{"id":385135,"date":"2024-06-29T05:55:45","date_gmt":"2024-06-29T05:55:45","guid":{"rendered":"http:\/\/savepearlharbor.com\/?p=385135"},"modified":"-0001-11-30T00:00:00","modified_gmt":"-0001-11-29T21:00:00","slug":"","status":"publish","type":"post","link":"https:\/\/savepearlharbor.com\/?p=385135","title":{"rendered":"\u0421\u043f\u0435\u0446-\u0438\u043d\u0442\u0435\u0440\u0432\u044c\u044e \u0434\u043b\u044f \u0425\u0430\u0431\u0440\u0430: \u0413\u0440\u0430\u043d\u0442 \u0421\u0430\u043d\u0434\u0435\u0440\u0441\u043e\u043d<\/span>"},"content":{"rendered":"
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\u0413\u0440\u0430\u043d\u0442 \u043e\u0442\u0432\u0435\u0442\u0438\u043b \u043d\u0430 \u0432\u0430\u0448\u0438 \u0432\u043e\u043f\u0440\u043e\u0441\u044b. \u041f\u043e\u0434 \u043a\u0430\u0442\u043e\u043c \u0432\u044b \u043d\u0430\u0439\u0434\u0451\u0442\u0435 \u0441\u043e\u0432\u0440\u0435\u043c\u0435\u043d\u043d\u044b\u0439 \u0432\u0437\u0433\u043b\u044f\u0434 \u043d\u0430 \u043e\u0431\u0443\u0447\u0435\u043d\u0438\u0435 \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0435, \u0442\u043e\u043c\u0443 \u043a\u0430\u043a \u043f\u0440\u0430\u0432\u0438\u043b\u044c\u043d\u043e \u043f\u0440\u0438\u043c\u0435\u043d\u044f\u0442\u044c \u043c\u0430\u0442\u0435\u043c\u0430\u0442\u0438\u043a\u0443 \u0432 \u043f\u0440\u043e\u0433\u0440\u0430\u043c\u043c\u0438\u0440\u043e\u0432\u0430\u043d\u0438\u0438, \u0438 \u0440\u0430\u0441\u0441\u043a\u0430\u0437 \u043e \u0442\u043e\u043c, \u043a\u0430\u043a \u043f\u0440\u0435\u0432\u0440\u0430\u0442\u0438\u0442\u044c \u043f\u0440\u043e\u0441\u0442\u0443\u044e \u0443\u0442\u0438\u043b\u0438\u0442\u0443 \u0434\u043b\u044f \u0430\u043d\u0438\u043c\u0430\u0446\u0438\u0438 \u0432 \u043e\u0433\u0440\u043e\u043c\u043d\u044b\u0439 \u043a\u0430\u043d\u0430\u043b. \u0418 \u043a\u043e\u043d\u0435\u0447\u043d\u043e \u0436\u0435, \u0432\u0430\u0441 \u0432\u0441\u0435\u0445 \u0436\u0434\u0451\u0442 \u0432\u0437\u044f\u0442\u0438\u0435 \u043f\u0440\u043e\u0438\u0437\u0432\u043e\u0434\u043d\u044b\u0445 \u043d\u0430 \u0445\u043e\u0434\u0443 \u0438 \u0440\u0430\u0441\u0441\u043a\u0430\u0437 \u043e \u0442\u043e\u043c, \u043a\u0430\u043a \u0441\u0447\u0438\u0442\u0430\u0442\u044c \u043f\u0435\u0440\u0432\u043e\u043e\u0431\u0440\u0430\u0437\u043d\u044b\u0435 \u0432 \u0443\u043c\u0435. <\/p>\n

\u041f\u043e\u0435\u0445\u0430\u043b\u0438! <\/a> <\/p>\n

\u0414\u043b\u044f \u0442\u0435\u0445, \u043a\u0442\u043e \u0445\u043e\u0447\u0435\u0442 \u043f\u0440\u043e\u043d\u0438\u043a\u043d\u0443\u0442\u044c\u0441\u044f \u043e\u0442\u0432\u0435\u0442\u0430\u043c\u0438 \u043d\u0430 \u0430\u043d\u0433\u043b\u0438\u0439\u0441\u043a\u043e\u043c \u044f\u0437\u044b\u043a\u0435, \u043c\u0438\u043b\u043e\u0441\u0442\u0438 \u043f\u0440\u043e\u0441\u0438\u043c \u043f\u043e\u0434 \u0441\u043f\u043e\u0439\u043b\u0435\u0440:<\/p><\/blockquote>\n

