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different ways to do algebra


lightburst
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Richard Feynman on The Pleasure of Finding Things Out said something about how when he was tutoring his cousin about algebra and said that his cousin was to find the value of 'x'. Feynman went on to the answer and then his cousin said 'but you did it with arithmetic, you have to do it in algebra'. The following was then about how algebraic rules were invented so kids could pass a class even though they didn't know what they were doing and that it didn't matter how you got the answer so long as your answer is correct.

 

My question is, apart from the 'traditional' rules of algebra, in what other ways could you 'do' algebra and also what did it mean when it was said that Feynman did it with arithmetic? Also, how did Feynman 'do' algebra math since he obviously didn't do it the same way that is taught in school (He taught his kids math and the teacher would complain and tell them 'thats not how to do it' and then some math techniques he told other los alamos scientists about).

Edited by lightburst
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Feynman did math as everyone else does it, only faster and better and in complete comprehension of what he was doing.

 

The issue of "arithmetic" was probably a reference to the fact that most high school algebra problems can be solved by putting in specific numbers and looking at the consequences, considering what you want. This is quite true, and like the fact that most introductory calculus problems can be solved by intelligent and competent applications of the min/max principle (if a quantity is split into shares, their product is maximized and their sum is minimized when they are all equal) difficult to apply by people of lesser abilities than Feynman.

 

Feynman's point that the specific rules are worthless (you will never be able to use them in real life) if you don't have enough understanding to handle simple problems without the specific manipulations they require is also valid, but there is a large grey area here: people who need more work to get a handle on the rules, and benefit from the experience of employing those rules in situations simple enough to allow the creation or inculcation of that understanding he values from scratch, so to speak. That is, practice with the "rules" can set up insight, is one way to acquire that understanding.

 

I once had a student who took an entire semester of introductory calculus (derivatives, without doing any calculus at all (that I know of), using instead the methods Isaac Newton used in Principia Mathematica to prove that his calculus techniques were valid and gave the same answers as the geometric methods then used. I passed him, based on his getting enough right answers on the homework and exams (and employing a considerable amount of mathematical insight and ability to do that), but it put me in an odd and interesting ethical position I have not settled in my own mind to this day.

 

The grade, the permanent record I signed my name to, will not be read by others as an estimation of that student's ability to solve problems of a particular type, but of his familiarity with a set of "rules" he in fact may not be familiar with. So which is more important? In some abstract sense, problem solving of course. But that unfamiliarity (if it does describe that particular student) may very well bite him in the ass some day - there's a reason for those rules.

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My question is, apart from the 'traditional' rules of algebra, in what other ways could you 'do' algebra ...

 

For some systems you can develop graphical notation and that can help you calculate and prove things. With Feynman in mind, we have Feynman diagrams in quantum field theory, but other systems do admit similar graphical notions. Another famous example is Penrose graphical notation for tensor fields as well as spinors and twistors.

 

I know other algebraic systems like Hopf algebras also have similar graphical notation. Also there are similar notations in parts of category theory and I am sure elsewhere.

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Different ways to do algebra?

 

Well in part at least it depends upon what you mean by algebra?

 

Many of the traditional 'algebraic' processes can be accomplished by pure drafting techniques.

 

These include addition, subtraction, multiplication, differentiation, integration, fractions, solution of equations and more.

 

Some of these were embodied in the old fashioned slide rule.

 

Again before calculators came, nomograms were popular.

 

Then there are more specialised techniques mostly to do with vector and tensor algebra - link polygons, funicular polygons, Mohr diagrams, Mohr circles, Maxwell diagrams.

 

So tell us what you mean by algebra?

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By algebra, I meant something a highschool or maybe 2nd year college student would understand. I think you guys are mistaking what I call algebra to 'stuff that use algebra', or maybe I'm just ignorant.

 

When I think of algebra, I think of 'finding x', satisfying equations (i.e. solving for f(x) = 0), simplifying, expressing a function in terms of x/y/z, solving for coefficients (i.e. partial fractions), factoring of functions (i.e. x2 + 2x + 1 = (x+1)(x+1)), 'maintaining equality' (i.e. rationalization or multiplying by 1 i.e. (x/x)), and then maybe a few I missed.

 

By 'traditional rules', I meant like 'if you add on one side you add on the other' or 'dividing both sides', which was what Feynman was criticizing. I grew up with these 'rules' and so the idea of 'different ways' to achieve these things in algebra is so confusing and new to me. If Feynman didn't do it this way, then how COULD he have done it?

 

The best 'different' I've come up with was in arithmetic which made it easier for me to count and stuff; counting the way I was taught is HARD. But in algebra, the notion that 'whatever you do to the other side, do to the other as well as to maintain equality' is such a natural idea to me now and 'any other way' is just unthinkable. So I'm asking. smile.png

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I don't think Feynman was criticizing the rules themselves, but rather criticizing the tendency to get caught up in symbol pushing without really understanding what's going on.

 

The fact that we can solve for x in an equation like x + 3 = 7 by subtracting 3 from both sides is useful, but immediately answering, "Well, x must be 4," rather than saying, "Well, x + 3 = 7, so x + 3 - 3 = 7 - 3, so x = 4," is fine (unless you're in a class and the teacher insists you show every step). As overtone mentioned (and as I'm sure Feynman recognized), the specific rules and methods taught in class are taught for a reason. Even for seemingly trivial exercises, the use of a new technique can provide insight into what's actually going on at a fundamental level. But we shouldn't get so caught up in any particular algorithm for finding a solution that we lose sight of the general idea.

Edited by John
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Well the solution of quadratic equations is something a high school student should be learning in algebra.

 

However here is an interesting non algebraic method of solution by geometrical drawing (not by a graph).

 

I have chosen to demonstrate using a specific example but the method may be used with any quadratic and, what is more interesting, extended to an algebraic polynomial of any degree.

 

post-74263-0-01484000-1366918191_thumb.jpg

Edited by studiot
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