Find square root of 6 by the Babylonian algorithm:
Since x^2 = 6,
x = 6/x, or taking the average of those two,
x = (x+6/x)/2
We can try a starting guess of 2: the right side produces
(2+6/2)/2=(2+3)/2 = 5/2 : this is an improvement! Use it as a new starting point. The right side produces
(5/2 + 6/(5/2))/2 = (5/2 + 12/5)/2 = 49/20 = 2.45, not all that different from 5/2=2.5, and BETTER. Use it as a new starting point. The right side produces
(49/20 + 6/(49/20))/2 = (49/20 + 120/49)/2 = 4801/1960 = 2.4494897959..., very close to 2.45, and even better. Use it as a new starting point. The right side produces
(4801/1960+6*1960/4801)/2 =46099201/18819920=2.44948974278...
The square of this is 6.0000... (at least 13 zeros!), so it is extremely accurate.
If you try this scheme on something like square root of 9, which we know is 3, and use 3 as a starting guess, you just confirm that 3 is right:
Evaluate (x+9/x)/2 with x=3: you get (3+9/3)/2=(3+3)/2=3. No surprise there.
But try it with a starting guess of 2:
(2+9/3)/2=5/2 = 2.5
(5/2+9/(5/2))/2=(5/2+18/5)/2=61/20=3.05 (getting close!)
(61/20+180/61)/2 = (7321/1220)/2=7321/2440=3.0004098...
(7321/2440+21960/7321)/2=(107179441/17863240)/2=3.00000002799...
It approaches the true answer 3, but never actually gets there. It will eventually look like 3 on any finite machine, if you use a computer or calculator, because the difference will be too small to be kept.