Infinities

[petrushkagoogol]

When no two infinities are equal then how is it that they converge on one value when a limit is applied? eg) lim x-> infinity x^0 = 1

[LaurieAG]

When no two infinities are equal then how is it that they converge on one value when a limit is applied? eg) lim x-> infinity x^0 = 1

Anything, apart from zero, to the zeroth power equals one, even different infinities and everything else in between.

[CraigD]

eg) lim x-> infinity x^0 = 1

I don’t think this is a good example, because [math]x^0 = 1[/math] for all real numbers [math]x \not= 0[/math].

[math]\lim_{x \rightarrow \infty} x^0 = 1[/math]

implies (per the usual definition of a limit)

[math]x^0 = 1 + \epsilon[/math] for some [math]\epsilon \not= 0[/math]

But for [math]x^0 = 1 + \epsilon[/math], [math]\epsilon = 0[/math].

 

A better example is

[math]\lim_{x \rightarrow \infty} x^{-1} = 0[/math]

 

When no two infinities are equal then how is it that they converge on one value when a limit is applied?

There are many ways to answer this question.

 

One is that transfinite numbers and limits are not the same as finite numbers and ordinary functions. [math]\lim_{x \rightarrow \infty} f(x)[/math] is not the same as [math]f(\infty)[/math].

 

Mathematicians going back 300+ years have been uncomfortable with the use of transfinite and infinitesimal numbers. The notion of limits can be considered a way of avoiding their use by describing limits as a process in which finite numbers increase arbitrarily.

 

Anything, apart from zero, to the zeroth power equals one, even different infinities and everything else in between.

Which begs the question, what is the value of 0⁰?

[LaurieAG]

Which begs the question, what is the value of 0⁰?

 

That's a good question Craig as [math] 0^n = 0[/math] regardless of what n is (similar to how [math]\frac {0}{0} = 0[/math] while [math]\frac {n}{n} = 1[/math]).

[GreenBeast]

Hello everyone, 

 

That's a good question Craig as [math] 0^n = 0[/math] regardless of what n is (similar to how [math]\frac {0}{0} = 0[/math] while [math]\frac {n}{n} = 1[/math]).

 

Point A: I think that there are real number expressions, and functions which can use 0 effectively using multiplication only.  Not division.

 

Also of course as the starting point to find the exact value of a variable, such as x + y = 0.  Now we know x = -y.  If we know x or y we know the other one now. 0 is great here as the ultimate problem solver.

 

When we have imaginary numbers in advanced math which do not really even make sense, if you want to use only real numbers in your math, that can be verified by measuring lines for example, with an internationally standardized ruler or something,

 

When I saw that 0^n = 0, I was thinking the real question is:

Should all equations which include 0, be on a master list of functions and equations using variables in a textbook, so only certain equations or functions that multiply by 0 which are usually provable in math using real numbers (no imaginary numbers), now can be used in other proofs, which math scholars will accept?

 

Why has this never been done yet?

 

Also a definitive list of equations and functions which cannot use 0, for folks who want to start there.

 

Also, interestingly, 0^2 = 0.  I guess everyone can agree on that.

 

And when we square root 0, everyone is good with that, the answer is 0.

 

However start dividing both sides by zero, and that is where the problems begin.

 

 

 

When no two infinities are equal then how is it that they converge on one value when a limit is applied? eg) lim x-> infinity x^0 = 1

 

Maybe the list of the equations or functions that definitively should not use 0 would also include a third list title Parts of Functions or Expressions Without Proof.

 

Maybe x^0 should be on that list.

 

Maybe this would be in a textbook called Real Number Math that did not have (i) or imaginary numbers.

 

I would keep infinity though.

Edited April 8, 2016 by GreenBeast

[GreenBeast]

Sorry, I was not able to answer the question about two infinities not being equal converging on the same point when a limit is applied. I hope someone with more knowledge can still answer.

 

Maybe infinity equations would always be more provable, if every equation that used infinity was able to balance only a single infinity to one side by itself, probably with a single variable approaching it. And the other side could have any function or expression probably. [Not having expressions of multiple infinities in the same equation or function. For example: not using functions like f(infinity) = infinity + (x)infinity].

 

This would allow graphs showing parabolas to be more correct I believe.

 

I think dividing by infinity is less correct in the same way that dividing by irrational numbers like pi without limiting infinite place numbers is not going to be accepted by math scholars as 100% correct.

 

It seems like the only link is like for example pi radians is equal to 180 degrees. Even though pi is irrational and may be infinite, when you multiply a single radian times pi to 5 places you get very close to 180 degrees.

 

A second solution may be a new variable expressing infinity to five places.

Edited April 9, 2016 by GreenBeast

[CraigD]

[math]\frac {0}{0} = 0[/math] while [math]\frac {n}{n} = 1[/math]

[math]\frac {n}{0}[/math] is usually considered undefined (or indeterminate), which is to say the expression doesn’t tell us what its value is, so it could be anything.

 

The special case [math]\frac {0}{0}[/math] is also usually considered undefined, but one can argue that [math]\frac {0}{0}=0[/math], since [math]\frac {0}{0}[/math] can be algebraically transformed into [math]0 \cdot \frac {1}{0}[/math], and [math]0 \cdot n = 0[/math] is a basic property of all the common types of numbers.

 

You can also argue it’s infinite, with some weird and not widely accepted math, like:

[math]\frac {1}{0} = \infty[/math]

[math]n \cdot \infty > 0[/math]

so

[math]0 \cdot \infty > 0[/math]

and

[math]0 \cdot \frac {1}{0} > 0[/math]

 

The two opposing ideas here is the common-sensical “zero times anything is zero” and the weird but intuitively sensible “infinity time anything is still infinite” (which I’ve weakened to just “greater than zero”).

