The Escapist's Math Corner

[kurokotetsu]

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 really doesn't. One statement that they're making is that for any ratio 1/n there exists a 1:1 decimal representation. That's generally, though admittedly not universally, accepted. You might reject that. You might even reject that you're allowed to do multiplication on any infinite series without changing its cardinality. You might say that if someone's going to sum an infinite series it constitutes a fixpoint of a function - which appears less problematic from the 'this number is not that number' perspective.

There are a bunch of things you might do to get out of it by accepting different rules or interpretations. But it doesn't exist in its own special bubble. Maths is about having a set of internally consistent rules and seeing what those imply. You can assert whatever you like, but you've got to explain the set of rules that make it so.

Like there's the proof that goes:

x=.9...
10x = 9.9...
10x = 9 + .9...
9x = 9
x = 1

If you want to say there's a .0...1 difference between .9... and 1 what step do you reject there that yields that answer instead of the one given? What rule do you jettison?

Naturally, it's possible that everybody views math ever-so-slightly differently. That's like...Given. Heh.

I don't see how this equation follows, though. The rebalancing of the numbers is not performed correctly. In order to change the value of the number attached to x, it must be done on its terms. You would have to place one of the other numbers in division under it, not over it. Equal values are dispensed all around, so that would be 10/0.9-etc. The answer is not 9. When calculating with variables, you do not interchangeably treat the variable as its value all at once. It's either x or it's the value of x put in immediately throughout the equation. To wit, if it's the value, you basically have it set that one side is equal to the value of each other (that 0.9 = 0.9, therefore 1 = 1), or...you find that the value of 9 is x-times-a-long-repeating-decimal-number, and therefore X does not equal 1.

Snip

You're quite lengthy, so I hope you don't mind the snippage.

The thing about it is that mind mind rejects not because there's anything wrong with me, but because it doesn't logically follow. The math doesn't follow because you actually have to fudge the numbers (like above) to even get close, and it doesn't technically work. The proof of the '0.0...01' number is that it's simply what you get when you logcially subject 0.9-etc. from 1. Finite or infinite doesn't actually matter. You are infinitely valued in decimal places, not infinite in actual number. The beginning of this number starts with a zero, meaning less than one, less than infinite. You're not reaching infinite values, just taking up infinite space to write it on, in which the remainder between 1 and 0.9-Unto-The-Infinitesimal-Place is logically 0.0-Unto-The-Same-Infinitesimal-Place-Until-One. In short, the value would be 1/Infinity (One-Numerator-Value-Over-An-Infinite-Denomonator-Value) Basically, a url=http://www.merriam-webster.com/dictionary/infinitesimal]textbook[/ur] infinitesimal number is the number between 0.9-etc. and 1, the absolute smallest value that can exist.

THe thing is, as I expressed in the OP and after in several posts, the infinitesimals do not exist in the normally constructed reals. They are in other fields, such as the hyperreals, but the reals don't have infnitesimals and in the reals the calue 1/infinity is not defined, because infinite is not aprt of the reals. I have posted a proof of why if the reals are dense, an infinitesimal as you describe it can't exist. ANd finite and infite matter a lot, ebcause the reason they are equalin the standard reals is because there exists an infnite string, which excludes a "smallest number". here I'll copy paste my own proof.

The proof about the infinitesimal, was based on the fact that the reals are desne. Lets us have a number x, and an infinitesimal ω, 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.

About the proof Nemmerle gave, it is also in the OP so I'll copy paste it too:

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...

And you can defenitely do that in an euqation. You can subsitute any value (that is why a lot of integrals are solved using trigonometric identities or the cases of some clever manipulations) that is euqal at that point. I like to write down a few extra steps so it is extra clear what is ebing done. And the amnipulation can be doen exactly because there are an infnite number of 9s, which means that loign one nine, it still is the same size.

ALso, I never said there was anything worng with you. I mean humans, in general are terrible at understandign the infinite. Also, yes, I'm quite lengthy, but I really love the subject of math.

 

[Nemmerle]

Naturally, it's possible that everybody views math ever-so-slightly differently. That's like...Given. Heh.

I don't see how this equation follows, though. The rebalancing of the numbers is not performed correctly. In order to change the value of the number attached to x, it must be done on its terms. You would have to place one of the other numbers in division under it, not over it. Equal values are dispensed all around, so that would be 10/0.9-etc. The answer is not 9. When calculating with variables, you do not interchangeably treat the variable as its value all at once. It's either x or it's the value of x put in immediately throughout the equation. To wit, if it's the value, you basically have it set that one side is equal to the value of each other (that 0.9 = 0.9, therefore 1 = 1), or...you find that the value of 9 is x-times-a-long-repeating-decimal-number, and therefore X does not equal 1.

