The Escapist's Math Corner

[kurokotetsu]

Well, I hope this has an audience. Recently I had a discussion here where I had to delve inot Game Theory and studying and sharing things was fun. Now I saw a thread about the question if 1=0.9... is true. So I want to opem thread about discussing mathematics, theories, interpretations, theorems. All the good stuff that you are curious, want to share,or just ask. It is not an AMA, as I'm not a professional mathematician, I have taken several courses on several topics of Math and love to read about it. I've made proofs and want to do my disaertation in Applied Math. SO let's talk about math and ask anything you want to and I'll discuss it to the best of my abilities. Put your most interesting results, mathematical riddles, all about Math

So let's discuss. Topology? G?del's First Incompleteness Theorem and it's misconceptions? Euler's equation? Cantor's Set Theory and bigger infnites? New developments? Fermat's Last Theorem? The MIlenium Problems? Goldbach's COnjecture? Ramanujan's Series? PDE? Chaos THeory? Fractals? What should we talk about?


Also I will talk about the 1=0.9..., as someone that has had PhD in mathematics teach the subjecto to him. Just because it will keep bugging me if I don't give my piece.

Spoiler

Thing is more complicated that it looks. In that thread we had several answers. Yes and no for different reason. What is the real thing?

Well, let's give some proofs.

Spoiler: Fraction proof

Each rational number has an "exact" decimal representation, which means that it can be expressed as either a decimal with an periodic queue of zeros at the end or a periodic set of numbers that repeat at infinitum. That can be proven as an if and only if (which is a way to proof that the square root of 2 is irrational). So the expanion 1/3=0.3... is an exact equality, not an aproximation. as such the simple algebaraci manipulation 1=3/3=3*(1/3)=3*0.3...=0.9... proofs that it is exact interpretation.

Spoiler: A simple equation

Let's say x=0.9...
Now Let's multiply by ten: 10x=9.9... and get a new expression.

This 10x=9+0.9... by just taking the whole and the decimal part of the right hand term.

By hypotheisis 0.9...=x so let's replace it in the above expresion and get 10x=9+x

Now let's solve the equation and we get 9x=9 so x=1, exactly. Then 1=0.9...

Spoiler: A series proof

First we have that 0.9... can be expressed (in a similar way any decimal number can be expressed) as 9*(1/10)+9*(1/10[sup]2[/sup])+9*(1/10[sup]3[/sup])+...+9*(1/10[sup]n[/sup])+...

Which by the distributive law and be expressed as 0.9...=9*((1/10)+(1/10[sup]2[/sup])+(1/10[sup]3[/sup])+...+(1/10[sup]n[/sup])+...)

The second term there is a gemoetric series with initial term a=1/10 and a ratio r=1/10. As r is less than one strictly the series has a limit and can be found as a/(1-r) which is (1/10)/(9/10)=1/9.

Subsitute this in the previous expression and you get 0.9...=9*((1/10)+(1/10[sup]2[/sup])+(1/10[sup]3[/sup])+...+(1/10[sup]n[/sup])+...)=9*(1/9)=1

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

ANd many more. The wikipedia article gives a good rundown of several of those: https://en.wikipedia.org/wiki/0.999...

So why I said that it is complicated? Because we are talking about only the standard construction of the real numbers. There are two branches of calcules, the Newtonian and the nonstandard. All of the proofs above are based on limits and how they are constructed in the standard way. It is based on density, and the lack of infinitesimals, which is a conclusion of the least upper bound property or supremum property, which is a necessity for the construction of reals with the normal fomralisation, dedekind's cuts.

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.

 

[IceForce]

Just for the benefit of people who might be unaware, that thread you speak of was just someone trying to start a flamewar. We get those here occasionally, and it's always the same topic. Because apparently people like to go around reading old Cracked articles.

Don't know what I'm talking about? Here, see for yourself: http://archive.is/fyx3X#selection-3179.0-3179.41

 

[Guffe]

I suck at math...
Mathematics and numbers make me dizzy xD
I like the show Numb3rs though, and all the theories they throw around there.

 

[Redlin5_v1legacy]

I enjoy seeing the fruits of math labour but the actual breakdown of the numbers isn't something of interest to me. I leave that to you super-nerds. Go and math! Make the world a better place!

