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« Last post by STxAxTIC on Yesterday at 09:08:00 PM »
Hi Steve (and all)
I just stumbled onto the forums today and saw this. It made me wonder if I knew a cool method to get square roots that does not involve iterative calculus, graphs, symbols - just good old fashion farm math. I'm happy with what I came up with and will scribble it here before I embed it in a PDF.
To review the problem, we are pretending not to know the square root of 60, but meanwhile we know:
7 * 7 = 49
8 * 8 = 64
Using the method Steve describes, we come up with sqrt(60) ~ 7.7333, where squaring this gives some hint at the accuracy:
:7.7333^2
:+59.80392889
59.8/60 isn't bad, but I would rather see something much closer to 60.0. So that's the mission now.
I will start with the same starting information. The final answer will be a formula that has fewer side calculations that the Steve method, and I could just blurt that answer out and drop the microphone - but I will derive the answer first, and put a box around it. I insist on explaining why it works though.
Let me rewrite the known information, along with a way to frame the question:
7 * 7 = 49
(8-x)*(8-x) = 60
8 * 8 = 64
Like Steve's setup, the first key here is already knowing the square root of the "next" integer. In this case, we need to know 8*8=64, and that our answer is going to be something less than 8. Whatever the answer is, I write it as 8-x, so I know x is a number less than one. Read that twice: The number x is less than one. The number 8-x solves the problem.
To proceed, take that middle equation and FOIL out the left side:
(8-x)*(8-x) = 60
64 - 16x + x^2 = 60
Now, since x<1 as given, it follows that x^2 is much less than all of the other numbers in the equation, and we effectively ignore x^2. At this point, all "equal" signs are really approximations, just to be clear. Anyway, we can solve for x with no quadratic funny business:
16x = 64-60 = 4
... And it's evident that x = 1/4 = 0.25. And we're done. The solution is 8-0.25 = 7.75. Testing this number, we get:
:7.75^2
:+60.0625
Which is much closer to 60.
And that's it - the whole method is demonstrated. In the general case, instead of 60, suppose want to find the square root of Q^2. If the solution is proposed as N-x, then x is equal to N/2 - Q^2/2N, or N-x = N/2 + Q^2/2N. In summary - here is the box with the answer:
Q = N/2 + Q^2/2N
where N is the integer just greater than Q.
To see the sqrt(60) example all over again, just identify Q^2=60 and N=8. Then Q = 4 + 60/16 = 7.75.
For another example, calculate the square root of 1000. So Q^2=1000, and N has to be 32, because 32^2=1024 (but 31^2=961). Plugging these into the formula:
Q = N/2 + Q^2/2N = 16 + 1000/64 = 31.625
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