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Math for Programmers: 3D Graphics, Machine Learning, and Simulations with Python

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Here are a couple of books that I would highly recommend for anyone who wants to learn math in the context of some interesting problems : The math computer scientists use regularly, in real life, has very little overlap with the list above. For one thing, most of the math you learn in grade school and high school is continuous: that is, math on the real numbers. For computer scientists, 95% or more of the interesting math is discrete: i.e., math on the integers. I would put teaching/learning of prime-numbers up as one of the firstmost things to do. I'd like it to be taught before simplification of fractions, for example -- if you break the numerator and denominator into their prime factorization then simplifying the fraction becomes a trivial matter of removing what is in both numbers. Frequently one finds that what one learns in school has direct application to one's vocation. And when this is true, education provides economic value. Ecomomic prosperity does play a significant role in securing freedom. But even if this were not true, education would still be both the first and the last line of defense of a democracy. (The argument is too long to present here; read Montesquieu.http://www.constitution.org/cm/sol-02.htm) Let's take geometry as an example of a common language. In a strict sense, geometry is almost useless as vocational training - except to mathematics students who go on to teach the subject. But in a broader sense, the construct of geometry is deductive reasoning. And the sort of reasoning used here is crucial in a host of disciplines. It may be courtroom law that draws on the mental disciplines and processes inculcated by geometry.

the doubter: I think Wikipedia has vastly surpassed Mathworld in terms of usefulness and quantity of information. to begin with, i always think of myself as good at math. I have been doing a program called Kumon (it started in Japan, it sums up to being 25 minutes of math a day, everyday, 365 days a year) and i had a HUGE start on it. It is quite logically structured, and makes everything quite easy to learn.the only arguement i would make about it is that is disallows calculators, however i think that for basic arithmetic one should be able to do it mentally anyways. To make a long story short I can do linear algebra in my head while the teacher starts writing the first step on the board. One reservation though: It seems to me that your approach to mathematics ( the just-in-time method?) leads to only surface understanding that can be dangerous--literally deadly. Probability and statistics, is an area, for example, where these errors can be especially egregious since for some it seems deceptively simple to some. Subtleties that come from deeper knowledge are lost. I think it is safe to say that your ability to move easily from one subject to another owes much to the teachers that spent countless hours drilling into our heads the fundamentals of mathematics. You can learn algebra without calculus, but not vice versa. 7:42 PM, March 17, 2006 Anonymous said... It really seems that in the past calculus and differential equations were at the heart of various engineering edeavors. Now, however, it seems that probability and discrete math are becoming more important, espeically given the new interdisciplinary mash-ups like 'genetic algorithms/genetic programming' (combining evolutionary biology with algorithm design), various social-insect based search methods (ACO, PSO - another biology meets engineering mashup), bioinformatics, etc. Probability and statistics play a big role in genetics and biology in general and now we're seeing biology and engineering combining. 5:59 PM, March 17, 2006 Anonymous said...You might want to read up in the areas of the learning sciences, instructional design, cognitive psychology, and related fields. See for example the book "How People Learn" and an instructional design book called "Understanding by Design". I think it would be much more beneficial for students in high school to get a good foundation in statistics, then calculus or pre-calculus. 10:58 PM, March 17, 2006 Anonymous said... I had some trouble with math up through the 8th grade. So, when most of my peers were taking Algebra 1, I took "Arithmetic II". And then I "got it" and things were better. And yes, in many ways math was fun. To put this in perspective, think about long division. Raise your hand if you can do long division on paper, right now. Hands? Anyone? I didn't think so.

From reading technical/research papers and following references one can survey the terrain and find the important topics. i don't recall the lessons in school when i was told how to do long division, or which trigonometric functions to apply to which triangles, or which combinatorial functions to apply to which problems. but if i had to do any of this right now, i could easily figure it out, because it's nothing more complicated than middle school math. So now come university, And i'm doing discrete mathematics as a first unit, rather easy to understand straight off the bat unlike many others who thought they were exelent at maths the last 2 years..

about the book

I think Knuth's "Concrete Mathematics" contains most of the math we'd need as programmers, and is also the most enjoyable math book I've come across. Now if only I can get around to studying it ... :) Perhaps it's another a bad generalization, but I think people learn much better when they can imagine what's happening and apply their intuition, rather than just pushing symbols around (which is also interesting later, in computer science). So, I have given up on justifying math other than stating that what they are accomplishing here is to prove to someone (the people who run the education system) that they can perform long and tedious procedures without making mistakes at least 70% of the time. It's about the best you can hope for in terms of the general population. I would emphasize building things and geometry very early on. Make it as important as arithmetic. It should be as physical as possible.

So they're right: you don't need to know math, and you can get by for your entire life just fine without it. So the list's no good anymore. Schools are teaching us the wrong math, and they're teaching it the wrong way. It's no wonder programmers think they don't need any math: most of the math we learned isn't helping us. You can follow this link to get a good free ebook about mathematical logic, written by Stefan Bilaniuk: Just because you don't use the math doesn't mean it's useless, and it sure as hell doesn't mean that the curriculum is designed wrong. 12:41 PM, March 17, 2006 Anonymous said...One STILL cannot get away with saying "I read it on wikipedia" (get some public official endorsement for it and that will probably change - then: get it and probably wikipedia will lose editors) I agree and yet disagree. It depends upon the programming being done. As one person mentioned, graphics requires a lot of math if you program from the ground up, such as the the transforms between 3d to 2d planes. It would be nice if you could post a list of books that you are currently reading/have read/will read 12:34 AM, March 20, 2006 Unknown said... By the way, my hand is up. I can do long division on paper, right now 2:56 PM, March 17, 2006 Anonymous said... And you know what? They're absolutely right. You can be a good, solid, professional programmer without knowing much math.

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