That is not basic math. The article talks about a discovery in a field of things called eigenvalues and eigenvectors – and I don't feel that stupid for not knowing whatever the hell those are,...
That is not basic math.
The article talks about a discovery in a field of things called eigenvalues and eigenvectors – and I don't feel that stupid for not knowing whatever the hell those are, 'cause those seem to be beyond what my high school-level education had to offer.
I think what the article is trying to convey is that it's basic for people who work in the field, and... I can see how it's difficult to differentiate what I would consider "basic" and what Terence Tao would consider "basic". Is it "fundamental"? Perhaps – but isn't also 1 + 1? The article uses the word "bedrock" in there somewhere, which seems most apt: fundamental to mathematics in the same sense as... bit is to computer science? sememe is to linguistics?
Pardon the rant. I find the prospect of getting the descriptor just shy of being on-point surprisingly frustrating.
There are plenty of high school students that learn linear algebra in their senior year of high school. Linear algebra is the field of math that concerns itself with "eigenvalues" and...
Exemplary
There are plenty of high school students that learn linear algebra in their senior year of high school. Linear algebra is the field of math that concerns itself with "eigenvalues" and "eigenvectors". Linear algebra does not require any calculus knowledge, either, and technically you learn pretty much all the tools you need to understand linear algebra during your second year algebra class in grade school, a course most high schoolers in North America will take before the end of their Junior year. Linear algebra, in a lot of ways, can be thought of as nothing more than "matrix algebra" instead of "number algebra".
So to understand what an "eigenvector" is, you have to understand what matrices are, and what vectors are. What's nice, is that you can usually think of vectors as just being a special kind of matrix, so we can easily multiply matrices by vectors. If you give me a matrix A, and a vector x, and it's true that Ax = c*x, where c is just some plain old number, then we call x an "eigenvector" of A. Now, what's an eigenvalue? Well, that's what "c" is! The idea is that if you multiply an eigenvector by its matrix, then the result will just be the same vector except longer or shorter. And the eigenvalue of the eigenvector is the scale factor by which the length of x changes. Meanwhile, if you multiply a non-eigenvector by the same matrix, it might change the direction of the vector.
I actually think the real reason for your comment is that we don't teach kids math, we teach them how to be human calculators.
One of my instructors often remarked that the best way to approach math was to think of it as a language for describing reality. If I'd heard that perspective on it in primary school I'd have...
One of my instructors often remarked that the best way to approach math was to think of it as a language for describing reality. If I'd heard that perspective on it in primary school I'd have probably been a lot better at it. Instead I heard it in my final year of college, during calc-based physics rather than in a math class. Completely changed how I think about the subject.
I got a degree in computer science (which comes along with about a minor in math's worth of math classes). Throughout my education I am surprised by how rarely I actually got to do math, rather...
I got a degree in computer science (which comes along with about a minor in math's worth of math classes). Throughout my education I am surprised by how rarely I actually got to do math, rather than just computation. Moments where the equations are unknown and you need to compose your own solution are so rare. We waste so much time teaching things in a way that is focused on memorization.
I think they mean "basic" in the sense that it isn't ultra abstract and has direct impact in applied maths and engineering. It is not high school level but you learn about eigenstuff in any first...
I think they mean "basic" in the sense that it isn't ultra abstract and has direct impact in applied maths and engineering. It is not high school level but you learn about eigenstuff in any first year of a engineering course.
Here's the question: am I expected to have some level of hard-science knowledge to read articles from Quanta? If I am, I might wanna brush up on them sciences, 'cause that was a well-written...
Here's the question: am I expected to have some level of hard-science knowledge to read articles from Quanta? If I am, I might wanna brush up on them sciences, 'cause that was a well-written article.
And if I'm not... they should pick a more-precise descriptor?
But fuck do I know. I'm just a guy alone in the apartment, coding all day.
I’d like to point out that the target audience for Quanta isn’t so much laypeople (eg, random person off the street) as much as it is people with a vested interest in the subject (eg, students or...
