1/2 <-- not a number. Two numbers and an operator. But also a number.
this post was submitted on 01 Jul 2025
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The thing is that it's legit a fraction and d/dx actually explains what's going on under the hood. People interact with it as an operator because it's mostly looking up common derivatives and using the properties.
Take for example ∫f(x) dx to mean "the sum (∫) of supersmall sections of x (dx) multiplied by the value of x at that point ( f(x) ). This is why there's dx at the end of all integrals.
The same way you can say that the slope at x is tiny f(x) divided by tiny x or d*f(x) / dx or more traditionally (d/dx) * f(x).
We teach kids the derive operator being ' or ·. Then we switch to that writing which makes sense when you can use it properly enough it behaves like a fraction
clearly, d/dx simplifies to 1/x
If not fraction, why fraction shaped?
Having studied physics myself I'm sure physicists know what a derivative looks like.
This very nice Romanian lady that taught me complex plane calculus made sure to emphasize that e^j*theta was just a notation.
Then proceeded to just use it as if it was actually eulers number to the j arg. And I still don’t understand why and under what cases I can’t just assume it’s the actual thing.
It is just a definition, but it's the only definition of the complex exponential function which is well behaved and is equal to the real variable function on the real line.
Also, every identity about analytical functions on the real line also holds for the respective complex function (excluding things that require ordering). They should have probably explained it.
Let's face it: Calculus notation is a mess. We have three different ways to notate a derivative, and they all suck.
I've seen e^{d/dx}
Why does using it as a fraction work just fine then?
I still don't know how I made it through those math curses at uni.
Mathematicians will in one breath tell you they aren't fractions, then in the next tell you dz/dx = dz/dy * dy/dx
Have you seen a mathematician claim that? Because there's entire algebra they created just so it becomes a fraction.
This is until you do multivariate functions. Then you get for f(x(t), y(t)) this: df/dt = df/dx * dx/dt + df/dy * dy/dt
Brah, chain rule & function composition.
(d/dx)(x) = 1 = dx/dx
It was a fraction in Leibniz’s original notation.
And it denotes an operation that gives you that fraction in operational algebra...
Instead of making it clear that d is an operator, not a value, and thus the entire thing becomes an operator, physicists keep claiming that there's no fraction involved. I guess they like confusing people.
I found math in physics to have this really fun duality of "these are rigorous rules that must be followed" and "if we make a set of edge case assumptions, we can fit the square peg in the round hole"
Also I will always treat the derivative operator as a fraction
I always chafed at that.
"Here are these rigid rules you must use and follow."
"How did we get these rules?"
"By ignoring others."
2+2 = 5
…for sufficiently large values of 2
Found the engineer
i was in a math class once where a physics major treated a particular variable as one because at csmic scale the value of the variable basically doesn't matter. the math professor both was and wasn't amused
Engineer. 2+2=5+/-1
I mean as an engineer, this should actually be 2+2=4 +/-1.
Computer science: 2+2=4 (for integers at least; try this with floating point numbers at your own peril, you absolute fool)
0.1 + 0.2 = 0.30000000000000004
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Derivatives started making more sense to me after I started learning their practical applications in physics class. d/dx was too abstract when learning it in precalc, but once physics introduced d/dt (change with respect to time t), it made derivative formulas feel more intuitive, like "velocity is the change in position with respect to time, which the derivative of position" and "acceleration is the change in velocity with respect to time, which is the derivative of velocity"
Possibly you just had to hear it more than once.
I learned it the other way around since my physics teacher was speedrunning the math sections to get to the fun physics stuff and I really got it after hearing it the second time in math class.
But yeah: it often helps to have practical examples and it doesn't get any more applicable to real life than d/dt.
I always needed practical examples, which is why it was helpful to learn physics alongside calculus my senior year in high school. Knowing where the physics equations came from was easier than just blindly memorizing the formulas.
The specific example of things clicking for me was understanding where the "1/2" came from in distance = 1/2 (acceleration)(time)^2 (the simpler case of initial velocity being 0).
And then later on, complex numbers didn't make any sense to me until phase angles in AC circuits showed me a practical application, and vector calculus didn't make sense to me until I had to actually work out practical applications of Maxwell's equations.
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Division is an operator
Except you can kinda treat it as a fraction when dealing with differential equations
Oh god this comment just gave me ptsd
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What is Phil Swift going to do with that chicken?
The will repair it with flex seal of course
To demonstrate the power of flex seal, I SAWED THIS CHICKEN IN HALF!
Chicken thinking: "Someone please explain this guy how we solve the Schroëdinger equation"
Look it is so simple, it just acts on an uncountably infinite dimensional vector space of differentiable functions.
fun fact: the vector space of differentiable functions (at least on compact domains) is actually of countable dimension.
still infinite though
Doesn't BCT imply that infinite dimensional Banach spaces cannot have a countable basis
Uhm, yeah, but there's two different definitions of basis iirc. And i'm using the analytical definition here; you're talking about the linear algebra definition.
So I call an infinite dimensional vector space of countable/uncountable dimensions if it has a countable and uncountable basis. What is the analytical definition? Or do you mean basis in the sense of topology?
Uhm, i remember there's two definitions for basis.
The basis in linear algebra says that you can compose every vector v as a finite sum v = sum over i from 1 to N of a_i * v_i, where a_i are arbitrary coefficients
The basis in analysis says that you can compose every vector v as an infinite sum v = sum over i from 1 to infinity of a_i * v_i. So that makes a convergent series. It requires that a topology is defined on the vector space fist, so convergence becomes well-defined. We call such a vector space of countably infinite dimension if such a basis (v_1, v_2, ...) exists that every vector v can be represented as a convergent series.
i just checked and there's official names for it:
- the term Hamel basis refers to basis in linear algebra
- the term Schauder basis is used to refer to the basis in analysis sense.
It's not even a fraction, you can just cancel out the two "d"s
"d"s nuts lmao
When a mathematician want to scare an physicist he only need to speak about ∞
When a physicist want to impress a mathematician he explains how he tames infinities with renormalization.
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