Seems to work fine for other numbers

x = 5
Multiply both sides by ten
10x = 50
Subtract x from both sides
9x = 50 - 5
Simplify
9x = 45
Divide both sides by 9
x = 5

y = .456
Multiply both sides by ten
10y = 4.56
Subtract y from both sides
9y = 4.56 - .456
Simplify
9y = 4.104
Divide both sides by 9
y = .456

Why do you think it yields, (what I assume you'd agree is,) a valid result in the above but not in the case of x = .9... ?

 

[The Almighty Aardvark]

Why do you think it yields, (what I assume you'd agree is,) a valid result in the above but not in the case of x = .9... ?

I agree with you, but I don't know if that's the best justification, just because it works with other numbers doesn't mean it works with all numbers.

Let x = a.

10x = 10a
10x = 9a + a
10x = 9a + x
9x = 9a
x = a

For whatever value you put as 'a', this series of operations does nothing to violate x = a.

EDIT:

Though, if you really want a version of this in which you substitute the x's all at once, how about:

x = 0.9..
10x = 9.9..
9x + x = 9 + 0.9..
9x = 9 + 0.9.. - x
9(0.9..) = 9 + (0.9.. - 0.9..)
9(0.9..) = 9
0.9.. = 1

 

[FalloutJack]

Naturally, it's possible that everybody views math ever-so-slightly differently. That's like...Given. Heh.

I don't see how this equation follows, though. The rebalancing of the numbers is not performed correctly. In order to change the value of the number attached to x, it must be done on its terms. You would have to place one of the other numbers in division under it, not over it. Equal values are dispensed all around, so that would be 10/0.9-etc. The answer is not 9. When calculating with variables, you do not interchangeably treat the variable as its value all at once. It's either x or it's the value of x put in immediately throughout the equation. To wit, if it's the value, you basically have it set that one side is equal to the value of each other (that 0.9 = 0.9, therefore 1 = 1), or...you find that the value of 9 is x-times-a-long-repeating-decimal-number, and therefore X does not equal 1.

Seems to work fine for other numbers

x = 5
Multiply both sides by ten
10x = 50
Subtract x from both sides
9x = 50 - 5
Simplify
9x = 45
Divide both sides by 9
x = 5

y = .456
Multiply both sides by ten
10y = 4.56
Subtract y from both sides
9y = 4.56 - .456
Simplify
9y = 4.104
Divide both sides by 9
y = .456

Why do you think it yields, (what I assume you'd agree is,) a valid result in the above but not in the case of x = .9... ?

Well, you can't treat variable x as value x interchangeably. That isn't procedure. If you want to remove the value of x, you're not placing its given numerical value. You're using x. Only in an equation where you are told to replace the variable with its value in order to finish the equation do you input its value as value, not variable. And when you do that, it's not quite the same problem. I mean, of course once side equals the other, but telling us 5 = 5 doesn't make 0.9 equal 1. Additionally, when you remove values from either side, you do not subtract on one side when the other is in a multiplicative state. 10x is a different value from a regular 10, or a 50. Those numbers have no x value, because the variable is not attached to their number. You can only touch it with a common denomenator.

Snip

Well, I don't have to accept the statements of the OP as given if it doesn't follow logically. That is, if we're throwing out a straightforward concept such as an infinitesimal value and using special rules, then it's metaphysics, no longer operating in the actual real, but in conception only where the unyielding numbers can be changed to suit the argument, but not the truth except in a world where it occurs 'this way'. In short, we've pretty much reached the point where we're separated by a matter of opinion which will not yield in one direction or the other, since we're both fairly certain that we're making sense, but we don't agree with each other. Basically, an impasse.

 

[kurokotetsu]

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.

Great answers! If you don't mind I have another question, or rather, two questions around the same topic. The first is whether or not you think the Riemann Hypothesis is valid. The second is what you think of the apparent relationship between the non-trivial zeroes of the zeta function and eigenvalues in quantum mechanics.

Glad that you like the answers. So now to the questions.

First, I do believe the Reimann Hypotheiss is valid. It is one of the wierd mathemaical turth that looks incredibly wierd but somehow works. THe zeta function also has a lot of interesting properties and relations and it would ba far more fun world (and it was the first Millenium problem I knoew of and so it carries a certain sentimental value). I of course have no way of knowing if it is valid, but I would love it to be, and more because of what you ask in your second question.