I’d like to point out that the target audience for Quanta isn’t so much laypeople (eg, random person off the street) as much as it is people with a vested interest in the subject (eg, students or scientists). So yes, you’re right in a sense -- eigenvalues and eigenvectors aren’t “basic math” for most people. But for the target audience of this article (say, a math undergrad), this would be considered basic math.
Edit: Not that I mean to be dismissive. Thankfully you don't need to be a scientist to be a fan of science, and there are plenty of media outlets that cater to that audience. This particular magazine, however, bridges the gap between popular science and the actual journal article.
They're basic in the sense of the word as "to form the basis for something" or "fundamental", perhaps. Whether or not they're basic in the "simple and easy math 101" sense is relative, I suppose....
They're basic in the sense of the word as "to form the basis for something" or "fundamental", perhaps. Whether or not they're basic in the "simple and easy math 101" sense is relative, I suppose. I can say in my experience at least I didn't learn them in high school but I did learn about them pretty early on in university—but that's only because I had to take some linear algebra courses. Of course I couldn't tell you much about them now though because when was the last time I sat down and did some linear algebra?
(Sidenote: I feel like getting into math and physics could be a fun hobby, but it's a lower priority than other hobbies I want to fit into the spare time I have. Still, it's always something in the back of my mind.)
Yeah, looking at the article again, I think this is the meaning they were trying to convey through "basic". When I was in middle school, I thought calculus was higher-level math. By the time I...
They're basic in the sense of the word as "to form the basis for something" or "fundamental", perhaps.
Yeah, looking at the article again, I think this is the meaning they were trying to convey through "basic".
Whether or not they're basic in the "simple and easy math 101" sense is relative, I suppose.
When I was in middle school, I thought calculus was higher-level math. By the time I graduated college, I hardly thought of calculus as being math at all (as you astutely noted elsewhere, math isn't just about memorizing algorithms, which is largely what you do in calculus classes). Point being: I totally agree with this sentiment.
(Sidenote: I feel like getting into math and physics could be a fun hobby, but it's a lower priority than other hobbies I want to fit into the spare time I have. Still, it's always something in the back of my mind.)
Personally, physics is one of those things that I find to be very rewarding at a high level but less so when you're working on the minutiae (those ah-ha! moments are nice, though). Still, if it's a hobby, then you have the pleasure of pursuing whatever catches your fancy. Frankly, I find yours a fascinating perspective -- I've been in grad school for so long that I find the idea of physics as a hobby completely alien. Could you elaborate?
Part of it is just a curiosity in wanting to follow my interests in seeing how things work. I can read a layperson pop-science book but I always feel like something is missing from those in terms...
I've been in grad school for so long that I find the idea of physics as a hobby completely alien. Could you elaborate?
Part of it is just a curiosity in wanting to follow my interests in seeing how things work. I can read a layperson pop-science book but I always feel like something is missing from those in terms of understanding. As soon as those get to the really interesting parts it either becomes muddied and metaphorical or it starts going over my head, e.g. I can read an article that says "maybe time is an emergent phenomena" because it's not needed in these equations that describe this other thing, and okay I can file that away as a neat fact, but I don't really understand that. Understanding something in physics takes a way of thinking I haven't learned or practised that is based on mathematical description.
Then, there's all the unknown areas and open questions.1 It would be fun to work my way out to some of that and start thinking about those questions on my own. I wouldn't expect to make any major progress (that way lies crank-ism and perpetual motion machines) but it's maybe a bit like going up to Everest, taking a look at it, and then hiking a bit in the foothills just for fun and exercise.
Like I said though it's not a priority for me. It's something I'm curious about but I'm curious about a lot of things and I can't study all of them. Realistically I can barely study any of them. It's also not "productive" in the sense that I can't necessarily create2 anything out of my interest, and that's a priority for me too.