 

[kurokotetsu]

Just for the benefit of people who might be unaware, that thread you speak of was just someone trying to start a flamewar. We get those here occasionally, and it's always the same topic. Because apparently people like to go around reading old Cracked articles.

Don't know what I'm talking about? Here, see for yourself: http://archive.is/fyx3X#selection-3179.0-3179.41

The thing is, it is an interesting subject, since it can explore some fringes of math which are unkown to a lot and can have some interesting things. The flamewar is irrelevant, since the thing is pretty clear cut, but it can be a foot for omsthing very interesting.

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.

I suck at math...
Mathematics and numbers make me dizzy xD
I like the show Numb3rs though, and all the theories they throw around there.

NUmb3rs was fun, and here we can discuss waht it presente, since it talks about real mathematical theories, just in a very silly way.

I enjoy seeing the fruits of math labour but the actual breakdown of the numbers isn't something of interest to me. I leave that to you super-nerds. Go and math! Make the world a better place!

Well, Math is more than numbers. That is just one of the multiple Theories. There is a lot of Math to go arround! It is awesome.

 

[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.

 

[Pirate Of PC Master race]

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.

 

[kris40k]

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.

 

[Pirate Of PC Master race]

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.

 

[Zombie_Fish]

Do you think we will ever solve another Millennium Prize Problem [https://en.wikipedia.org/wiki/Millennium_Prize_Problems]? And if so, do you have any thoughts on which one?

Also, since you asked for riddles:

You have a set of balance scales and eight identical marbles. One marble is slightly heavier than the rest, but not enough that you can pick it up and notice the difference. You can put as many marbles on either side of the scales as you want, but you can only use it twice. How can you find the heavier marble?

EDIT: To quote DJ Khaled: "Another One":

What do you think of the proof that the following summation equals -1/12?

\sum_{i=0}^{\infty}i = 1 + 2 + 3 + 4 + ...

(apologies if you don't use L[sup]A[/sup]T[sub]E[/sub]X regularly, the sum I'm referring to is this [https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF]. A proof of the statement is also there.)

 

[Pirate Of PC Master race]

Do you think we will ever solve another Millennium Prize Problem [https://en.wikipedia.org/wiki/Millennium_Prize_Problems]? And if so, do you have any thoughts on which one?

Also, since you asked for riddles:

You have a set of balance scales and eight identical marbles. One marble is slightly heavier than the rest, but not enough that you can pick it up and notice the difference. You can put as many marbles on either side of the scales as you want, but you can only use it twice. How can you find the heavier marble?

EDIT: To quote DJ Khaled: "Another One":

What do you think of the proof that the following summation equals -1/12?

\sum_{i=0}^{\infty}i = 1 + 2 + 3 + 4 + ...

(apologies if you don't use L[sup]A[/sup]T[sub]E[/sub]X regularly, the sum I'm referring to is this [https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF]. A proof of the statement is also there.)

First one is easy.

Spoiler

Div stones to 3/3/2.

Measure 3 : 3. Leave 2 to the side.

IF UNBALANCED
{
Div heavy side stone to 1:1:1
Measure first 2(1 : 1)
IF UNBALANCED
{
Heavy stone is the one left behind
}
IF BALANCED
Leftover stone is heavy one
}
IF BALANCED

Only 2 stone left. Measure the heavy one.

 

[Zombie_Fish]

snip

I would suggest putting your response in spoiler tags for the benefit of people who want to figure out an answer to it themselves, but yes that is correct.

 

[Pseudonym]

I think an intuitive (though what seems intuitive to me now did not seem so a year ago, so this might not work for you) way to think about whether .99999...=1 is to identify each real number with a dedekind cut. That is, with a set X of rational numbers, such that X is not empty but not the set of all rational numbers either, if X contains y and z<y than X contains z and X contains no largest rational number. So a real number x would be identified with the dedekind cut containing all rational numbers smaller than x. This means the real number 1 would be identified with the dedekind cut of all rational numbers smaller than 1. Now any rational number is smaller than 1 if and only if it is also smaller than .999... and hence these two numbers are the same dedekind cut and hence the same number.

