Matrix representations in general, if that counts?
Complex numbers, polynomials, the derivative operator, spinors etc. they’re all matrices. Numbers are just shorthand labels for certain classes of matrices, fight me.
Would you elaborate on the last statement?
I’m just being silly, but I mean that if everything can be represented as a matrix then there’s a point of view where things like complex numbers are just “names” of specific matrices and the rules that apply to those “names” are just derived from the relevant matrix operations.
Essentially I’m saying that the normal form is an abstract short hand notation of the matrix representation. The matrices are of course significantly harder and more confusing to work with, but in some cases the richness of that structure is very beautiful and insightful.
(I’m particularly in love with the fact one can derive spinors and their transforms purely from the spacetime/Lorentz transforms. It’s a really satisfying exercise and it’s some beautiful algebra/group theory.)
I have a strong relationship to what you get when you divide by zero.
Good ol’ NaN
I like writing swear words into the mantissa of NaN numbers
You only get NaN for division by zero if you divide 0 by 0 in IEEE floating point. For X/0 with X ≠ 0, you get sign(X)•Inf.
And for real numbers, X/0 has to be left undefined (for all real X) or else the remaining field axioms would allow you to derive yourself into contradictions. (And this extends to complex numbers too.)
Negative zero, comes up in comp sci.
i=sqrt(-1) is nice, but im hoping someone finds a use for the number x where |x| = -1 or some nonsense like that because it looks fun to mess with
Those exist in the split-complex number system which adds the number j, where j^2=1 (but j does not equal 1 or -1). The modulus of j is -1.
j = √-1
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Set phasors to “confuse”
ε, the base of the dual numbers.
It’s a nonzero hypercomplex number that squares to zero, enabling automatic differentiation.
Came here to say this, but since it’s already here, I’ll throw in a bonus mind-melting fact: ε itself has no square root in the dual numbers.
As a not good at math person, advanced math sounds hella fake sometimes. Like almost a parody of math. I accept that it’s real but part of me will always think it’s some inside joke or a secret society cult for the lamest god.
A lot of “advanced” maths comes from asking “What if this was a thing, how would that work? Would it even work?”, so you could say there’s a deliberate sense of “fake” about it.
Dual numbers come from “What if there was another number that isn’t 0 which when multiplied by itself you get 0?”
Basic dual numbers (and complex numbers) are no harder than basic algebra, shallow end of the pool kind of stuff, but then not everyone is comfortable getting in the water in the first place, and that’s OK too.
Thank you for the info!
Puts on floaties and a brave face, then advances to the shallow end
What is the value in learning about “What if there was another number that isn’t 0 which when multiplied by itself you get 0?”
Are there any practical applications IRL for dual numbers?
Edit: Screw Screw theory. Wikipedia says dual numbers have applications in mechanics and to see Screw Theory. I tried reading about it and my eyes glazed over so quickly. Math so isn’t for me lol.
I can’t answer for dual numbers, but I can answer for imaginary numbers in circuit design.
Imaginary numbers are those that include an imaginary component, that squares into a negative number. Traditionally, i^2 = -1, but electrical engineers like to use j instead (I tends to be a variable used to describe electrical current).
Complex numbers, that include a real component and an imaginary component, can be thought of as having an “angle,” based on how much of it is imaginary and how much of it is real, mapped onto a 2-dimensional representation of that number’s real and imaginary components. 5 + 5j is as real as it is imaginary, so it’s like having a 45° angle. The real number 5 is completely real, so it has a 0° angle.
Meanwhile, in alternating current (AC) circuits, like what you get from your wall outlet, the voltage source is a wave that alternates between a maximum peak of positive voltage and a bottom trough of negative voltage, in a nice clean sinusoidal shape over time. If you hook up a normal resistor, the nice clean sinusoidal voltage also becomes a nice clean sinusoidal current with the exact same timing of when the max voltage matches up with the max current.
But there’s also capacitors, which accumulate charge so that the flow of current on the other side depends on its own state of charge. And there are inductors, that affect current based on the amount of energy stored magnetically. These react to the existing current and voltage in the system and manipulate the time relationship between what moment in time a peak current will happen and when the peak voltage was.
And through some interesting overlap in how adding and subtracting and delaying sinusoidal waves works, the circuit characteristics line up perfectly with that complex angle I was talking about, with the imaginary numbers. So any circuit, or any part of a circuit, can be represented with an “impedance” that has both an imaginary and real component, with a corresponding phase angle. And that complex number can be used to calculate information about the time delay in the wave of current versus the wave of voltage.
So using complex phase angles makes certain AC calculations much, much easier, to represent the output of real current from real voltage, where the imaginary numbers are an important part of the calculation but not in the actual real world observation itself.
So even though we start with real numbers and end with real numbers, having imaginary numbers in the toolbox make the middle part feasible.
Since there’s a reasonably strong link to calculus, and mechanics as you’ve already found, it could theoretically help in physics simulations either in a computer or on paper.
As for practical application, well, emulating physics is pretty important in a lot of computer games, or getting robots (assembly line arms, androids, automated vacuum cleaners) around the place and to do what they need without accidentally catapulting themselves into next Tuesday.
How that’s actually programmed might not involve dual numbers at all, but they’re one way of looking at how those calculations might be done.
Complex numbers 🤝 Split-complex numbers 🤝 Dual numbers
All super rad.
I’m a big fan of 10-adics, especially this one.
j is cool too, as is (1+j)/2.
So complex or quaternion I imagine? ‘i’ it is!
√-4 = 2
It’s all fun and games until someone loses an i.
∞
Which one?
i?
Clavdivs?
j or ĵ just a base vector
Meow
Elevendy
I always like seeing π.
Pi is a real number, though. It’s irrational, but real.
Pi written out is a real number, yes, I’m referring to the symbol representing Pi. Does that not count?
Nnnnno.
Chaitin’s constants.
Not even a number.
They are still real numbers. Specifically uncomputable, normal numbers. Which means their rational expansion contain every natural number.
Oops, I misunderstood what an uncomputable is…
In that case, I would say Infinity-Infinity. This time, it’s truly not a number.
