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infinity?


Guest DrBelfrey
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2+2=4 because x=x; prove x=x and i'll admit you as a reasonable man/woman' date=' unless it's proven unreasonably.

 

As you can notice, you so called Mat Grime, x/0 is not a constant value, as it is R, and all imaginary numbers, because if 0/0=x, then y, where y doesn't =x, would be y*0/0, meaning y*x=x, and x=any number greater than equal to or less than x, which is certainly a contradiction for the givens.[/quote']

 

 

x/0 does not represent any real number, nor does it represent every real number, it isn't a number, it isn't a set, it is a symbol that has no mathematical meaning per se especially since you are using it for arithmetic. I don't think I'll respond to the ad hominem attacks.

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Dividing by zero is undefined and cant be defined' date=' since such an operation or notation would not have the normal sense. 1/x is a short hand notation for [math']x^{-1}[/math], the unique element in [math]\mathbb{R}[/math] s.t. [math]x^{-1}x = xx^{-1} = 1[/math]. If you want to give a sense to 1/0, in no way it would be the inverse of zero thus making this notation totally superfluous. The same holds for your argument 123rock, since like i just said, no way 1/0 will be the inverse of 0, so your argument with x and 0/0 is nonsense.

 

Another exemple : Often people write [math]i = \sqrt{-1}[/math], where i is the complex number. This notation also has no sense. I will let you guys try to find the error in the following reasoning and see for yourself why it is meaningless to write [math]i = \sqrt{-1}[/math]

:

[math]-1 = i^2 = (\sqrt{-1})^2 = \sqrt{-1}\sqrt{-1} = \sqrt{-1 -1}= \sqrt{1}=1[/math]

 

Mandrake

 

of course it makes sense, picking the principal branch of the square root function

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of course it makes sense, picking the principal branch of the square root function

 

Yeah without specifying this sort of stuff; meaning it is not the same sqrt function !

That is the whole point of my post, that you cannot use this as the classical sqrt function. So NO the notation doesnt have a sense when you do not specify with branch you are using.

 

Mandrake

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Maths no matter what you say is an abstract subject. If it were not so abstract it wouldn't be so difficult to define numbers. The number n is something of a property our mind tends to associate with certain collections. It is more in the mind and not so inherent in nature.

 

It is not that I make this statements out of instantaneous thought, I have thought over this issue long and hard. One point I must add is that if maths were indeed derived from nature, mathematical laws would have the characteristics all natural laws carry. This is untrue, natural laws tend to be descriptive of real objects or even ideal objects. Maths describes abstract objects. Eg. Look at topics like group theory, ring theory, vector spaces, metric spaces etc.

 

Lastly I would say that if it were so inherent in nature then each time a new physics theory is developed it wouldn't be associated with an accompanying mathematical theory. Science uses maths as an abstract tool, which it modifies as per requirements and needs, you can't do that to things that are inherent in nature. Maths is more inherent in the mind than in nature. As the ancient Greeks would have said it is an inate type of knowledge.

 

2drops+2drops=4drops of the same size, assuming the original 4 are the same size. You're a big dumbass

Atleast read carefully what I said. I have studied enough maths and given it enough thought before I said what I said.

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mathematics has a theoretical background, engineering is the practical form of mathematics. without mathematics, engineering cannot exist. so if in mathematics infinity exist.then it does exist in reality

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Yeah without specifying this sort of stuff; meaning it is not the same sqrt function !

That is the whole point of my post' date=' that you cannot use this as the classical sqrt function. So NO the notation doesnt have a sense when you do not specify with branch you are using.

 

Mandrake[/quote']

 

But you don't have to specify it, as it is assumed.

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But you don't have to specify it, as it is assumed

 

In this context you assume merely because otherwise square root would become 2 valued and no longer be classifed as a function. The very fact that you refer to it as a function assumes that you talk of psitive square root. And note that positive square root has a range of 0 to infinity and hence its square will also lie in the same range thats one reason why the arguement holds. The other reason is that domain of square root is only positive numbers, you get weird results if you try to interpret square root of complex numbers or negative numbers that way because now the range is extended to complex numbers too.

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Math exists everywhere in nature. The rules of physics/biology/chemisty are based on mathematics. It is a property of the universe. Sure we use man made symbols to define its characteristics but this is no different from using words to describe objects, which definately exist in nature.

 

Take a single entity, add another identical entity and you have twice as many entities as you start with. It is a concept that would universal, to any conscious creature throughout the universe and thus, while described by man made absrtactions, is fundamentally the same no matter where you are in the universe.

 

Maths was around before this planet spawned human consciousness and it will be around long after we are gone. So while the nature and direction of mathematics is guided by abstract thought, the processes that govern it are fundamental to the universe and therefore NOT man-made.

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You can extend to the complex plane by demanding that the argument lies in the interval [0,pi) without any problems at all, it ceases to be continuous but that is not important.

 

Let me remind you that the argument of -1 is pi !

It is always the best to specify which branches you are using when using complex functions (with branches), since otherwise you are omitting something in your argumentation.

 

I will emphasize again that the goal of the post was to show that using notations that have a specified context in another context can be dangerous, since you can not blindly apply the same rules ! You can't calculate with the complex sqrt function as easily as you can with the classical real sqrt function.

