Posted: 6/2/2011 4:19:28 PM EDT
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√72
No calculators, no cheating! 
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I never did get high marks in geometry class because I never got the point of simplifying the equation when it was less effort to simply solve the damned thing. Geometry proofs were hard for me as a result. Getting the right answer....easy. Writing it as a proof...not so much. I just never saw the POINT. Try working the answer back to the original question. If you get the original question, the answer must be the right one. All that stage-by-stage simplification is just mental masturbation with no practical application. I guess it proves that you understand the order of operations, but once you've shown that, why keep doing it? CJ |
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Quoted:
I never did get high marks in geometry class because I never got the point of simplifying the equation when it was less effort to simply solve the damned thing. Geometry proofs were hard for me as a result. Getting the right answer....easy. Writing it as a proof...not so much. I just never saw the POINT. Try working the answer back to the original question. If you get the original question, the answer must be the right one. All that stage-by-stage simplification is just mental masturbation with no practical application. I guess it proves that you understand the order of operations, but once you've shown that, why keep doing it? CJ Because it very well might be faster and simpler to mentally break it down to 6*rt2. I know that the square root of 2 is 1.414. More importantly, rt2 appears in everything from geometry to electrical circuits. Quite often, in fact. It might very well be advantageous to keep the rt2 factor in place, as it will become easier to solve further down the line. For example, let's say you had an isoscles right triangle (for the sake of argument, let's say it's a roof frame), and you knew the hypotenuse had to be rt72. You want to find the length of a side so you can build the thing. You could whip out your calculator and turn that into 8.485. The multiply by Sin45, and you get 5.999 because of the rounding error. If you only use 3 significant digits of Sin45, your value drops to 5.998. And those errors compound the less accuracy you carry. Or you could just reduce it to 6rt2. Since I know Sin45 is rt2 over 2, I just mentally cancel (rt2 * rt2)/2 =1. And I know each side is exactly six. Or maybe I just don't have a calculator and I need an approximate decimal measurement. I don't know the square root of 72. No idea. But I can remember rt2 = 1.414. rt3 = 1.732 Just commit those to memory and you're gold. Sure, occasionally you run into other prime roots, but for most sciences, rt2 and rt3 will do most of it. Now now, instead of finding the square root of 72 in my head, I'm just trying to multiply 6 times 1.414. Which is a hell of a lot easier. Even if you just do 6*1.4 = 6+6*0.4 = 6+2.4 = 8.4 that gets you really close. If we're talking an inch measurement on a noncritical part, that very well may be accurate enough. Do it longhand and you retain four significant digits. Not bad for a mental calculation, right? |
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Quoted:
I never did get high marks in geometry class because I never got the point of simplifying the equation when it was less effort to simply solve the damned thing. Geometry proofs were hard for me as a result. Getting the right answer....easy. Writing it as a proof...not so much. I just never saw the POINT. Try working the answer back to the original question. If you get the original question, the answer must be the right one. All that stage-by-stage simplification is just mental masturbation with no practical application. I guess it proves that you understand the order of operations, but once you've shown that, why keep doing it? CJ Two reasons: 1) Before the age of scientific calculators, computers, etc. it was simpler to solve by breaking it down into it smore simpler components. That way slide rules, data tables that included the square roots of certain numbers, etc. could all be, well, more portable. 2) Learning to do proofs and other functions is important to training the mind to solve problems in a logical, method based process. Kinda like when they teach computer programming you usually start with a very basic programming language with simple logic, and go from there. |
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Quoted:
I never did get high marks in geometry class because I never got the point of simplifying the equation when it was less effort to simply solve the damned thing. Geometry proofs were hard for me as a result. Getting the right answer....easy. Writing it as a proof...not so much. I just never saw the POINT. Try working the answer back to the original question. If you get the original question, the answer must be the right one. All that stage-by-stage simplification is just mental masturbation with no practical application. I guess it proves that you understand the order of operations, but once you've shown that, why keep doing it? CJ Then you've never done a moderately long problem. When you have a bunch of expressions like sqr 72 that are in the same problem, you want them to be consolidated at the end to one expression, otherwise the rounding errors compound. Furthermore, you can eliminate the rounding errors entirely if you can move the expressions around in a more complicated equation and either cancel them out or simplify them into something without an irrational solution. If we used your mindset and just stopped, we would never have progressed to just understanding that the acceleration of gravity is always 9.8 m/s^2, but instead would be multiplying in the weight of the object to determine the force, then dividing it back out to get the acceleration. |
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Quoted: Quoted: I never did get high marks in geometry class because I never got the point of simplifying the equation when it was less effort to simply solve the damned thing. Geometry proofs were hard for me as a result. Getting the right answer....easy. Writing it as a proof...not so much. I just never saw the POINT. Try working the answer back to the original question. If you get the original question, the answer must be the right one. All that stage-by-stage simplification is just mental masturbation with no practical application. I guess it proves that you understand the order of operations, but once you've shown that, why keep doing it? CJ Then you've never done a moderately long problem. When you have a bunch of expressions like sqr 72 that are in the same problem, you want them to be consolidated at the end to one expression, otherwise the rounding errors compound. Furthermore, you can eliminate the rounding errors entirely if you can move the expressions around in a more complicated equation and either cancel them out or simplify them into something without an irrational solution. If we used your mindset and just stopped, we would never have progressed to just understanding that the acceleration of gravity is always 9.8 m/s^2, but instead would be multiplying in the weight of the object to determine the force, then dividing it back out to get the acceleration. I do perform the operations, to reduce to simpler terms, but I never was much of a fan of bothering to write out the proofs. It's mostly a case of "I know how to do it but won't be bothered with laying it out for you in the form of a formal proof." As is typical, I always scored well on the tests that relied on getting the right answer. Doing the reductions and iterations, no problem. Bothering to write the proofs? I never really cared to do it. CJ |
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Quoted: I just never saw the POINT. Try working the answer back to the original question. If you get the original question, the answer must be the right one. All that stage-by-stage simplification is just mental masturbation with no practical application. I guess it proves that you understand the order of operations, but once you've shown that, why keep doing it? CJ Because you can find the root of pretty much any number in your head by memorizing the roots of a few small numbers. |