neither. Because Tarsky's proof of the completness and consistency of classical Euclidean Geometry, without violating G?del's theorem (the Tarsky axioms for Geometry do not define an arithmetic). But as I said before I'm not a Geometry man.

And I certainly prefer consistency over completeness. Being able to proof all the conjectures in a system, is less fun when there can be contradictions, and having conjectures be undiciable is ina sense fun, as you are walking into unkown territory, not knowing if you are walking into an unresolvable proposition. I find it kind of fun.

 

[FalloutJack]

Well, in the strictest sense, 0.999999... can never be 1. Simply pushed towards that gray area in mathematics where the human says "Eh, close enough." and rounds up.

In the strictest sense, in the standard reals, there is no discussion. 0.9... is always exactly 1, there are many mathematical proofs of it. The human limitation comes more from the fact of denying it, not understanding in the full what infnite means and how it works, since it looks counter intuitive. There is no rounding up. It is in the reals always 1. If you disagree, please give a mathematical proof, suhc as the ones presented above. I would be delighted to discuss it.

Well, when I think of a decimal number that forever stretches out towards 1, yet is always underneath it, I think of a curving line that is arcing upwards indeed towards infinity. But the truth is that no matter how long past the decimal point your nines run, you still have a zero before the point. You run within one-billionth of 1, one-trillianth, one-quadrillianth. You still can't reach it. And a graph bears this out, so long as you're prepared to magnify the arcing line forever. That line, and the border between less-and-one and more-than-one will not, and cannot, meet. There is no intersection point, from now until the end of time. It's infinitely close to 1, less than the width of a nanite from 1, but it is not actually 1. In calculations in math, it is allowed to behave as 1 because it's so close. Allowed, by human reasoning, because we decided it so. And for whatever practical applications this has, I'm sure it doesn't matter. But just speaking of it intellectually insofar as the numbers go, it's exactly what your calculator says, a nigh-infinitely repeating decimal of less than 1.

 

[Nemmerle]

That's a more general problem with how we think of infinity though. You can't get there by finite addition. Even 9[sup]9[sup]9[sup]9[sup]9[/sup][/sup][/sup][/sup] doesn't come anywhere close to it. If you're trying to carry over the results you get from conventional arithmetic into something with an infinite series it's going to get weird in a hurry.

 

[Yopaz]

How would one suggest the learning of statistics, for a would-be biologist? I am interested in genomics and bioinformatics but lack a strong math or coding background.

I'd suggest not doing microarrays and stick with dtudying the effects of a few genes when you influence a process in which those genes are involved.

Honestly though, start with learning python rather than learning statistics. It will be a valuable tool and you can learn the statistics later. For the statistics I wouls say that for the methods this site might be a good start http://www.biostathandbook.com/ I would also advice you to look for publications where they have made raw data accessible to the public, publications with sequencing data is probably most relevant for you.

If you're up for a challenge:
http://www.ncbi.nlm.nih.gov/pubmed/21349924 This is a (open access) paper (quality debateable) where they sequenced the T cell receptors of humans and obtained a ton of data from the variability of the receptors and made the data available for us to analyze them. You could try to use the methods they explain to analyze their results and see if you can get the same results.

 

[kurokotetsu]

Well, in the strictest sense, 0.999999... can never be 1. Simply pushed towards that gray area in mathematics where the human says "Eh, close enough." and rounds up.

In the strictest sense, in the standard reals, there is no discussion. 0.9... is always exactly 1, there are many mathematical proofs of it. The human limitation comes more from the fact of denying it, not understanding in the full what infnite means and how it works, since it looks counter intuitive. There is no rounding up. It is in the reals always 1. If you disagree, please give a mathematical proof, suhc as the ones presented above. I would be delighted to discuss it.

Well, when I think of a decimal number that forever stretches out towards 1, yet is always underneath it, I think of a curving line that is arcing upwards indeed towards infinity. But the truth is that no matter how long past the decimal point your nines run, you still have a zero before the point. You run within one-billionth of 1, one-trillianth, one-quadrillianth. You still can't reach it. And a graph bears this out, so long as you're prepared to magnify the arcing line forever. That line, and the border between less-and-one and more-than-one will not, and cannot, meet. There is no intersection point, from now until the end of time. It's infinitely close to 1, less than the width of a nanite from 1, but it is not actually 1. In calculations in math, it is allowed to behave as 1 because it's so close. Allowed, by human reasoning, because we decided it so. And for whatever practical applications this has, I'm sure it doesn't matter. But just speaking of it intellectually insofar as the numbers go, it's exactly what your calculator says, a nigh-infinitely repeating decimal of less than 1.

