Questions about Infinity

[mathscurious1]

Hi there, I was thinking about zero and infinity.

I tried to find the multiplicative inverse of infinity and this is what I got.

Infinity x Infinity = Infinity x 1/Infinity = 0?

For the above equation, can you actually divide 1 by infinity? If not, does this mean it's impossible to find the multiplicative inverse of infinity?


Also, not sure if this is of only value but I was playing with zero and set theory and found another way that zero can create contradictions;

Set of positive single-digit even numbers = Set of PSDEN

Set of PSDEN = {0, 2, 4, 6, 8}

Set of PSDEN - Set of PSDEN = 0 = Set of PSDEN Subtraction

Set of PSDEN = {Set of PSDEN Subtraction, 2, 4, 6, 8}

Set of PSDEN less than two or Set of PSDEN LTT = {0}

Set of PSDEN x Set of PSDEN LTT = Set of PSDEN LTT (i.e. {0, 2, 4, 6, 8} x 0 = 0)

Set of PSDEN LLT divided by Set of PSDEN = Set of PSDEN LTT (Eq 1)

i.e. 0 divided by {0, 2, 4, 6, 8} = 0

Set of PSDEN LLT divided by (Set of PSDEN - Set of PSDEN) = Set of PSDEN LTT (Eq 2)

or 0 divided by ({0, 2, 4, 6, 8} - {0, 2, 4, 6, 8}) = 0


Meaning Eq1 = Eq2, meaning Set of PSDEN = Set of PSDEN - Set of PSDEN, meaning {0, 2, 4, 6, 8} = 0

Did I make some errors here or is this another way of showing the contradictions of multiplying and dividing by zero?

 

[blamocur]

I did not understand most of the post, but: I am not aware of any meaningful algebras involving infinity, or allowing divisions by zero. One can use infinity for specifying limits in calculus, or for building projective spaces, but even there infinity is a convenience symbol rather than an actual "thing".

 

[Steven G]

Infinity x Infinity = Infinity x 1/Infinity. Just like 3*4 = 3*1/4????

 

[topsquark]

Hi there, I was thinking about zero and infinity.

I tried to find the multiplicative inverse of infinity and this is what I got.

Infinity x Infinity = Infinity x 1/Infinity = 0?

For the above equation, can you actually divide 1 by infinity? If not, does this mean it's impossible to find the multiplicative inverse of infinity?

Unfortunately, no. You are treating [imath]\infty[/imath] like it is a number and it is not. There is a system, called the hyper-real numbers, that you can do this sort of work with but there are many details.

-Dan

 

[pka]

Unfortunately, no. You are treating [imath]\infty[/imath] like it is a number and it is not. There is a system, called the hyper-real numbers, that you can do this sort of work with but there are many details.

In Elementary Calculus: An Infinitesimal Approach​

In Elementary Calculus: An Infinitesimal Approach by Jerome Keisler in chapter 3, there is good proof of this. The chapters and whole book is a free down-load at http://www.math.wisc.edu/~keisler/.

 

[JeffM]

I did not understand most of the post, but: I am not aware of any meaningful algebras involving infinity, or allowing divisions by zero. One can use infinity for specifying limits in calculus, or for building projective spaces, but even there infinity is a convenience symbol rather than an actual "thing".

Absolutely, Georg Cantor was a loony, locked up for the safety of the public. Complete nut job, literally.

How do we define a "thing"? Can we demonstrate the physical reality of an irrational number? How about lines without breadth?

Kronecker was correct. Only the non-negative integers correspond to what is physically observable. Everything else is the product of the human mind.

People (like Wolfram Mathworld) who talk about projectively extended real numbers or affinely extended real numbers are simply propagating "meaningless" misinformation like flat earthers. https://mathworld.wolfram.com/ProjectivelyExtendedRealNumbers.html

Alternatively, we could say that infinity is not a real number but that it can logically be defined as a number with an arithmetic and algebra different from that of the real numbers. Here is one such arithmetic: https://en.wikipedia.org/wiki/Projectively_extended_real_line

We define [imath]\hat \mathbb R[/imath] as the real numbers plus an additional number symbolized as [imath]\infty[/imath]. In addition to the standard arithmetic operations on the subset

of

, the following operations are defined for

, with exceptions as indicated:

In this system, the above are axioms, not something to be proved. Notice that arithmetic operations where both arguments are infinite are not defined. Here is an example of why we do not define them. There are an infinite number of integers greater than five. There are an infinite number of integers greater than two. If we look at the set of integers that are greater than two but not greater than five, we have 3. 4, and 5, or three such integers. That suggests that [imath]\infty - \infty = 3[/imath]. But the number of integers greater than six is also infinite. Thus, the number of integers that are greater than two but not greater than six is [imath]\infty - \infty = 3[/imath], which is clearly the wrong answer, because the set contains four such integers, namely 3, 4, 5, and 6.

