Question

What is the approximate value of `\sqrt { 107}`?

Answer 1

`sqrt(107) ~~ 31/3 ~~ 10.33`

Explanation:

Note that:

`10^2 = 100`

`11^2 = 121`

`107` is exactly `1/3` of the way between `100` and `121`.

That is:

`(107-100)/(121-100) = 7/21 = 1/3`

So we can linearly interpolate between `10` and `11` to find:

`sqrt(107) ~~ 10+1/3(11-10) = 10+1/3 = 31/3 ~~ 10.33`

(To linearly interpolate in this example is to approximate the curve of the parabola of the graph of `y=x^2` between `(10, 100)` and `(11, 121)` as a straight line)

Bonus

For more accuracy, we can use:

`sqrt(a^2+b) = a+b/(2a+b/(2a+b/(2a+...)))`

Putting `a=31/3` we want:

`b = 107-(31/3)^2 = 963/9 - 961/9 = 2/9`

Then:

`sqrt(107) = 31/3+(2/9)/(62/3+(2/9)/(62/3+(2/9)/(62/3+...)))`

So as a first step of improvement:

`sqrt(107) ~~ 31/3+(2/9)/(62/3) = 31/3+1/93 = 962/93 ~~ 10.3441`

If we want more accuracy, use more terms:

`sqrt(107) ~~ 31/3+(2/9)/(62/3+(2/9)/(62/3)) = 31/3+(2/9)/(62/3+1/93) = 31/3+(2/9)/(1923/93) = 31/3+62/5769 = 59675/5769 ~~ 10.34408043`