For those wondering: identifying reals with dedekind cuts is fairly standard practice (though some people prefer using cauchy sequences of rational numbers) that is justified by proving that the real numbers are a complete linearly ordered archimedean field and any complete linearly ordered archimedean field is isomorfic to the dedekind cuts (if we define a suitable multiplication and addition).

So why I said that it is complicated? Because we are talking about only the standard construction of the real numbers. There are two branches of calcules, the Newtonian and the nonstandard. All of the proofs above are based on limits and how they are constructed in the standard way. It is based on density, and the lack of infinitesimals, which is a conclusion of the least upper bound property or supremum property, which is a necessity for the construction of reals with the normal fomralisation, dedekind's cuts.

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.

Ehm, I think this is not quite right. Both Newton and Leibniz used infinnitessimals (or fluxions which were kind of like infinitessimals but intended to be less weird). The epsilon delta stuff you'd learn in a modern analysis course was invented in the nineteenth century by Dedekind, Weierstrass and some others. As I understand the history here non standard analysis was standard untill around the moment somebody constructed a function that was continuous everywhere but nowhere differentiable. (intuitively, a function that can be drawn without taking the pencil of the paper that sharply changes direction everywhere) That was the moment mathematicians wanted a more secure foundation of calculus than infinitessimals. In the decades thereafter modern 'standard' analysis was invented.

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.

While a lot of mathematicians have this attitude, it isn't really that easy. If I define addition in some insane way than '2+2=5' might become true. If I define 'banana' to mean person than 'you are a banana' becomes true. But no mathematician seriously considers the option that 2+2=5 ans saying you are a banana is confusing and pointless at best. Typically redefining your terms halfway through a conversation or defining them differently than other people is considered a fallacy. One called equivocation. I don't really know the ins and outs but I was under the impression that there is a more substantial debate to be had about analysis than just 'pick a system and run with it'. While both systems have interpretations according to ZFC that doesn't mean they are both equally worthwhile. One might be vastly more interesting or useful to use so that provides at least a pragmatic argument for defining the reals one way or the other. And once we have settled on a definition of the reals we should certainly not use the other one halfway through and have that be the reals too. That is just confusing. This is only using pragmatic arguments before going into the idea that 'real number' is a rigid designator and that we can't just define it any way we like.

 

[kurokotetsu]

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.

Well, you'll need algebra and calculus at least, especially since integrals are fundamental to statistics. THere are a lot of biostatistics courses, I believe there is one in EDX form Harvard, which means it will probably fulfill a lot of questions and mechanisations. I would reocmmend at least a course of probability too, since a lot of statistics is based on probability and probaility density functions, and knowing the differences will help you and will allow you to not make the standard error of thinking everything has anormal distributution.

First, I'm curious if you're an P=NP-complete, or P&#8800;NP-complete man/woman? Whichever answer, why? Second, are you familiar with the mathematics of quantum mechanics, and the difficulty of resolving it with the mathematics of relativity? There is some speculation that in the way of Godel, a single set postulates couldn't define the laws of physics everywhere. One thought is that you'd need to have a number of approximations mapping the laws, which are accurate for certain regions of the map. Creative combination of those overlapping regions would be your "grand unified theory". Of course some people think that would be impossible, or believe that only an ever more accurate series of approximations are. What do you believe, and why?

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.

Do you have a background that includes a little algebra, geometry, and basic calculus? Khan Academy might be something you should explore, either to get that background, or to study statistics. If you have that background down pat though, you could also do a course online at Harvard or MIT, which do a great job for that.

I'm of the idea that it probably P&#8800;NP-complete. It seems very unlikely that every single complex problem can be reduced to a much simpler one. It would have very interesting consequences if it was, but it the underlying complexities of all NP-complete problems seem to many to all have a P solution.

I have some basico understanding of Schr?dinger's equation (derived it at one point in a class) and have looked at Einstein's Field Equations, but the mathematical efforts at unification such as Q.E.D. and more extensive forms I have not studied. I remember that Heisenberg's Uncertainty principle in a seocndary form was used as an argument against the hypothesis (or axiom) of the smoothness of space and time in Eisntein's General Relativity (the formation of what I read as quantum foam), but in a mathematical sense I have not enough knowledge to say what is exactly the complexity more than the consequences of that foam as a violation of the hypotehisis of the theory. I also understand that some mathematical manipulations of Feynman in his unification are enought to give heart attacks at the more formal mathematicians, were we cancels some infnite amounts, but I haven't read the appers myself.