 

Though it’s fun and mind-expanding to play with arguments like these, you gotta keep in mind that they all are flirting with indeterminancy, so are not much removed from nonsense. :)

[CraigD]

Welcome to hypography, GreenBeast! :) Please feel free to start a topic in the introductions forum to tell us something about yourself.

 

Point A: I think that there are real number expressions, and functions which can use 0 effectively using multiplication only. Not division

:thumbs_up This is conventionally accepted. In the language of number theory and abstract algebra, you would say that the set of real numbers are not closed under the operation of division, that is

“for real numbers a and b, there exists a number a divided by b that is not a real number”

which we can render in LaTeX/Math in a standard mathematical shorthand

[math]a, b \in \mathbb{R}[/math], [math]\exists c = a \div b[/math], [math]c \not \in \mathbb{R}[/math].

 

 

Also of course as the starting point to find the exact value of a variable, such as x + y = 0. Now we know x = -y. If we know x or y we know the other one now. 0 is great here as the ultimate problem solver.

0 is a special number, as is 1. Using number theory language again, they are identity elements, the additive and multiplicative.

 

Identity elements are what make the sort of ordinary algebra people who use algebra use possible, via the basic “you can add 0 to anything or multiply anything by 1 without changing it” rule.

 

When we have imaginary numbers in advanced math which do not really even make sense, if you want to use only real numbers in your math, that can be verified by measuring lines for example, with an internationally standardized ruler or something,

I consider the math of real numbers pretty advanced, only slightly less than that of complex numbers. Though the imaginary numbers ([math]\forall a \cdot i[/math], where [math]a, \in \mathbb{R}[/math], [math]i=\sqrt{-1}[/math]) are a well-defined set like the intergers, rationals, and reals, they’re not much used, other than to build the complex ([math]a + b \cdot i[/math], [math]a, b \in \mathbb{R}[/math], [math]i=\sqrt{-1}[/math]].

 

The misconception that real numbers are somehow much less advanced than complex numbers I blame mostly on the name “real”, which suggests they’re more down-to-Earth than other kinds of numers, and the occasional computer programming convention of using “real” as a synonym for “floating point” for numeric data types, which suggests that mathematical real numbers are always easy to represent. That’s not the case, even is constraints are put on their size. An infinite number of real numbers can’t be represented by any finite representation, which, to my thinking, makes such reals even more “imaginary” as [math]\sqrt{-1}[/math], which can be easily represented.

 

The most useful numbers that I’d consider less advanced than the reals are the rationals, [math]\forall \frac{a}{b}[/math] where a and b are integers.

 

It’s helps “make sense” of complex numbers to consider that they’re equivalent to points or vectors on a 2-dimensional, real numbered plane, the real part being the coordinate on one axis, the imaginary part on the other axis. The sequence {1, i, -1, -I, 1, i, -1, -I, ...} generated by a_n=i^n-1 can be visualized as rotating a vector by 90 degrees.

 

Should all equations which include 0, be on a master list of functions and equations using variables in a textbook, so only certain equations or functions that multiply by 0 which are usually provable in math using real numbers (no imaginary numbers), now can be used in other proofs, which math scholars will accept?

Since there an infinite number of functions and equations can be written using any types of numbers, I can’t imaging making a “master list” of them.

 

Mathematicians (“math scholar” isn’t much used to describe mathematicians, I’d say because they consider themselves to do math rather than study it, in the way, say, that a “Proust scholar” studies the writing of Marcel Proust), in my experience, will accept any notation or approach that’s useful. Math is very free-form, with most mathematicians having an every-growing collection of private and public notational schemes.

[LaurieAG]

The two opposing ideas here is the common-sensical “zero times anything is zero” and the weird but intuitively sensible “infinity time anything is still infinite” (which I’ve weakened to just “greater than zero”).

 

Though it’s fun and mind-expanding to play with arguments like these, you gotta keep in mind that they all are flirting with indeterminancy, so are not much removed from nonsense. :)

 

Nice one CraigD, it's a good idea to state the obvious at the beginning to make sure everybody is in the same mathematical sand pit. ;)

 

It's also interesting if you compare mathematical structures/conceptions that contain infinity and how their use has evolved in modern science.

 

In standard calculus If you have an integral that does not converge at its limits you have an indefinite integral and regard the result as undefined while if it does converge at its limits you can call it an improper integral and can use the result. Improper integrals even play an important role in relativity. 

 

Nina Byers wrote a very informative paper in 1998 titled "E. Noether’s Discovery of the Deep Connection Between Symmetries and Conservation Laws:". Unfortunately the normal arXiv link in the PDF of the paper links to a different paper called "Dimensional Reduction" for some reason so try http://xxx.lanl.gov/abs/physics/9807044.

 

In the early days, Hilbert wrote about this problem as ‘the failure of the energy theorem ’. In a correspondence with Klein [3], he asserted that this ‘failure’ is a characteristic feature of the general theory, and that instead of ‘proper energy theorems’ one had ‘improper energy theorems’ in such a theory. This conjecture was clarified, quantified and proved correct by Emmy Noether.

 

The only problem I can see is if you have a sub function of a higher level function that cycles between + infinity and - infinity at its limits (symmetric), and you regard this as an improper integral even though the sub function may be indefinite in isolation (i.e. if only less than one cycle completes, never greater or equal to one complete cycle).

 

To me this seems to be the place where the sand pits change.