About that, it was a property I never heard of. From what I'm researching it seem to be related to an idea to approach the Riemann Hypotheisis, the Hilbert-Polya conjecture from what I found out ( https://en.wikipedia.org/wiki/Hilbert%E2%80%93P%C3%B3lya_conjecture ). That connection seem to be connnected to expressing the potential a particle goes through, and that potential is deeply related to the non-trivial zeroes of the zeta-fucntion. This relateds to the eigenvalue of the energy. SO the zeta-functions defines de eigenvalues of the energy. And this seems incredibly lovely. That there is a relation between sucha theoretical construct and quantum mechanics is awesome, and more so, because it is also related to the prime numbers, a mathematical construct which so far has not of my knowledge been used in any physical model. And I find that incredibly interesting. I have to read more on it, because that is so awesome. Thanks for the tip.

Naturally, it's possible that everybody views math ever-so-slightly differently. That's like...Given. Heh.

I don't see how this equation follows, though. The rebalancing of the numbers is not performed correctly. In order to change the value of the number attached to x, it must be done on its terms. You would have to place one of the other numbers in division under it, not over it. Equal values are dispensed all around, so that would be 10/0.9-etc. The answer is not 9. When calculating with variables, you do not interchangeably treat the variable as its value all at once. It's either x or it's the value of x put in immediately throughout the equation. To wit, if it's the value, you basically have it set that one side is equal to the value of each other (that 0.9 = 0.9, therefore 1 = 1), or...you find that the value of 9 is x-times-a-long-repeating-decimal-number, and therefore X does not equal 1.

Seems to work fine for other numbers

x = 5
Multiply both sides by ten
10x = 50
Subtract x from both sides
9x = 50 - 5
Simplify
9x = 45
Divide both sides by 9
x = 5

y = .456
Multiply both sides by ten
10y = 4.56
Subtract y from both sides
9y = 4.56 - .456
Simplify
9y = 4.104
Divide both sides by 9
y = .456

Why do you think it yields, (what I assume you'd agree is,) a valid result in the above but not in the case of x = .9... ?

Well, you can't treat variable x as value x interchangeably. That isn't procedure. If you want to remove the value of x, you're not placing its given numerical value. You're using x. Only in an equation where you are told to replace the variable with its value in order to finish the equation do you input its value as value, not variable. And when you do that, it's not quite the same problem. I mean, of course once side equals the other, but telling us 5 = 5 doesn't make 0.9 equal 1. Additionally, when you remove values from either side, you do not subtract on one side when the other is in a multiplicative state. 10x is a different value from a regular 10, or a 50. Those numbers have no x value, because the variable is not attached to their number. You can only touch it with a common denomenator.

Snip

Well, I don't have to accept the statements of the OP as given if it doesn't follow logically. That is, if we're throwing out a straightforward concept such as an infinitesimal value and using special rules, then it's metaphysics, no longer operating in the actual real, but in conception only where the unyielding numbers can be changed to suit the argument, but not the truth except in a world where it occurs 'this way'. In short, we've pretty much reached the point where we're separated by a matter of opinion which will not yield in one direction or the other, since we're both fairly certain that we're making sense, but we don't agree with each other. Basically, an impasse.

How does the OP not follow logically? So far I haven't seen anything that is indeed logic or mathematical about your arguments. "Common sense" is not logical, and that is the closet I've seen.

And an infinitesimal is not a striaght forward. While the concept has been in mathematics as long as that of limits, and it was more popular with the Bernoullis and Euler, the formalization went in favor of limits and the frormalization of numbers towards the Dedekind cuts and Cauchy sequences, and other such. The fromalization of Dedekind ame in mid 19th Century (Continuity and Irrational Numbers) while Abraham Robinson?s semianl work is form the sxities, mid 20th century (the more formal treatment of infinitesimals, otehr previous formalisations were from the fifties). The field that includes the infinitesimals is called non-standard analysis", "non-standard calculus" and so forth exactl because it is not the contruction of the reals. "Non-standard analysis[1][2][3] instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers." ( https://en.wikipedia.org/wiki/Non-standard_analysis ) form Wikipedia and form Field Medal winer Terence Tao (https://terrytao.wordpress.com/2012/04/02/a-cheap-version-of-nonstandard-analysis/ ) "one can place some useful additional definitions and constructions on these universes, such as defining the concept of an infinitesimal nonstandard number (a number which is smaller in magnitude than any positive standard number).", as you see you will eb hard pressed to find infinitesiamls as a standrd cosntruction of the reals, it is not a straightfoward concept and as such not meaphysics nor changin the numbers. THat is how they've been defined for over a century and the rigurous concept of infinitesimals is much newer.

Please do provide a source or proof that infinitesimals exist within the reals. Because so far I've seen none.

And the expression used in the proof is not a function where x is a variable. It is jsut about inding the value of x that makes teh expression true and such manipulations can be done withou no problem.

 

[DoPo]

Well, you can't treat variable x as value x interchangeably.

