Showerthoughts
A "Showerthought" is a simple term used to describe the thoughts that pop into your head while you're doing everyday things like taking a shower, driving, or just daydreaming. A showerthought should offer a unique perspective on an ordinary part of life.
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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.
What do you mean by action? Like, how running or writing are actions people can take? So maybe infinity would be an action a group of numbers can take?
Well yes and no. With running or writing those actions will stop at some point because we aren’t capable of doing them forever. Infinity doesn’t stop.
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.
So Descartes coined the term specifically as a dig because he didn't see any geometric possibilty to the concept. The concept seems to have roots going back to ancient Egypt, but the modern inquiry goes to the Renaissance. I think Gauss wanted to call them laterals.
Thank you for adding some facts to my vague conflated memories.
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
Complex numbers are just a way of representing an additional degree of freedom in an equation. You have to represent complex numbers not on a number line but on the complex plane, so each complex number is associated with two numbers. That means if you create a function that requires two inputs and two outputs, you could "compress" that function into a single input and output by using complex numbers.
Complex numbers are used all throughout classical mechanics. Waves are two-dimensional objects because they both have an amplitude and a wavelength. Classical wave dynamics thus very often use complex numbers because you can capture the properties of waves more concisely. An example of this is the Fourier transform. If you look up the function, it looks very scary, it has an integral and Euler's number raised to the negative power of the imaginary number multiplied by pi. However, if you've worked with complex numbers a lot, you'd immediately recognize that raising Euler's number to pi times the imaginary number is just how you represent rotations on the complex plane.
Despite how scary the Fourier transform looks, literally all it is actually doing is wrapping a wave around a circle. 3Blue1Brown has a good video on his channel of how to visualize the Fourier transform. The Fourier transform, again, isn't inherently anything quantum mechanical, we use it all the time in classical mechanics, for example, if you ever used an old dial-up model and wondered why it made those weird noises, it was encoding data as sound wave by representing them as different harmonic waves that it would then add together, producing that sound. The Fourier transform could then be used by the modem at the other end to break the sound back apart into those harmonic waves and then decode it back into data.
In quantum mechanics, properties of systems always have an additional kind of "orientation" to them. When particles interact, if their orientations are aligned, the outcome of the interaction is deterministic. If they are misaligned, then it introduces randomness. For example, an electron's spin state can either be up or down. However, its spin state also has a particular orientation to it, so you can only measure it "correctly" by having the orientation of the measuring device aligned with the electron. If they are misaligned, you introduce randomness. These orientations often are associated with physical rotations, for example, with the electron spins state, you measure it with something known as a Stern-Gerlach apparatus, and to measure the electron on a different orientation you have to physically rotate the whole apparatus.
Because the probability of measuring certain things directly relates to the relative orientation between your measuring device and the particle, it would be nice if we had a way to represent both the relative orientation and the probability at the same time. And, of course, you guessed it, we do. It turns out you can achieve this simply by representing your probability amplitudes (the % chance of something occurring) as complex numbers. This means in quantum mechanics, for example, an event can have a -70.7i% chance of occurring.
While that sounds weird at first, you quickly realize that the only reason we represent it this way is because it directly connects the relative orientation between the systems interacting and the probabilities of certain outcomes. You see, you can convert quantum probabilities to classical just by computing the distance from 0% on the complex plane and squaring it, which in the case of -70.7i% would give you 50%, which tells you this just means it is basically a fair coin flip. However, you can also compute from this number the relative orientation of the two measuring devices, which in this case you would find it to be rotated 90 degrees. Hence, because both values can be computed from the same number, if you rotate the measuring device it must necessarily alter the probabilities of different outcomes.
You technically don't need to ever use complex numbers. You could, for example, take the Schrodinger equation and just break it up into two separate equations for the real and imaginary part, and have them both act on real numbers. Indeed, if you actually build a quantum computer simulator in a classical computer, most programming languages don't include complex numbers, so all your algorithms have to break the complex numbers into two real numbers. It's just when you are writing down these equations, they can get very messy this way. Complex numbers are just far more concise to represent additional degrees of freedom without needing additional equations/functions.
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.
electrical impedance is often represented with a complex number
The imaginary numbers and real numbers cross at infinity (on the Riemann sphere).
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. :)
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 number which famously has an infinite number of digits? I thought we were arguing against the real-ness of infinity.
Also note: the method I was describing is one of the ways in which pi can be calculated.
Instead of "has infinite digits", I prefer to say that it CANNOT be expressed as a base10/decimal number. If you choose a different base (base-pi for example), then it very much has finite digits... :)
It can't be expressed in any integer-based notation without an infinite number of digits. Only when expressed in some bases which are themselves, irrational. It's infinity either way.
therefore ∞ = 0 😀👍
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.
imaginary numbers have a really bad name, that totally unnecessarily imply they dont somehow exist.
but tell me, how many numbers are there?
Obviously there is no number to quantify how many numbers there are, only a weaker concept of it being unbounded and limitless. null or undefined if you prefer.
NaN ?
Horses are more real than fruit.
A set is infinite exactly when there exists a proper subset whose cardinality is that of the set.