Was surprised to have that learning moment in the middle of the exam and not prior…
I sat my first exam for a university course I was teaching last year. I thought I needed to introduce some new ideas, so the students wouldn't be bored doing it.
From the evaluations, not all students agreed...
I always hated when my instructors put "important" results that we have never seen before on an exam. It was like adding insult to injury if I didn't know how to solve it.
It was different on homework assignments, because usually that you had time to work through the problem in detail and have the "aha" moment, without stress and time pressure.
You would have hated my 1977 quantum mechanics final; not a single question that had been directly covered in the course. Really sorted out those who had been paying attention from those who thought that memorization was enough.
As far as buzzwords, I think the weight of "MIT" is much heavier than any buzzwords that could be attached. (Though, I'm biased.)
The guy I studied with sat behind me, and at one point one of us started stress laughing. Then it was two of us in the middle of a lecture laughing like our gun just jammed while the horror movie cereal killer was almost within striking distance.
There were a lot of pissed off people in class for the next couple of weeks.
Better than at the end of the exam...
What's wrong with calling it ln x? The way this is written in the article implies there's something weird about calling it that. The name 'log' can mean log2, log10 or natural logarithm depending on the field.
Removing ambiguities from math notation should be considered a good thing.
The author expressed a worry about math education. Consider that a clear non ambiguous notation would help.
In school (and engineering or physics I guess) you often are made to use ln for natural log and you are taught a way to pronounce that name (somewhere between lun and l’n)
It feels like the point is “this person had not been exposed to university style mathematics”.
Imho math is about logic and reasoning, not about what group you're part of
Mathematicians decide what is interesting, and that's not a matter of logic. A computer can bang out new theorems at light speed but nobody cares. Mathematics, like science and programming, is as much about humans as about the raw logic and data.
You're welcome to be a group of one and please only yourself. But then you wouldn't care if it were published, and it wouldn't be unless you showed it to someone and they took an interest.
Similar to those who mock people for saying a word incorrectly that they only learned from reading.
Whereas in uni log is generally assumed to be the natural log, or else it's specified, or else the base is unimportant (like in big O notation)
As someone who has done a lot of mathematics in their life, I've never found this perceived ambiguity to be an issue.
The issue with being able to derive the formulas for derivation yourself is that it's not very useful. You simply don't have time to make those derivations during a test. It's like trying to use grammar rules in a conversation - conversations happen at a pace where you cannot apply grammar rules. You'll just have to know the patterns.
You learn things in school to do a test. The usefulness of the vast majority of the knowledge they attain is purely to help them do the test. Later in life you might wish you knew more about this or that, but that's not at all apparent to the student.
And schools have figured out that rather than teaching the subject from first principles, it's easier to get students to get high grades by teaching them each of the structures. Eg. "Whenever there is a question about differentiating x^7, just put 7x^6 as the answer." They then get the students to try a few examples (x^3 becomes 3x^2, x^77 becomes 77x^76, etc), and thats the way every science-y subject is taught.
I often think it leads to students who do well in exams, but can't solve many real world problems.
It could be solved by having a part of every exam paper be never-seen-before applied problems. For example, for differentiation, one might ask "A road's height in meters as a function of the horizontal distance along the road in kilometers is defined as sin(x)cos(x)tan(x). At what points are the steepest uphills? Would you describe the slope of the road as 'very hilly', and why?"
- the physics course can’t depend on the concurrent maths course because you are allowed to take physics without taking maths, so you just learn weird equations full of exponential a instead of the ODE
- I think the maths course doesn’t even teach differential equations. They are in FP1 (from a separate ‘further maths’ course) but definitely not in AS (penultimate year of school) maths. Possibly a few turn up in A2 (final year) but then they can’t have any good examples from physics because not everyone doing maths will be able to depend on knowledge about what a capacitor is or how nuclear decay works. But I guess population models might work.