AS for locla laws. AS far as the observable Universe the laws seem to hold pretty weel. I've heard of a similar concpet, but mostly directed to constants in the Universe, in specific the speed of light in a vaccum. If I recall correctly that was related to a possible eplanation of the asymetry in the early Universe, but I heard the theory a long time ago. The asyomptotic approach to reality though is one that I have pondered myself a few times and one that I like as an idea.

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.

Do you think we will ever solve another Millennium Prize Problem [https://en.wikipedia.org/wiki/Millennium_Prize_Problems]? And if so, do you have any thoughts on which one?

Also, since you asked for riddles:

You have a set of balance scales and eight identical marbles. One marble is slightly heavier than the rest, but not enough that you can pick it up and notice the difference. You can put as many marbles on either side of the scales as you want, but you can only use it twice. How can you find the heavier marble?

EDIT: To quote DJ Khaled: "Another One":

What do you think of the proof that the following summation equals -1/12?

\sum_{i=0}^{\infty}i = 1 + 2 + 3 + 4 + ...

(apologies if you don't use L[sup]A[/sup]T[sub]E[/sub]X regularly, the sum I'm referring to is this [https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF]. A proof of the statement is also there.)

Ahh the Millenum Problem's. I do think we will find solutions (even if the solution is that there is no solution) to all of them. In our lifetime? I think maybe another one after Poincare's conjecture if we are extremely lucky. In terms of which one. I like a lot Navie-Stokes and would love to see the work on it (nad there was a tentative proof a few years back), and the aforementioned P vs. NP-complete is probably the most famous and widely researched so it is a good candidate. Yang-Mills has some interesting conotations, but I don't know of much work done into it. Rieman's Hypothesis also has applications into crypthography for its relation with prime number sitribution (I would guess, since knowing the density of prime numbers in any given set is valuable information by itself) and is extemely interesting (and for waht I recall related to your second riddle). The other two probelms, the Hodge conjecture and the Birch and Swinnerton-Dyer conjecture I have only read their statements and know no furthir thing about it. SO my guess would be either Navier-Stokes or P vs. NP-complete.

AS for your riddles

Spoiler

Ridlle 1: Since the riddle specifies it is heavier, it can be done in two moves (can't be done if we only know if the wight is different I believe). You take three six balls and wight three vs. three. If the balance doens't move, the heavier ball is one of the other two and you just compare those two to indentify it in your second wieghting. If the balance is not equal, take the heavier side and wight two against each otehr. If the are balanced, the heavier is the third ball, If they are unbalanced, the heavier just choose the heavier one and that is. AS noted, it is fundamental for the riddle to specify.

Riddle 2: Well, I've heard of that before. And looked into it. And was baffled by the proof. I was introudced by the NUmberphile and looked into it after. And the more rigorous proof still baffles me. It is an interesting manipulations but what I consider the more sastifying proof, going into the zet function, while I coudn't follow it fully, it shows much more limitations tahn simple algebraic manipulation and that it might be true in a more specific context. But how it works in general still baffles me, even when I think it can be expressed that way. But I haven't seen a technical fault, even when it looks wierd as hell. A taste of the wierd genius that Ramanujan was.

I think an intuitive (though what seems intuitive to me now did not seem so a year ago, so this might not work for you) way to think about whether .99999...=1 is to identify each real number with a dedekind cut. That is, with a set X of rational numbers, such that X is not empty but not the set of all rational numbers either, if X contains y and z<y than X contains z and X contains no largest rational number. So a real number x would be identified with the dedekind cut containing all rational numbers smaller than x. This means the real number 1 would be identified with the dedekind cut of all rational numbers smaller than 1. Now any rational number is smaller than 1 if and only if it is also smaller than .999... and hence these two numbers are the same dedekind cut and hence the same number.

For those wondering: identifying reals with dedekind cuts is fairly standard practice (though some people prefer using cauchy sequences of rational numbers) that is justified by proving that the real numbers are a complete linearly ordered archimedean field and any complete linearly ordered archimedean field is isomorfic to the dedekind cuts (if we define a suitable multiplication and addition).

