Posted by mysterychicken on Friday, June 5, 2009 at 12:18pm.
You can use the same method of interpolation requested by the earlier question.
A higher accuracy required can be obtained by choosing a better initial guess.
In this case, we choose 33 (*33=1089, <1100) and 34 (*34=1156, >1100).
33*33=1089
34*34=1156
sqrt(1100)
=33+(34-33)*(1100-1089)/(1156-1089)
=33.164
The precise value is 33.166.
More accuracy and less work can be obtained by Newton's method.
First we choose a close estimate, for example, 33*33=1089
Then we improve the estimate by adding a correction, which is obtained by subtracting the square of the estimate from the target, and divide the difference by twice the estimate.
This can be repeated to get still better results.
Startin estimate = 33
33*33=1089
next estimate
33+(1100-1089)/(2*33)=33.1667
The precise value is 33.1662
Thanks a million!!
-MC
mc, what is the name of your text? The NCTM almost 17 years ago decided that a calculator based curriculum was the way to go in math education, and these types of problems (and the skills) were decided to be forgone. Somehow, your text has ignored this. Frankly, I dont know of anyone teaching this anymore. I stimulates my curiosity. Apparently, some homeschoolers are still doing it, and preaching mastery.
http://www.homeschoolmath.net/teaching/square-root-algorithm.php
Thanks for letting me know the text name.
MC,
Are you supposed to find the square root by manual method similar to a long division? I am curious, because this is the best method to be used if a known precision is required, and the question did ask for a precision of a hundredth.
If was not clear in this question how it should be done. Same as Bob's case, I am not sure if the method is in the curriculum any more.
rough appr. of √1100 is 30
1100/30 = 36.6666
average of 30 and 36.666 = 33.3333
1100/33.333 = 33
average of 33.333 and 33 is 33.16666
1100/33.1666 = 33.1658
average of 33.1658 and 33.16666 is 33.166
so √1100 to the nearest hundreth is 33.17
I used Newton's Method.
And of course a calculator to do the divisions. Of course this didn't make much sense, why not just use the calculator to find √1100 if I use the calculator anyway.
I can see the point of the above question had they asked for the approximation to the nearest whole number, but to ask for an "approximation" accurate to 2 decimal places sort of defeats the purpose of the approximation concept.
I'm just doing the problems my homeschooling gives me...i have no clue how outdated it is.
-MC
Glad to see 3 math enthusiasts all agreeing on this one !
MC,
Here's a link to the manual square root method, a process similar to a long division. It is a method worthwhile learning because you can get the answer to a specified accuracy without a calculator.
Here's the link:
http://www.itl.nist.gov/div897/sqg/dads/HTML/squareRoot.html
Thanks
-MC
You can find many different algorithms here:
http://en.wikipedia.org/wiki/Methods_of_computing_square_roots
"The digit by digit method" mentioned by MathMate and Bob is explained here too.
Another method that is mentioned here that can be used without a calculator is the Newton-Raphson method for
1/sqrt(x), as this is free of divisions.
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