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Domain: sosmath.com
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Comments · 10
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First. World. Problems: Paper isn't wasted
First. World. Problems: We no longer waste paper to print archaic Mathematical tables
You know what else is "dead" ?
* Slide rule
* Tables of common Logarithms
* Tables of Trigonometric functionsGuess what, nobody is stopping you from buying those tables from old CRC Handbook of Chemistry and Physics which have them.
Apparently you didn't get the memo that a cheap calculator is "good enough."
What's next?
Whining that we don't have rotary telephones? Black and White televisions?
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Re:Next up: Lag emulation
Try construction. I'm talking about old-school architectural engineers here. I recently saw somebody do it with a fairly simple formula to give an approximation on something. The younger engineer had to find and unwrap his scientific TI calculator and then was figuring out typing in the correct syntax of the formula when the old guy was already done doing it on paper. He kept a sheet (one can be found here: http://www.sosmath.com/tables/trigtable/trigtable.html) in the back of his notepad. The old guys on the project meeting were then reminiscing for a while about the good ol' days and how those papers are cheap to replace and can be used in any construction environment.
Funny to say but there are notepads still being sold (I forgot where, I should've picked it up when I saw it) with a trig table printed in the back. My wife is doing Six Sigma and with her training she got a neat little pocket book that has all sorts of conversion tables, trig tables, translations, formulas and standards in it so apparently people still use good-ol paperback to make quick calculations on the go without having to find a calculator.
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Re:Navigating by compass is obsolete?
That's got nothing to do with log tables. It requires understanding of logarithms, but that's not the same thing.
This is a log table. It's the equivalent of a sine or cosine or tangent table - it's almost totally useless if you have a working calculator. -
Re:3 ideas
I like sosmath.com.
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Re: A better question:
How does this function compute 1/x^(1/2)?
It starts by taking a guess at the right answer, and then improving the guess until it's accurate enough to use.
The first step depends heavily on the fact that a floating point number on a computer is represented as a significand (aka mantissa) and an exponent (a power of two). For the moment, consider taking just the square root of X instead of its inverse. You could separate out the exponent part of the floating point number, divide it by two, and then put the result back together with the original significand, and have a reasonable starting point.
From there, you could improve your guesses to get a better approximation. The simplest version of that would be like a high-low game -- you split the difference between the current guess and the previous guess, and then add or subtract that depending on whether your previous guess was high or low. Eventually, you'll get arbitrarily close to the correct answer.
This can take quite a few iterations to get to the right answer though. To improve that, Newton-Raphson looks at the curve of the function you're working with, and projects a line tangent to the curve at the point of the current guess. Where that line crosses the origin gives you the next guess. That's probably a lot easier to understand from picture.
In this case, we're looking for the inverse square root, which changes the curve, but not the basic idea. As a general rule, the closer your first guess, the fewer iterations you need to get some particular level of accuracy. That's the point of the:
i = 0x5f3759df - (i>>1);
While the originator of this constant is unknown, and some of it is rather obscure, the basic idea of most of it is fairly simple: we start by shifting the original number right a bit. This divides both the mantissa and the exponent part by two, with the possibility that IF the exponent was odd, it shifts a bit from the exponent into the mantissa. The subtraction from the magic number then does a couple of things. For one thing, if a bit from the exponent was shifted into the mantissa, it removes it. The rest of the subtraction is trickier. If memory serves, it's based on the harmonic mean of the difference between sqrt(x) and (x/2) for every possible floating point number of the size you're using.
This is where the fact that it's 1/sqrt(x) instead of sqrt(x) means a lot: 1/sqrt(x) is a curve, but it's a fairly flat curve -- much flatter than sqrt(x). The result is that we can approximate a point on the curve fairly accurately with a line. In this case, it's really two lines, which gets it a bit closer still.
From there, the number has had a bit of extra tweaking done -- it doesn't actually give the most accurate first guess, but its errors are often enough in the opposite direction from those you get in the Newton-Raphson iteration steps that it gives slightly more accurate final results. -
Re:Know your math department
What Gauss did do was prove was The Fundamental Theorem of Algebra, showing that there is no need for other numbers next to the real and imaginary.
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Re:Population Control
"Animals have the ability to continously procreate until all resources are consumed, however, most don't."
BullShiznit. Animals do not limit themselves, they are limited their environment. Resources are not the only factor. Predators, disease, birth rate, death rate, territorial insticts, and many other factors play a part. Please see http://dieoff.org/page80.htm for a concrete example of what happens when a species is introduced into an environment where resources are the only limiting factor. This is the classic example that I guaruntee will be brought up, should you take a university level environmental studies class.
The growth of animal populations can be predicted: http://www.sosmath.com/diffeq/first/application/po pulation/population.html. Accuracy varies on how many of the environmental factors that will limit population growth are known, and whether the known factors are the most important.
Even humans do not consciously maintain a population that will not exceed resources. Other factors limit human populations, such as food, water, disease, all of the limitations on animals - plus economics, birth control, and numerous other societal factors. But now we're getting into Sociology 101 - lets get back to Environmental Studies 101...
"Would machines follow this same type or universal standard of population control or would they just envelope every item they could?"
Would machines follow the same universal standard of population control? Yes. That universal standard is that they will continue to reproduce until some environmental factor limits them. Resources, predators, death rate, birth rate, etc. So if the machines were very robust, lasted a long time, had a high reproduction rate, could eat anything, could migrate long distances, and had no predators - then its grey goo for us.
This seems a rather unlikely scenario to me. Could grey goo be a serious problem when the conditions are right? I would say its likely that someday, somewhere, grey goo will be a problem. And then they'll send out some guy in a truck to spray machine poison, and things will get back to normal. -
Re:Some I gave during interviews
Perhaps you could post the answer for us? I honestly was never taught how to PROVE the theorem, I was only taught to memorize it...in 8th grade, I distinctly remember, to the tune of "pop goes the weasel". Now that I'm suddenly realizing that I don't have a clue how to solve it, it's disturbing me! I've had years of calculus and just finished fourier analysis, I should really know this.
Found the solution here. Whew, that's definitely pretty simple, I've been taking complex math too long... -
SOSmath.com
Though I'm not in the same situation as you, SOSMath is a GREAT reference that I've used many times to "remember" things such as how to solve differential equations and matrix multiplication properties. good luck!
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smartplanet
Back when Smart Planet was still ZDnet's "ZD University" or www.ZDU.com, I took a free interactive PERL class. I was very impressed, especially since it was free. I don't believe that any classes there are still free, but they were reasonalble priced. Unfortunately their prices have gone up. recently, but I will seriously consider taking more classes online some time in the future.
It would be great if there were more interactive sites for learning stuff because I think it's a great way to learn, (especially computer programming)
Another place that might be good (haven't tried it yet) is www.sosmath.com math website. It appears to be a free website where you can learn stuff like Trig, Calculus, and Diff. E.Q.
Great topic. I can't wait to see what others have to say.