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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”.
A circle has an infinite number of corners.
Or zero…
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. :)
The imaginary numbers and real numbers cross at infinity (on the Riemann sphere).
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”
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.
Better term. But using o as the imaginary unit would be even worse than i.
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.
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.
A set is infinite exactly when there exists a proper subset whose cardinality is that of the set.