\u041e\u0440\u0438\u0433\u0438\u043d\u0430\u043b\u044c\u043d\u044b\u0439 \u0442\u0435\u043a\u0441\u0442 \u0432\u043e\u043f\u0440\u043e\u0441\u043e\u0432 \u0438 \u043e\u0442\u0432\u0435\u0442\u043e\u0432<\/b> <\/p>\n
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Did you start learning math as entertainment, or did you know that you would make your living doing math?<\/i><\/p><\/blockquote>\n

Grant S.(3Blue1Brown): <\/font><\/b>Like anyone else, I started learning math simply because it’s a common part of the school. My dad did take a special interest in supplementing learning for both my brother and me, and I do remember a number of math-based games he would play. When I was very little, this included things like stacking sugar cubes in interesting arrangements and having me figure out how many there were (with the reward of eating a sugar cube if I got it right). A little bit later in elementary school, he volunteered to teach a \u00abmath olympiad\u00bb seminar at our school, which involved creative problem-solving. Also, at home, he’d often share little tips and tricks he knew. I’m struggling to remember many specifics, but one that comes to mind is how if you memorize more square numbers, it can help to multiply numbers. E.g. 16 * 18 = 17^2 \u2014 1 = 288. This made me want to memorize more square numbers beyond the usual first 12 taught in our multiplication tables.<\/p>\n

I wouldn’t have called this \u00abentertainment\u00bb at the time, but somehow I did come to really enjoy math as a hobby. Part of it might have been ego. It feels good to be ahead of your peers, so I wonder if some of my earlier desire to read more math or do more problems stemmed from the less-than-pure goal of staying ahead. In either case, by the time I was in high school, I think it had evolved into a more genuine and less ego-driven interest. I didn’t have very definitive career ambitions, but if you had asked me then, the answer \u00abmathematician\u00bb would have certainly been somewhere on the list. At the very least, I knew for sure that I’d be studying math once I got to university.<\/p>\n\n

How often would you be googling for a math formula?<\/i><\/p><\/blockquote>\n

Grant S.(3Blue1Brown): <\/font><\/b>It depends on what I’m doing, but pretty frequently. A lot of the day-to-day math I end up doing is in the service of making the graphics for the videos, which sometimes requires some specific formula to get some visual to work (or to just gut check my own calculation). When I’m just learning math, I tend to get more from books than Google.<\/p>\n\n

How much math can you apply to a business? (Often in the University we would have to study higher math and many people would be questioning this. \u201cYou don’t actually need math at work\u201d. But can you prove this wrong? How an actual regular company can benefit from applying something that you know?)<\/i><\/p><\/blockquote>\n

Grant S.(3Blue1Brown): <\/font><\/b>There’s no doubt that a strong number sense can be beneficial for business, say in understanding the influence of compounding growth, or making quick judgments on if a deal is profitable. But honestly, that’s rarely the deep and interesting math, so maybe the more interesting question is how often the kind of math found in an undergrad math major’s curriculum would actually be useful. <\/p>\n

I think it’s fair to say there’s a strong correlation between how much math someone knows, and how much they can earn, but it’s unclear how direct the applications are, or even whether this correlation is causal. The clich\u00e9 is that we teach math not just because of its direct applications, but because it teaches you how to think. For example, you’re unlikely to ever use the fundamental theorem of calculus in a very literal way in a job (unless you have a weird job), but for some reason, those who have grappled with the nuances and technical demands of calculus and analysis in their school years also have a tendency to end up in higher-paying technical positions in their careers. It’s tempting to say it’s because of this math education, but maybe there’s a lurking cause between the two, like how much value a particular culture places on technical know-how.<\/p>\n

One clear example where I think math is quite useful is that knowing more math seems to make it quicker to learn how to code and to understand the architecture of a piece of software more broadly. Knowing about proofs by induction makes recursion quite easy, having written and read many proofs can make the detail-oriented task of debugging more natural, etc. There are also plenty of software tasks that require very direct applications of lots of math, like how machine learning requires linear algebra and probability, or how computer graphics requires a strong command of linear algebra, calculus, and geometry. But even the ones that don’t seem so inherently mathematical seem to benefit from a strong math background anyway.<\/p>\n\n

Are there any criteria which could distinguish a good mathematician? How would you say if someone is good at math?<\/i><\/p><\/blockquote>\n