Still, though, the idea keeps coming to me and it's appealing. To be able to learn about and understand the natural world better at a fundamental level and to be able to think about some of the questions we can't answer yet would be fun. And, it wouldn't be completely unproductive—Neal Stephenson was obviously able to take a 'basic' understanding of orbital mechanics and turn it into some pretty good writing in the middle section of Seveneves, after all.3
But, life is finite and we need to prioritize and math+physics beyond qualitative descriptions in popular books/articles would be such a time sink.
1 Mostly thinking of the "big" questions but there are lots of interesting smaller things to study or things that no one has really thought to consider at all.
2 Defining "create" in kind of a loose sense that could include even something like being able to speak with someone in another language or having a healthier and more capable body through exercise. Basically, being able to create something permanent or transient to share in a way that's positive for other people, or being able to do something that improves my physical or mental well-being.
3 I don't know if he got into the math of it though. I felt like I could follow along and understand what was going on just from having played Kerbal Space Program.
That is not basic math.
The article talks about a discovery in a field of things called eigenvalues and eigenvectors – and I don't feel that stupid for not knowing whatever the hell those are, 'cause those seem to be beyond what my high school-level education had to offer.
I think what the article is trying to convey is that it's basic for people who work in the field, and... I can see how it's difficult to differentiate what I would consider "basic" and what Terence Tao would consider "basic". Is it "fundamental"? Perhaps – but isn't also
1 + 1? The article uses the word "bedrock" in there somewhere, which seems most apt: fundamental to mathematics in the same sense as... bit is to computer science? sememe is to linguistics?Pardon the rant. I find the prospect of getting the descriptor just shy of being on-point surprisingly frustrating.
There are plenty of high school students that learn linear algebra in their senior year of high school. Linear algebra is the field of math that concerns itself with "eigenvalues" and "eigenvectors". Linear algebra does not require any calculus knowledge, either, and technically you learn pretty much all the tools you need to understand linear algebra during your second year algebra class in grade school, a course most high schoolers in North America will take before the end of their Junior year. Linear algebra, in a lot of ways, can be thought of as nothing more than "matrix algebra" instead of "number algebra".
So to understand what an "eigenvector" is, you have to understand what matrices are, and what vectors are. What's nice, is that you can usually think of vectors as just being a special kind of matrix, so we can easily multiply matrices by vectors. If you give me a matrix A, and a vector x, and it's true that Ax = c*x, where c is just some plain old number, then we call x an "eigenvector" of A. Now, what's an eigenvalue? Well, that's what "c" is! The idea is that if you multiply an eigenvector by its matrix, then the result will just be the same vector except longer or shorter. And the eigenvalue of the eigenvector is the scale factor by which the length of x changes. Meanwhile, if you multiply a non-eigenvector by the same matrix, it might change the direction of the vector.
I actually think the real reason for your comment is that we don't teach kids math, we teach them how to be human calculators.
I feel like I was in my early 20s before I really grokked that math wasn't about memorizing algorithms and science wasn't about memorizing facts.
One of my instructors often remarked that the best way to approach math was to think of it as a language for describing reality. If I'd heard that perspective on it in primary school I'd have probably been a lot better at it. Instead I heard it in my final year of college, during calc-based physics rather than in a math class. Completely changed how I think about the subject.
I got a degree in computer science (which comes along with about a minor in math's worth of math classes). Throughout my education I am surprised by how rarely I actually got to do math, rather than just computation. Moments where the equations are unknown and you need to compose your own solution are so rare. We waste so much time teaching things in a way that is focused on memorization.
Jesus, this is a better explanation of eigenvectors and eigenvalues than I heard in a whole year of LA.
I think they mean "basic" in the sense that it isn't ultra abstract and has direct impact in applied maths and engineering. It is not high school level but you learn about eigenstuff in any first year of a engineering course.
Here's the question: am I expected to have some level of hard-science knowledge to read articles from Quanta? If I am, I might wanna brush up on them sciences, 'cause that was a well-written article.
And if I'm not... they should pick a more-precise descriptor?
