Infinity

Definition

In mathematics, infinity refers to a concept of something that is limitless or boundless. It is often represented by the symbol ∞. Infinity can be used to describe a number of different things, such as an unbounded set or an unbounded limit. It is also used in calculus to describe the concept of a limit approaching infinity.

Who

Calculus doesn’t really use infinity, it just an example of where we’ve developed a meaningful way of talking about the limit of a process. Equivalently I suppose, it lets us use an infinitely long process in some situations.

Infinity as a notion crops up all over maths, but we can use it specifically to solve problems in many areas like:

Real Analysis: This is essentially the more general version of limits you’ve come across in calculus
Geometry: Just as working over the complex numbers simplify things algebraically by adding in roots of polynomials, we can often simplify geometric problems by adding in points at infinity in a rigourous fashion. Doing both at once, for example, gives us the really nice result that an n degree curve and an m degree curve intersect at exactly nm points (counting multiplicities).
Functional Analysis: When doing linear algebra about 90% of your proofs probably start along the lines of ‘Let V be a vector space of dimension n<∞’ however many naturally occurring vector spaces (like say the continuous functions on [0,1][0,1]) are not finite dimensional. We have to be very careful with our treatment of these spaces, but you can get some very useful and powerful results out of them.

There are of course many other places in maths that use some concept of infinity, these are just the ones that spring to mind immediately.

In general infinity is an idea that is extremely powerful, though like many powerful ideas we must be careful in how it is applied. ³

There’s abstract algebra, topology,
non-Euclidean geometry—
don’t forget all cohomology,
de Rham, Galois, and all the rest.

I could teach you number theory.
You’ll see that models aren’t dreary,
groups and rings won’t make you weary,
and p-adics are the best.

Categories, sets and more,
sheaves and topos, furthermore,
homotopies all lie in store,
for those that study far enough.

Do you wish to learn perspective?
When projections are surjective?
To make lambda calculus selective?
This is all much more than fluff.

There is power in real analysis,
complex and functional analysis—
proving what you learned in calculus,
but also going far beyond.

There is much I didn’t mention.
How to find a field extension,
and calculate relative dimension,
I will not say; I won’t drag on.

Sheydvasser, Senia. “Do infinities ever come up in calculus or other math classes? If so, what are they called?” 2023. Quora. https://qr.ae/py7GOh.

What

Infinity is without doubt one of the hardest mathematical concepts to understand. Many ideas that we find intuitive when working with normal numbers don’t work anymore, and instead there are countless apparent paradoxes. Is there a largest number? Is there anything bigger than infinity? What is infinity plus one? What is infinity plus infinity?

The symbol for infinity is ∞, a horizontal 8. It was invented by John Wallis (1616–1703) who could have derived it from the Roman numeral M for 1000. But one thing we know for certain: infinity is a lot bigger than 1000.

The German mathematician David Hilbert (1862–1943) tried to explain some of the properties of infinity using a hotel with infinitely many rooms. Hilbert is also well-known for presenting a list of 23 mathematical problems which he considered the most important ones at the start of the 20th century. Some of these problems, like the Riemann Hypothesis, are still unsolved. ²

Absolutely, infinity has countless (:P) practical applications.

Here’s one way to think about it: do negative numbers have any practical applications?

I mean you can’t really have a negative amount of anything, can you? You can’t have negative five apples.

If your bank balance is negative, that’s just another way of saying you owe the bank a (positive amount of) money, rather than the other way around. When we say a particle’s charge is negative, we mean only, that it has more of the charge that comes from electrons than the opposite kind that comes from protons. And so on.

Nonetheless, the abstraction of negative numbers – which is to say of the Integers, and of additive inverses and number rings and so on generally – is hugely useful. It pervades our understanding of numbers, both as applied to the real world, and in many of the most theoretical branches of pure mathematics.

The same is true of other mathematical abstractions whose ontology, and thus utility, you might similarly question, from complex numbers to (the very many different forms of) infinity. For instance, infinities underlie all of traditional real analysis, the foundation of modern calculus and related fields. Whether or not the real numbers are in fact real, in the sense of somehow existing within the universe, they have proven to be at least an incredibly useful approximation for modelling all sorts of things that “may as well” vary continuously at the scales we measure them. There are ongoing efforts to replicate the results of the field using weaker postulates about infinities, in Constructivism and even Finitism, but they are far from being “complete”, and probably never will be.

Likewise infinity is at the core of measure theory, on which our current construction of probability is based. Hilbert Spaces, used in the formulation of quantum mechanics, are infinite not just in size, but in dimension. And there are deep links between even the exotic transfinite cardinals of Cantor, and the areas of logic that deal with the most foundational issues of mathematics (and indeed, with those finite but extremely fast growing functions others have mentioned in the context of numbers that are “infinite for any practical purpose.”)

Rastrick, Jordan. “Does The Concept Of Infinity Have Any Practical Applications?”. 2013. Mathematics Stack Exchange. https://math.stackexchange.com/a/891827.