- there can be some better stuff in the further maths course (e.g. I think they might even have the ‘exponentiate a matrix’ solution to systems of first order linear ODEs)
So I retook the course the next year. Taught by a new teacher fresh from teachers college, theoretically with a specialty in math since they were teaching an upper level math course.
I don't think the new teacher even knew how to do derivatives from first principles. Just rote memorization of the different types of differentiation.
I got an A in that class the second time, having learned nothing.
It contains Euler, the imaginary unit, the unit, the zero and some hidden trigonometry.
P.S. does anyone know why the unicode symbol for the Euler constant render as a weird E when it is usually represented as a slightly italicized e ?
There is a footnote on https://en.wikipedia.org/wiki/Letterlike_Symbols that says:
> It's unknown which constant this is supposed to be. Xerox standard XCCS 353/046 just says 'Euler's'.
See also this discussion on math stackexchange: https://math.stackexchange.com/a/3123704
Which are the hyperoperations of rank 1, 2 and 3:
> In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called hyperoperations in this context) that starts with a unary operation (the successor function with n = 0). The sequence continues with the binary operations of addition (n = 1), multiplication (n = 2), and exponentiation (n = 3).
(I would like to increase the count by e^0 btw)
Mathematician. Thought about a new (at least to me) transcendental number ...
Stop playing once you’ve seen at least n/e of the available options and the current one is acceptable.
(ei)^0 = 1^pi
e isn't important, the exponential function is. e shows up so often because we've chosen to write exp(x) as e^x. It's a result of a notational choice - the fact that exp(1) = 2.718.. and we call that e is pretty insignificant and boring.
The constant itself is still pretty interesting. Using e as a base for all numbers yields optimal information density IIRC. Binary (base 2) is close to e so it's information density is not bad, but this also tells us that trinary (base 3) would be even better on this metric since it's closer.
There are lots of interesting properties like this that end up linked to e.
The GP comment reads as either a grab at elite character at best or flat out anti-intellectual at worst. No need to bring it in here.
Take this headline: The function exp(x) = 1 + x + x^2/2 + x^3/6 + ... is the most beautiful function in mathematics. It is its own derivative, has "product linearity", i.e. exp(x+y) = exp(x) exp(y), and is related to trig functions through complex numbers.
The number e isn't doing the heavy lifting, it is the function. The number e comes from the function, not the other way around. Even the famous equation with pi and e is a consequence of the function. And the Taylor series is the easiest way to see the relationship with trig functions.
To be fair, there might be a difference in dispensation at play. Those who prefer a more causal or "active" feel to mathematics would prefer the function framing while those who prefer a more platonic or "mystical" feel would prefer the constant framing.
e^iπ = −1
The special case of x=pi... it's like being excited that sin(pi)=0 or cos(pi)=-1. It doesn't really say anything meaningful or consequential, people like it only because of the symbols it includes. It feels kind of like a math meme that people like to repeat and I can't get behind it.
Maybe it's just not for me and I should just let other people like what they like.
The impressive aspect of that version of the equation is simply the idea that you can obtain a plain old integer (-1) using nothing but simple arithmetic operations on two random-looking transcendentals.
The title of the link currently is "Why E, the Transcendental Math Constant, Is Just the Best".
But the article really is about Euler's constant - the lower case e - and not about any of the capital E:s out there (like the capital E sometimes used in scientific notation, or the expected value in probability theory).
This is not the e you know and love from school:
This one is:
Worth coming up with some better way to talk about this.
Yes, this problem is touched on in the Wikipedia "List of Things named after Leonhard Euler" (https://en.wikipedia.org/wiki/List_of_things_named_after_Leo...)
I particularly like the remark in the introduction:
> In an effort to avoid naming everything after Euler, some discoveries and theorems are attributed to the first person to have proved them after Euler.
I kind of wish we had a holiday of some kind to appreciate either Euler himself or even a month to discuss the historical contributions to knowledge by philosophers and scientists alike.
My math teacher from high school, who I still keep in touch with, sends out a reminder every April 15th.
Why Euler's number (e), the Transcendental Math Constant, Is Just the Best