You are talkng about an asymptotic behaviour ( https://en.wikipedia.org/wiki/Asymptote ). And that is interesting idea, but easily debunked.

Let us examine the most well know asymptote 1/x. for every real x, 1/x > 0. But let us look at x + &#949; where &#949;>0. Then x 0. as such, there is a number such that 0 < 1/(x + &#949 < 1/x, as it can be constructed (it can be alsoe proven from density of the reals). And that can be donde for any asymptote.

Such a number and construction can't be done between 1 and 0.9... as I wrote on the OP. There is no difference between the numbers. IN an asymptote you can always find a number closer to the asymptote. Between 0.9... and 1 there is no closer number. As there is no closer number, the numbers are equal. That is also becuase a number can only be less (it was also shown in the OP why it can't be less),equal or greater than another number. It isn't greater, if it was less there would be another number between those two numbers (there is not) so it isn't less, so it is equal. The zero before isn't a proof. It isn't asymptotic. It is exact.

That's a more general problem with how we think of infinity though. You can't get there by finite addition. Even 9[sup]9[sup]9[sup]9[sup]9[/sup][/sup][/sup][/sup] doesn't come anywhere close to it. If you're trying to carry over the results you get from conventional arithmetic into something with an infinite series it's going to get weird in a hurry.

There are a lot of problems with our understading of the infinite. Gragam's number, a number so large that it can't be written in full with with all the space in the observable Universe is as far away form the infnity as the number 1. YOu can't express it. And yes, the results form it convetntional arithmetic doesn't work in infnite quantities, our perception of it is terribly off. ANd it is incredibly fun how wierd things get with the infninte for example the coutnable and non-countable infinites.

 

[FalloutJack]

The sum of all problems betweem 0.9 and 1, and infinities.

I understand where you're going with this (They covered this in college, I swear!), but I must differ (on the first part) and confuse (on the infinite part). I understand how math can insert equations into the problem to bridge the gap, so to speak, but I must view these scenarios as metaphysical thought experiments. The equation exists within a bubble put aside from the normal course of things, in terms of physical space and math. It's a hypothetical reasoning of interest which, naturally, many mathematicians flock to in order to discuss. We are doing that, even though I don't consider myself an incredible mathematician. A good one, but I've known better than myself.

Now then, the line on the graph is a means to illustrate this, but you state the notion that unto approacing infinite, there being no closer number to one, it becomes one. My mind rejects, because of course there IS a difference that exists in a non-repeating value that can be written thusly: 0.00000...01. Pi is a number that can hardly fit the space of writing, but mathematicians, of course, find shortcuts to express the number in a manner understandable. The difference between 0.9 and 1 being, of course, zero-point-however-many-zeroes-it-takes-to-put-a-one-at-equal-placement-to-the-last-nine. Those two numbers are right beside each other, and yet, separated by a forever gap that exists...probably only within whatever lies beyond higgs-boson in quantum flux and foam.

Now, comes the part where I start to sound like the guy who writes XKCD stuff. The problem of 0.9 and 1, as expressed in percentage by a vote tally. There is a vote called by 100 people to make 100%, to be effectively one in all things. Arguments are made, people are convinced and cajoled and bribed into towing the line. The vote is called. One guy says no. Crap. Okay, so arguments and whatnot to try and get him in line are made, but for various reasons - same reason he can't be ousted from the percentage itself - he can't be convinced, begged, bought, etc. He is a lynchpin, an immovable object, a contant. For whatever reason, even if it's just because you said yes and he's an asshole, he says no. So, they try to work around him. Phrase the issue differently. No. Use metaphor and analogy, being philosphical even. No. Add more numbers who are willing to say yes. No. 100% not reached. More people from all over the country. No. Allies and underlings. No. World peace is declared and everyone else on the planet lives in harmony. No. Aliens from all over the cosmos come not as conquerors, but as well-wishers to this big thing going on here, trying to fill the universe with...well...intergalactic unity. The one guy still says no, so it's not 100%. It's not the definition of a unanimous vote, where everybody, every single person in a group says the same thing. It's merely a massivelt overwhelming majority. And if they exclude the one guy from the vote, that's not making it unanimous EXCEPT by changing the rules and invalidating the problem for a viable answer.

Of course, then we approach infinity, which is a mess. I feel as though it is for reasons like this that the term 'nigh-infinity' exists, just to express something that is so high that we cannot conceive of it, but we're pretty sure it's not actually infinite. Must like how it's very very VERY hard to find out what exists deeper and deeper inside the tiniest existence of what's inside the universe, it's naturally hard to conceive of a value so tiny to separate two numbers that are very extremely close, but it's in there. It must exist because 0.9 did not become 1 naturally. It needed executive meddling, and that's just cheating.