Yes you can. They are the same thing. That's how you solve milti-variable equations

2x + 7y = 38
5x + 10y = 65

solve one to get the value of X (or Y), substitute in the other and solve for Y (or X). That's exactly how you solve them - by treating them interchangeably. Because they are. Once you find out X and Y substitute them in the original equation and you would get the EXACT same result. To save the typing, I'll tell you that X is 5 and Y is 4. Let's substitute

2*5 + 7*4 = 38
5*5 + 7*4 = 65

You get the same result. You have to get the same result because X and 5 are the same thing.

 

[The Almighty Aardvark]

Well, I don't have to accept the statements of the OP as given if it doesn't follow logically.

I've already provided two different derivations, one that shows that the set of steps done do nothing to affect the value of 'a', and another that shows 1 = 0.9.. while substituting all the x's at once, even though that's not actually necessary. As Dopo says, x is 0.9.. you can substitute back and forth between them without care because swapping the two numbers does nothing to unbalance the equation. If 10x = 6x + 4, and x = 1, then 10x = 6 + 4x. x is nothing more than a symbol that refers to the value of 1. To say that you can't substitute a value with something that's exactly equal to it gets rid of even the most basic means of manipulating equations.

If you're going to stick to your interpretation, you're going to need to find a new problem with the proofs given before, particularly when you haven't given a mathematical proof of your own.

Let x = a.

10x = 10a
10x = 9a + a
10x = 9a + x
9x = 9a
x = a

For whatever value you put as 'a', this series of operations does nothing to violate x = a.

EDIT:

Though, if you really want a version of this in which you substitute the x's all at once, how about:

x = 0.9..
10x = 9.9..
9x + x = 9 + 0.9..
9x = 9 + 0.9.. - x
9(0.9..) = 9 + (0.9.. - 0.9..)
9(0.9..) = 9
0.9.. = 1

EDIT:

And I have been thinking of your construction of the number equal to the difference between 1 and 0.9.., the infinite zeros with a 1 at the end, trying to figure out what exactly's wrong with it. I'm pretty sure that it's an invalid construction. When you're talking about an infinite number of decimal places, there is no "last" decimal place. Each digit of number with infinite digits must have a finite placement. Whatever decimal place you say that this 1 is at, I can always make a number smaller by placing placing the 1 one digit further.

Just to be clear, what I'm saying is that there's an infinite number of decimal places, but each decimal place has a finite location. I'm pretty sure I'm correct here, but I am not sure exactly what the justification is for why this can't happen.

 

[Nemmerle]

Well, you can't treat variable x as value x interchangeably. That isn't procedure. If you want to remove the value of x, you're not placing its given numerical value. You're using x. Only in an equation where you are told to replace the variable with its value in order to finish the equation do you input its value as value, not variable. And when you do that, it's not quite the same problem. I mean, of course once side equals the other, but telling us 5 = 5 doesn't make 0.9 equal 1. Additionally, when you remove values from either side, you do not subtract on one side when the other is in a multiplicative state. 10x is a different value from a regular 10, or a 50. Those numbers have no x value, because the variable is not attached to their number. You can only touch it with a common denomenator.

Told to?... Okay, so how do you determine whether the person telling you to do it is making a legal move or not? - Whether what they're saying preserves the relationships expressed by the equation?

 

[kurokotetsu]

EDIT:

And I have been thinking of your construction of the number equal to the difference between 1 and 0.9.., the infinite zeros with a 1 at the end, trying to figure out what exactly's wrong with it. I'm pretty sure that it's an invalid construction. When you're talking about an infinite number of decimal places, there is no "last" decimal place. Each digit of number with infinite digits must have a finite placement. Whatever decimal place you say that this 1 is at, I can always make a number smaller by placing placing the 1 one digit further.

Just to be clear, what I'm saying is that there's an infinite number of decimal places, but each decimal place has a finite location. I'm pretty sure I'm correct here, but I am not sure exactly what the justification is for why this can't happen.

AS expresse by myself and Jack, such a number isn't a normal number. The idea of a "normal" number indeed is as you describe, ince there is no last place, wherever you place a one, there are an infninite string of nines after it (I did that proof on an above post). This is why it is wrong the construction of the reals as they usually are.

But is is why I went and added the adendum to the OP, about hyppereals and infinitesimals. In non-standard constructions, such a wierd number to concieve can be constructed. Infinitesimals ( https://en.wikipedia.org/wiki/Infinitesimal ) are constructed as the multiplicative inverse of a number such that any string in the form of: 1+1+....+1 < x SO every single number that can be wrriten in that form of adding ones (every single natural number) is smaller than that number and the inverse of that number is an infinitesimal. IN that construction one could argue for such a wierd number there is a difference between the two. It is fascinating concept, but not studied a lot. And a lot of things that engineers and physicist do while manipulating derivatives can be formalised with infinitesimals and non-standard calculus.