Question

How do you find a one-decimal place approximation for `sqrt 18`?

Answer 1

Use Newton Raphson method to find

`sqrt(18) ~~ 4.2` to one decimal place.

Explanation:

To find an approximation for the square root of a number `n`, start with a reasonable approximation `a_0` and apply the following formula to make a better approximation. Repeat to get better approximations.

`a_(i+1) = (a_i^2+n)/(2a_i)`

In our case `n = 18` and it makes sense to choose `a_0 = 4` since `4^2 = 16` is fairly close.

Then:

`a_1 = (a_0^2+n)/(2a_0) = (4^2+18)/(2*4) = (16+18)/8 = 34/8 = 17/4 = 4.25`

Unfortunately `a_1` is right in the middle between `4.2` and `4.3`, so let's try another iteration...

`a_2 = (a_1^2+n)/(2a_1) = ((17/4)^2+18)/(2*(17/4))`

`=(289/16+18)/(17/2) =(289+18*16)/(17*8)=(289+288)/136`

`=577/136 ~~ 4.24265`

That's more decimal places than we need, but at least we can say with confidence that `sqrt(18) ~~ 4.2` to one decimal place.

Alternatively, you can use the method described in http://socratic.org/questions/given-an-integer-n-is-there-an-efficient-way-to-find-integers-p-q-such-that-abs-
to find a continued fraction expansion for `sqrt(18)` ...

`sqrt(18) = [4;bar(4,8)] = 4+1/(4+1/(8+1/(4+1/(8+1/(4+...)))))`

Then you can approximate `sqrt(18)` by truncating the continued fraction.

For example:

`sqrt(18) ~~ [4;4,8] = 4+1/(4+1/8) = 4+8/33 = 140/33 = 4.dot(2)dot(4)`