In short, there are two valid options.

One is to say that mathematics is limited to discussing things, meaning what is physically observable. The other is to say that mathematics deals with ideas that are not limited in any way except that they must have consistent logical results. Under the latter approach, one can call infinity a number, but it is not a so called "real" number because it cannot consistently follow the rules that apply to real numbers.

Going back to the original question, infinity has no multiplicative inverse either because infinity does not exist or else because there is no determinate quotient, with infinity as the numerator or the denominator, equal to 1, the multiplicative identity.

 

[lookagain]

Absolutely, Georg Cantor was a loony, locked up for the safety of the public. Complete nut job, literally.

You brought him up, as if you were responding to someone asking about him in a previous post, but he was not explicitly mentioned.
You are disparaging him as a "complete nut job." George Cantor received his doctorate degree at the age of 22-years-old in 1867.
In about 1884 he had mental illness begin which he suffered from severe depression as one of his main problems. He did continue to
contribute to mathematics, and he was in sanitoriums at certain times.

Georg Cantor | Biography, Contributions, Books, & Facts | Britannica

Georg Cantor, German mathematician who founded set theory and introduced the mathematically meaningful concept of transfinite numbers, indefinitely large but distinct from one another. His work was fundamental to the development of function theory, analysis, and topology. Learn more about...

www.britannica.com

Georg Cantor - Wikipedia

en.wikipedia.org

 

[JeffM]

You brought him up, as if you were responding to someone asking about him in a previous post, but he was not explicitly mentioned.
You are disparaging him as a "complete nut job." George Cantor received his doctorate degree at the age of 22-years-old in 1867.
In about 1884 he had mental illness begin which he suffered from severe depression as one of his main problems. He did continue to
contribute to mathematics, and he was in sanitoriums at certain times.

Georg Cantor | Biography, Contributions, Books, & Facts | Britannica

Georg Cantor, German mathematician who founded set theory and introduced the mathematically meaningful concept of transfinite numbers, indefinitely large but distinct from one another. His work was fundamental to the development of function theory, analysis, and topology. Learn more about...

www.britannica.com

Georg Cantor - Wikipedia

en.wikipedia.org

I was responding (sarcastically) to the idea that infinity cannot be considered a number. I am well aware of Hilbert's comment that Cantor created a paradise for mathematicians.

I do believe with Kronecker that numbers, other than non-negative integers, are nothing more than ideas that spring from the human imagination. It does not follow that those ideas are meaningless or practically useless.

Infinity is not a real number, but the real numbers do not constitute the end-all and be-all of numerical ideas.

 

[topsquark]

Infinity is not a real number, but the real numbers do not constitute the end-all and be-all of numerical ideas.

I'm personally in love with Grassmann complex numbers.

-Dan

 

[JeffM]

I'm personally in love with Grassmann complex numbers.

-Dan

OK. Now I have something to look up. I presume that they are standard stuff for some investment bankers, but they don’t teach them to European history majors.

EDIT: Uh oh. My son keeps telling me that I am just poking around until I learn Clifford-algebra, and now it appears that there is a Grassmann-algebra too. Maybe I should stick to what I know and read a little more Mommsen. I doubt the Romans had much mathematics that I cannot master.

 

[topsquark]

OK. Now I have something to look up. I presume that they are standard stuff for some investment bankers, but they don’t teach them to European history majors.

EDIT: Uh oh. My son keeps telling me that I am just poking around until I learn Clifford-algebra, and now it appears that there is a Grassmann-algebra too. Maybe I should stick to what I know and read a little more Mommsen. I doubt the Romans had much mathematics that I cannot master.

Grassmann complex numbers come from the Grassmann Algebra. They are anti-commuting complex numbers, something that you can conceive of but I do not know of any way to represent them as a matrix or anything, which is rather strange.

-Dan