Why

See Theoretical Knowledge Vs Practical Application.

How

Many of the References and Additional Reading websites and Videos will assist you with understating infinity.

As some professors say: “It is intuitively obvious to even the most casual observer.“

What is infinity minus infinity?

There is a popular illustration called the Hilbert’s paradox of the grand hotel.

Suppose Hilbert’s hotel has an infinite number of rooms and infinite number of guests are booked into the hotel. By common sense, it seems like the hotel is fully booked right? Wrong. Infinite sets just defy logic. Suppose there was another guest who wanted to book into the hotel, all the hotel staff have to do is just shift guest in room number 1 to the next, the guest in room number two to the third and so on… So by this logic ∞+1=∞

Similarly, ∞−1=∞. Just remove the guest from room number 1 and shift the remaining guests to the predecessor of their room numbers. You still have an infinite number of guests.

Let’s apply the logic to your question. Seemingly ∞−∞=0.

But suppose we remove the guests which are present in the rooms having an odd number (1,3,5…..) we still have infinite number of guests. So we get ∞−∞=∞.

Let’s remove all the guests except the ones present in the first 50 rooms. So ∞−∞=50. You see where I’m going with this? Simply that ∞−∞ is indeterminable.

Hegde, Akash. “What Is Infinity Minus Infinity?” 2023. Quora. https://qr.ae/pr0QzK.

References

¹ “What is infinity in maths?” 2023. Quora. https://www.quora.com/What-is-infinity-in-maths.

² “Infinity | World of Mathematics – Mathigon”. 2023. Mathigon. https://mathigon.org/world/Infinity.

³ Stigant, Liam. “Other than calculus, is there any other mathematical method that uses infinity to find a solution?” 2023. Quora. https://qr.ae/py7G9J.

Additional Reading

Ali. “The Idea of Infinity: When Mathematics Conflicts with Physics”. 2023. Medium. https://ali.medium.com/the-idea-of-infinity-when-mathematics-conflicts-with-physics-dfd69f4da915.

Until the early 20th century, mathematics was divided into elementary and higher mathematics. Elementary mathematics was mathematics with addition, subtraction, multiplication, and division, that is, mathematics with four operations. On the other hand, higher mathematics was mathematics using the limit operation introduced by Augustin Cauchy in addition to the four operations.

Berry, Brett. “Hilbert’s Infinite Hotel Paradox”. 2018. Medium. https://medium.com/i-math/hilberts-infinite-hotel-paradox-ca388533f05.

Wolchover, Natalie. “How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer. | Quanta Magazine”. 2021. Quanta Magazine. https://www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/.

⭐ “Infinity”. 2023. math.dartmouth.edu. https://math.dartmouth.edu/~matc/Readers/HowManyAngels/InfinityMind/IM.html.

“Infinity in Maths (Definition, Meaning, Symbol & Properties)”. 2023. BYJUS. https://byjus.com/maths/infinity/.

In Mathematics, “infinity” is the concept describing something which is larger than the natural number. It generally refers to something without any limit. This concept is predominantly used in the field of Physics and Maths which is relevant in the number of fields. For the extended real number system, the term “infinity” can be also be used.

“Infinity | World Of Mathematics – Mathigon”. 2023. Mathigon. https://mathigon.org/world/Infinity.

“Infinity – Wikipedia”. 2023. en.wikipedia.org. https://en.wikipedia.org/wiki/Infinity.

“What Is Infinity?” 2023. mathsisfun.com. https://www.mathsisfun.com/numbers/infinity.html.

“What Is The Infinite Hotel Paradox?” 2022. Science ABC. https://www.scienceabc.com/pure-sciences/what-is-the-infinite-hotel-paradox-definition-examples.html.

Videos

“PBS Infinite Series”. 2023. youtube.com. https://www.youtube.com/@pbsinfiniteseries/videos.

Mathematician Tai-Danae Bradley and physicist Gabe Perez-Giz offer ambitious content for viewers that are eager to attain a greater understanding of the world around them. Math is pervasive – a robust yet precise language – and with each episode you’ll begin to see the math that underpins everything in this puzzling, yet fascinating, universe.

How An Infinite Hotel Ran Out Of Room

If there’s a hotel with infinite rooms, could it ever be completely full? Could you run out of space to put everyone? The surprising answer is yes — this is important to know if you’re the manager of the Hilbert Hotel.

The Infinite Hotel Paradox – Jeff Dekofsky

The Infinite Hotel, a thought experiment created by German mathematician David Hilbert, is a hotel with an infinite number of rooms. Easy to comprehend, right? Wrong. What if it’s completely booked but one person wants to check in? What about 40? Or an infinitely full bus of people? Jeff Dekofsky solves these heady lodging issues using Hilbert’s paradox.

The Riddle That Seems Impossible Even If You Know The Answer

The 100 Prisoners Riddle feels completely impossible even once you know the answer.

The featured image on this page is from Mathigon website.

⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.