ToohrVyk: I think you misunderstood my point. What I'm saying is that thinking the problem out, as the initial solutions centre on, is a far better general method, which will always be better than applying a solution that relies on a specific method.
These methods, while right and for many people useful, only limit what you can do.
stuck on extremely simple equation
[Argh, I just understood your point. You're right about the fact that you need to go beyond the particular solutions, and the larger the mathematical culture, the easier it is to adapt general solutions to particular cases, or to find a particular solution that is adapted. I'll leave my previous response for... entertainment purposes]
I entirely disagree with you, because of two things.
First, to consider a linear equation y = ax+b is also a general solution (it is, in fact, an interpolation polynomial method applied to a degree one polynomial) that also works for problems that linear algebra has many troubles [EDIT: solving] (such as interpolating between N points with a C-infinite easily integrable function). Sure, you may argue that usual problems in the computer world involve linear equations, not polynomials, but this restriction is just going against the generality of your solution ("it's more general in this particular case").
Also, even if it was a more general solution, this does not make it absolutely better than any other solution. Actually, most of the time it is the particular solutions that are used, rather than the general ones. Why? Because the more information you know about a problem, the faster it is to solve it, and by definition, general solutions "know" less about the problems than those particular solution that actively assume things.
I'd just like to finish by saying that using method A for one problem does not prevent you from using methods B, C, D... for other problems. There's nothing like a law in mathematics that says "thou shalst not change theorems once in a lifetime". For each problem, find the least general solution that applies to that problem (or even create one), and use it: in most cases it will be faster than the general method.
I entirely disagree with you, because of two things.
First, to consider a linear equation y = ax+b is also a general solution (it is, in fact, an interpolation polynomial method applied to a degree one polynomial) that also works for problems that linear algebra has many troubles [EDIT: solving] (such as interpolating between N points with a C-infinite easily integrable function). Sure, you may argue that usual problems in the computer world involve linear equations, not polynomials, but this restriction is just going against the generality of your solution ("it's more general in this particular case").
Also, even if it was a more general solution, this does not make it absolutely better than any other solution. Actually, most of the time it is the particular solutions that are used, rather than the general ones. Why? Because the more information you know about a problem, the faster it is to solve it, and by definition, general solutions "know" less about the problems than those particular solution that actively assume things.
I'd just like to finish by saying that using method A for one problem does not prevent you from using methods B, C, D... for other problems. There's nothing like a law in mathematics that says "thou shalst not change theorems once in a lifetime". For each problem, find the least general solution that applies to that problem (or even create one), and use it: in most cases it will be faster than the general method.
I'm not saying that you should choose a more general method. I'm saying that you shouldn't confine yourselves to methods at all. In doing so you leave yourself open to tripping up on an easy problem. The best maths exams try to test your mathematical ability and not how well you can carry out a certain method. They'll give problems that are certainly solvable using a given method, but there will be an easier way, a shortcut.
Clearly methods are important. But they should never be relied upon. They are clearly useful when mathematical ability isn't needed. For instance programming the algorithm into a computer. But there will always be situations that trip people up.
Clearly methods are important. But they should never be relied upon. They are clearly useful when mathematical ability isn't needed. For instance programming the algorithm into a computer. But there will always be situations that trip people up.
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