 

Mandrake

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Let me remind you that the argument of -1 is pi

 

So what? I'm talking about the argument in the range of the function, since that is where one must take care when making a single valued choice of the square root for complex numbers.

 

And you can easily and unambiguously calculate the square root of a complex number (on the principal branch), since sqrt(re^{it}) = sqrt®e^{it/2} where sqrt of a positive number is positive, the argument of the elements in the range is [0,pi).

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Let me remind you that the argument of -1 is pi !

It is always the best to specify which branches you are using when using complex functions (with branches)' date=' since otherwise you are omitting something in your argumentation.

 

I will emphasize again that the goal of the post was to show that using notations that have a specified context in another context can be dangerous, since you can not blindly apply the same rules ! You can't calculate with the complex sqrt function as easily as you can with the classical real sqrt function.

 

Mandrake[/quote']

 

-1 is pi radians, or 180 degrees. True without specifying anything can be everything, but what I meant about x/0 is that it is undefined because it has an infinite number of solutions, or all real and imaginary numbers can be multiplied by 0 to get 0.

 

1x0=2x0; this equation is definitely defined, but since it ends up as 0=0, doesn't mean that 1=2, it is simply the operations that make the values=.

 

Same thing with x/0. For any number x, x/0 is simply infinity which is definitely undefined, yet that's what undefined would be.

 

To Lumi: Engineering is an application of math, and math is not engineering.

There are infinities in math such as infinite geometric series, but this universe is still guided under the postulate x=x.

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A more practical value for x/0 would be the addition of an extended spatial dimension, which causes much controversy.

 

It takes an infinite number of lines to make a plane so how can anything have any dimension if with respect to the lower dimension it is infinitely big and how could we for example have a proportion other than infinity if we have infinite 2D planes incorporated in our 3D universe. This would obviously lead to the conclusion that only point universes can exist, otherwise a problem would occur, but that was fundamentally solved. So take this example:

If you have a line that has a length of 2, and you want to make it a plane with are 4 without changing the length of the original line, then you multiply 2 by 2/0 to get 4/0 or the area of the plane with respect to the line. Since the length of the line however with respect to the plane is 0. The are 4 would be divided by 2/0 to get the length of the line with respect to the plane.

 

It should be clearly noted that dividing by 1/0 isnot the same as dividing by 0/1 as could be misleading, since the definition of a reciprocal is that their product is 1, but 1/0 times 0/1 is 0/0, which is undefined or infinity.

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A number is finite iff |x| < n for some integer n. A number is infinite otherwise- we will call this number H. A number is infintesimal iff |x| < n for every integer n- we will call this number d. [0 is trivially infintesimal, and the only real number that is infintesimal.]

 

d = 1/H

 

Nonstandard analysis provides a means by which we can talk about infinite numbers and divisions by numbers smaller than any possible fraction 1/n, where n is an integer, in a rigorous way. It also turns calculus into arithmatic...how cool is that!

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A more practical value for x/0 would be the addition of an extended spatial dimension' date=' which causes much controversy.

 

It takes an infinite number of lines to make a plane so how can anything have any dimension if with respect to the lower dimension it is infinitely big and how could we for example have a proportion other than infinity if we have infinite 2D planes incorporated in our 3D universe. This would obviously lead to the conclusion that only point universes can exist, otherwise a problem would occur, but that was fundamentally solved. So take this example:

If you have a line that has a length of 2, and you want to make it a plane with are 4 without changing the length of the original line, then you multiply 2 by 2/0 to get 4/0 or the area of the plane with respect to the line. Since the length of the line however with respect to the plane is 0. The are 4 would be divided by 2/0 to get the length of the line with respect to the plane.

 

It should be clearly noted that dividing by 1/0 isnot the same as dividing by 0/1 as could be misleading, since the definition of a reciprocal is that their product is 1, but 1/0 times 0/1 is 0/0, which is undefined or infinity.[/quote']

 

wtf?

and how can a "number" be the same as an "operation" (last paragraph)

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A number is finite iff |x| < n for some integer n. A number is infinite otherwise- we will call this number H. A number is infintesimal iff |x| < n for every integer n- we will call this number d. [0 is trivially infintesimal' date= and the only real number that is infintesimal.]

 

d = 1/H

 

Nonstandard analysis provides a means by which we can talk about infinite numbers and divisions by numbers smaller than any possible fraction 1/n, where n is an integer, in a rigorous way. It also turns calculus into arithmatic...how cool is that!

 

 

you're implying there exists exactly one infinte number and one infinitesimal in non-standard analysis by saying it in that way. H, in your notation, is the smallest non-finite element in the extended system.

 

also your definition of infintesimal is wrong.

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you're implying there exists exactly one infinte number and one infinitesimal in non-standard analysis by saying it in that way. H' date=' in your notation, is the smallest non-finite element in the extended system.

 

also your definition of infintesimal is wrong.[/quote']

 

 

Well, that isn't what I mant to imply.

 

 

And my definition was actually stated twice. In the top part it says n, but in the bottom part I said 1/n- stated correctly. |x| < 1/n. Thanks for pointing that out.

 

[but where do you get off just blurting out something is wrong without mentioning the correction?]

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