So why I said that it is complicated? Because we are talking about only the standard construction of the real numbers. There are two branches of calcules, the Newtonian and the nonstandard. All of the proofs above are based on limits and how they are constructed in the standard way. It is based on density, and the lack of infinitesimals, which is a conclusion of the least upper bound property or supremum property, which is a necessity for the construction of reals with the normal fomralisation, dedekind's cuts.

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.

Ehm, I think this is not quite right. Both Newton and Leibniz used infinnitessimals (or fluxions which were kind of like infinitessimals but intended to be less weird). The epsilon delta stuff you'd learn in a modern analysis course was invented in the nineteenth century by Dedekind, Weierstrass and some others. As I understand the history here non standard analysis was standard untill around the moment somebody constructed a function that was continuous everywhere but nowhere differentiable. (intuitively, a function that can be drawn without taking the pencil of the paper that sharply changes direction everywhere) That was the moment mathematicians wanted a more secure foundation of calculus than infinitessimals. In the decades thereafter modern 'standard' analysis was invented.

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.

While a lot of mathematicians have this attitude, it isn't really that easy. If I define addition in some insane way than '2+2=5' might become true. If I define 'banana' to mean person than 'you are a banana' becomes true. But no mathematician seriously considers the option that 2+2=5 ans saying you are a banana is confusing and pointless at best. Typically redefining your terms halfway through a conversation or defining them differently than other people is considered a fallacy. One called equivocation. I don't really know the ins and outs but I was under the impression that there is a more substantial debate to be had about analysis than just 'pick a system and run with it'. While both systems have interpretations according to ZFC that doesn't mean they are both equally worthwhile. One might be vastly more interesting or useful to use so that provides at least a pragmatic argument for defining the reals one way or the other. And once we have settled on a definition of the reals we should certainly not use the other one halfway through and have that be the reals too. That is just confusing. This is only using pragmatic arguments before going into the idea that 'real number' is a rigid designator and that we can't just define it any way we like.

Well,I would argue that the Dedekind cuts themeselves are not that inutitive. It is deeply related to the supremum property (since you are defining a supremum of the cut) and while a beatiful constrution and it can be proven very nicely, it is not as intuitive for non-mathematical people, since it is a concpet and introducing the property in a but of an abrupt way. And yes, the Dedekind cuts is a fairly standard cosntruction of the reals (the most standard that I can think of) and pretty nice.

ABout the Newton and HIsotry of Math stuff. Well, let's get more into dpeth if you prefer. Newton wrote in the principia (quote form teh WIkipedia article, but veriefied searching for translations of the PRincipia in Google):

Those ultimate ratios ... are not actually ratios of ultimate quantities, but limits ... which they can approach so closely that their difference is less than any given quantity...

Which although it isn't a formal definition it implies a biut our modern understanding of limits. As such he can and often is credited witht he idea of "inventing" limits. HIs idea of flusions is more based in the idea of not doing an actual division of quantities (infinitesimals) but that of taking the concpet to the limit and that it can be approached as much as we can, a non-mathematical formulation of what today is the epsilon-delta definition that we use. Cauchy in the 19th century used it, but it was written by Bolzano explicitly, before mid 19th century, but it came better kown later. This was a aprt of formalistaion in the whole mathematcal field that also resulted in the works of Cantor and several other, but Cauchy was a great proponent of i (that was quoted from me by several professors and I think it is discussed in Morris Kline book, although which one I'm not sure right now). The Weirstrass example is indeed important, but the definition and the push can be cosnider to generally predate it. ANd standard and nonstandard calculus and analysis were a bit mxied before. The notation is of Leibniz and a lot of contemproaries used it, but we finally settled in Newton's idea of limit (and therefore fluxions, since that is what he called derivatives) but it is a usefull simplification to think of it as the "standard=Newton" and "non-standrad=Leibniz" as a way yo remember. The continuos everywhere example is very interesting, and mostly I think when it was found that Brownian motion is sucha curve.

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.

 

[Pirate Of PC Master race]

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 manual.

 

[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?

 

[Thaluikhain]

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.