2 Answers

Urb · Answer 1 · 2019-07-24T08:25:47+0000

Answer:

Estimate: Get as close as possible to the number you're trying to square root by finding two perfect square roots that give a close number. Divide: Divide your number by one of the square roots you've chosen from the previous step. Average: Take the average of step 2 and the root

Marking · Answer 2 · 2019-07-30T20:10:55+0000

Let's look at the first one. We're interested in approximating
$\sqrt n$ , the square root of some natural number. Here we have n=5, so we want to approximate
$\sqrt 5$

A good way to get at least the integer part of the approximation is to sandwich n between two perfect squares, i.e. to find a natural number x such that

x^2 < n < (x+1)^2

Once we do that we know

$x < \sqrt n < x+1$

The squares go

0^2=0, 1^2=1, 2^2=4, 3^2=9, 4^2=16, 5^2=25,

6^2=36, 7^2=49, 8^2=64, 9^2=81, 10^2=100

We see n=5 is between 4 and 9, i.e. between
2^2 and
3^2 so we conclude

$2 < \sqrt 5 < 3$

We might guess
$\sqrt 5 \approx 2.5$ and then adjust our guess downward because 5 is a lot closer to 4 than it is to 9. They went with 2.2, good enough I suppose.