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Does this mean the concept of infinity requires an infinite number of infinities?
Informally, yes. Formally, no. There is no cardinal sufficient such that every cardinal can fit in a set with that cardinality. Isn’t that fascinating? There’s too many infinities for us to mathematically express how many infinities there are.
I believe so. I wonder what is the ordinal of the set of ordinals.
Imaginary numbers have a specific value, just like all the normal numbers we use on a daily basis. Infinity is not a specific value. Instead, it’s a qualitative property like “flat”, “periodic”, or “symmetric”.
I think of it this way, infinity is an action. It’s not a “thing.” This is why it’s not countable. It doesn’t stop to be counted.
A circle has an infinite number of corners.
Or zero…
therefore ∞ = 0 😀👍
Probably More accurate to say it has an infinite number of edges
A circle has one edge/side, that is grade-school geometry. There is no reason to engender confusion by trying to make it into a polygon or introducing infinity. Your model of shapes does not seem to account for curved edges.
Consider a stereotypical pizza slice. One might plainly say that it is a “like a triangle but one edge is curved” without falling into a philosophical abyss. :)
It’s quite useful, though, to understand a curve or arc as having infinite edges in order to calculate its area. The area of a triangle is easy to calculate. Splitting the arc into two triangles by adding a point in the middle of the arc makes it easy to calculate the area… And so on, splitting the arc into an infinite number of triangles with an infinite number of points along the arc makes the area calculable to an arbitrary precision.
Or you could just enjoy your π
The imaginary numbers and real numbers cross at infinity (on the Riemann sphere).
Why? What does it mean for something to be real?
I believe pure mathematics isn’t concerned with its correspondence with reality.
I recall hearing a quote from the guy that coined the term “imaginary number”, and how he regretted that term because it conveys a conceptualization of fiction. IIRC, he would rather that they would have been called “orthogonal numbers” (in a different plane) and said that they were far more real that people tend to hold in their mind. I think he said “they are as real as negative numbers” along the same lines of one not being able to hold a negative quantity of apples, for example.
The stray shower thought (beyond simply juxtaposing the discordant terms of ‘imaginary’ and ‘real’) was that infinity by contrast is a much weaker and fantastic concept. It destroys meaningful operations it comes into contact with, and requires invisible and growing workarounds to maintain (e.g. “countably” infinite vs “uncountably” infinite) which smells of fantasy, philosophically speaking.
It destroys meaningful operations it comes into contact with, and requires invisible and growing workarounds to maintain (e.g. “countably” infinite vs “uncountably” infinite) which smells of fantasy, philosophically speaking.
This isn’t always true. The convergent series comes to mind, where an infinite summation can be resolved to a finite number.
Furthermore, it is meant to highlight the fact that people gleefully embrace the concept of infinity, but try their hardest to avoid and depreciate the concept of imaginary numbers. It would appear to me that the bias ought to be reversed.
Better term. But using o as the imaginary unit would be even worse than i.
I mean, complex numbers are important for quantum mechanics. In that sense, they are closer to reality as they are used to describe the underlying blocks of reality to our current best understanding
You don’t even have to go into quantum mechanics. I vaguely recall using a real/imaginary plane with a rotating vector to do something about electricity in first year engineering?
Don’t worry I’m not actually an electrical engineer.
But my point is that there are applications for imaginary numbers with very practical engineering applications. Foundational, even.
Electrical math is full of complex numbers.
something about electricity
It’s a usefull technique to model the symmetry between magnetic and electrical power.
I find this all to be very irrational. I need to have some pi and think about it.
Come on, be rational…
Let’s be real, it’s a complex topic.
Amazing. Your shower thought is incorrect on both counts. Perhaps you meant to say “conceivable?”
I’m guessing they maybe mean that they have a more trivial practical resolution to real numbers, in that i^2=-1?
Kinda like “yeah they’re imaginary but I understand that if I hit them with a certain stick they become real”
Infinity is a placeholder for uncountable large numbers and is used for stuff like either functions to describe the the craps behavior towards the “end” wich there is none.
A set is infinite exactly when there exists a proper subset whose cardinality is that of the set.
