Let's look at the first one. We're interested in approximating [tex]\sqrt n[/tex], the square root of some natural number. Here we have n=5, so we want to approximate [tex] \sqrt 5[/tex]
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
[tex]x^2 < n < (x+1)^2[/tex]
Once we do that we know
[tex]x < \sqrt n < x+1[/tex]
The squares go
[tex]0^2=0, 1^2=1, 2^2=4, 3^2=9, 4^2=16, 5^2=25,[/tex]
[tex] 6^2=36, 7^2=49, 8^2=64, 9^2=81, 10^2=100[/tex]
We see n=5 is between 4 and 9, i.e. between [tex]2^2[/tex] and [tex]3^2[/tex] so we conclude
[tex] 2 < \sqrt 5 < 3 [/tex]
We might guess [tex]\sqrt 5 \approx 2.5[/tex] 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.
I'll do a couple of more. We see from our list
[tex]64 < 75 < 81[/tex]
[tex]8 < \sqrt{75} < 9[/tex]
The 75 is closer to 81 than 64, so let's bump up 8.5 to say
[tex] \sqrt{75} \approx 8.7[/tex]
OK, one more, n=17
[tex]16 < 17 < 25[/tex]
[tex]4 < \sqrt{17} <5[/tex]
17 is real close to 16 so instead of 4.5 let's go with
[tex]\sqrt{17} \approx 4.1 [/tex]
