Anne Wacker
2021-12-14
Carl Swisher
Beginner2021-12-15Added 28 answers
There is no simpler form for 13's square root because it is a prime number, which is the explanation. is an irrational number somewhere between and
A good first approximation using linear interpolation would be:
From our initial guess, we can obtain more accurate ones. (call it ) utilizing a Newton-Raphson approach.
Typical formula for obtaining a closer approximation to would be:
I prefer to separate into numerator and denominator . So and we can iterate using the formula:
In our example and we find:
Our approximate figure if we stopped here would be:
Let's try one more iteration:
Stopping here, we have:
Using a calculator:
William Appel
Beginner2021-12-16Added 44 answers
Explanation:
Look for a generalised continued fraction of the form:
Multiply both ends by to find:
Hence:
Choose a reasonable approximation that is somewhat smaller than to ensure that our generalized continuous fraction converges rapidly .
Note that In order to find a good approximation, we use linear interpolation:
Note also that:
So let and to get:
To obtain an accurate estimate that is rational, we can truncate this continuous fraction.
For instance:
Or:
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