Grant S.(3Blue1Brown): <\/font><\/b>What I love about math is how it blends a detail-oriented mindset with a creative one. It’s not enough to merely be able to calculate and to avoid errors, you often need to be able to come up with the right perspective, or the right construction to add to your problem. To take a simple example, say you want to prove the Pythagorean theorem. On the one hand, you need to understand proofs in general, i.e. logical deduction, avoiding simple errors, identifying when there’s a gap in reasoning, etc. But just being able to do that could never be enough. The Pythagorean theorem isn’t something that directly falls out of definitions, you need to actually input some idea, or add the right lines to your diagram, to make any headway, and that’s where the creativity comes in.<\/p>\n\n

Have you ever heard of a habr.com<\/a>? What do you think about Russians? Do you know\/work with any Russian mathematicians?<\/i><\/p><\/blockquote>\n

Grant S.(3Blue1Brown): <\/font><\/b>I had not been familiar with Habr. <\/p>\n

The Russian mathematicians I’ve engaged with are always a delight. I have a lot of respect for the culture around math in Russia. The books by Russian authors I’ve read from seem to always place much more emphasis on the role of doing problems as a necessary part of learning, not just an incidental add-on sitting at the back of a chapter. Vladimir Arnold is one of my favorite mathematicians and explainers, and I feel like I’ve seen his style mirrored in some of the other Russian mathematicians I’ve interacted with.<\/p>\n\n

What are the non-solved math problems that could have a great potential in a current society?<\/i><\/p><\/blockquote>\n

Grant S.(3Blue1Brown): <\/font><\/b>Well, if it turned out that P=NP, that would have huge ramifications for all of cryptography. It seems quite unlikely that that would be true, but however the question is resolved, it would probably be significant. If someone manages to prove P\u2260NP, hopefully, the tactics of the proof would carry with them some insights beyond the direct result, in a way that could potentially be useful for constructing\/understanding NP problems more generally. If someone managed to prove the question is undecidable, that would also be highly relevant, and tell us something about the weakness of our current formalisms.<\/p>\n\n

How many digits of Pi do you know by heart? What is your viewpoint on those competitions where people would just remember digits of Pi?<\/i><\/p><\/blockquote>\n

Grant S.(3Blue1Brown): <\/font><\/b>Let’s see, not looking up anything now here’s what I have: 3.1415926535897932384\u2026<\/p>\n

As I type that, I’m not very confident it’s accurate. I remember learning around 20-30 digits when I was a kid and cared more about that kind of task. These days I think of it as pretty silly. Memorizing a constant reflects a lot of what’s wrong with current math education, where we have kids do rote tasks that don’t really matter. There are plenty of things where raw memorization is exceedingly helpful for actively doing math. For instance, kids should definitely memorize their multiplication tables. But there are also lots of pointless things they’re tasked with memorizing for no good reason, other than the fact that it’s seemingly easier to measure and test than creative math is.<\/p>\n

In fairness, it’s not like schools regularly have kids memorize digits of pi. But I do fear that people’s infatuation with that kind of task is rooted in the fact that the other things school rewards center on rote memorization. It becomes a way of flexing a muscle that was overemphasized by schools.<\/p>\n\n

How important it is to be able to calculate \u201cin your head\u201d when you have all those calculators and computers around you?<\/i><\/p><\/blockquote>\n

Grant S.(3Blue1Brown): <\/font><\/b>Despite what I just said about memorization, I actually think it’s quite helpful to be able to do quick computations in your head, especially if it’s for a kind of problem you work with a lot. The best way to build intuition is by drilling on many problems, and often this act of drilling comes with the added side-benefit that you’re familiar enough with certain subproblems to just do them in your head. And on the flip side, the more you can do in your head, the more readily you can get a rough intuitive sense of the problem before getting distracted by turning over to a calculator, where the intuition can be lost.<\/p>\n

Sure, for a lot of tasks you can (and should) just turn to a calculator or Mathematica, but the way you come to be able to do subproblems mentally seems to correspond with building up general mental constructs which are themselves handy for understanding the broader structures of a problem.<\/p>\n

Let’s say I give you the integral \\int_0^\\pi \\sin(\\theta) d\\theta. On the one hand, you can quickly go to Wolfram alpha to check it. But being able to just \u00absee\u00bb answer 2 by looking at this probably correlates with a better understanding of integrals and trig. And this is true at many levels:<\/p>\n