But fuck do I know. I'm just a guy alone in the apartment, coding all day.
I’d like to point out that the target audience for Quanta isn’t so much laypeople (eg, random person off the street) as much as it is people with a vested interest in the subject (eg, students or scientists). So yes, you’re right in a sense -- eigenvalues and eigenvectors aren’t “basic math” for most people. But for the target audience of this article (say, a math undergrad), this would be considered basic math.
Edit: Not that I mean to be dismissive. Thankfully you don't need to be a scientist to be a fan of science, and there are plenty of media outlets that cater to that audience. This particular magazine, however, bridges the gap between popular science and the actual journal article.
They're basic in the sense of the word as "to form the basis for something" or "fundamental", perhaps. Whether or not they're basic in the "simple and easy math 101" sense is relative, I suppose. I can say in my experience at least I didn't learn them in high school but I did learn about them pretty early on in university—but that's only because I had to take some linear algebra courses. Of course I couldn't tell you much about them now though because when was the last time I sat down and did some linear algebra?
(Sidenote: I feel like getting into math and physics could be a fun hobby, but it's a lower priority than other hobbies I want to fit into the spare time I have. Still, it's always something in the back of my mind.)
Yeah, looking at the article again, I think this is the meaning they were trying to convey through "basic".
When I was in middle school, I thought calculus was higher-level math. By the time I graduated college, I hardly thought of calculus as being math at all (as you astutely noted elsewhere, math isn't just about memorizing algorithms, which is largely what you do in calculus classes). Point being: I totally agree with this sentiment.
Personally, physics is one of those things that I find to be very rewarding at a high level but less so when you're working on the minutiae (those ah-ha! moments are nice, though). Still, if it's a hobby, then you have the pleasure of pursuing whatever catches your fancy. Frankly, I find yours a fascinating perspective -- I've been in grad school for so long that I find the idea of physics as a hobby completely alien. Could you elaborate?
Part of it is just a curiosity in wanting to follow my interests in seeing how things work. I can read a layperson pop-science book but I always feel like something is missing from those in terms of understanding. As soon as those get to the really interesting parts it either becomes muddied and metaphorical or it starts going over my head, e.g. I can read an article that says "maybe time is an emergent phenomena" because it's not needed in these equations that describe this other thing, and okay I can file that away as a neat fact, but I don't really understand that. Understanding something in physics takes a way of thinking I haven't learned or practised that is based on mathematical description.
Then, there's all the unknown areas and open questions.1 It would be fun to work my way out to some of that and start thinking about those questions on my own. I wouldn't expect to make any major progress (that way lies crank-ism and perpetual motion machines) but it's maybe a bit like going up to Everest, taking a look at it, and then hiking a bit in the foothills just for fun and exercise.
Like I said though it's not a priority for me. It's something I'm curious about but I'm curious about a lot of things and I can't study all of them. Realistically I can barely study any of them. It's also not "productive" in the sense that I can't necessarily create2 anything out of my interest, and that's a priority for me too.
Still, though, the idea keeps coming to me and it's appealing. To be able to learn about and understand the natural world better at a fundamental level and to be able to think about some of the questions we can't answer yet would be fun. And, it wouldn't be completely unproductive—Neal Stephenson was obviously able to take a 'basic' understanding of orbital mechanics and turn it into some pretty good writing in the middle section of Seveneves, after all.3
But, life is finite and we need to prioritize and math+physics beyond qualitative descriptions in popular books/articles would be such a time sink.
1 Mostly thinking of the "big" questions but there are lots of interesting smaller things to study or things that no one has really thought to consider at all.
2 Defining "create" in kind of a loose sense that could include even something like being able to speak with someone in another language or having a healthier and more capable body through exercise. Basically, being able to create something permanent or transient to share in a way that's positive for other people, or being able to do something that improves my physical or mental well-being.
3 I don't know if he got into the math of it though. I felt like I could follow along and understand what was going on just from having played Kerbal Space Program.