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To approximate , we can use this equation
In this case our radicand is R = 17, so the equation will update to
See this page for more info. In that link, scroll down until you reach the subheading labeled "Heron's Method". This is not to be confused with the Heron triangle area formula.
The initial x value is a guess.
Let's say we pick x = 4.5 since it's between 4 and 5 (note that 4^2 = 16 and 5^2 = 25; and 16 < 17 < 25)
We feed that guess into the equation. The result is then the next guess to try.
The process keeps going until both columns agree on the same decimal digits.
Table of values
x
y = 0.5*(x+17/x)
4.5
4.13888888888889
4.13888888888889
4.12313571961223
4.12313571961223
4.12310562572749
I have highlighted in red where the two columns agree on the partial decimal representation. We can confidently conclude that sqrt(17) = 4.123 when rounding to 3 decimal places.
Note that
4.123^2 = 16.999129
4.1231^2 = 16.99995361
both of which can be found by hand (given enough time)
(