How is this a 'lost art'? I had to do this in college. Didn't we all?
https://en.wikipedia.org/wiki/Binary_search_algorithm
rather than representing numbers or performing arithmetic in base 2.
sqrt(7720.17)
first guess 50:
2500
Go up
second guess 100:
10000
too high go down
third guess 75
5625
too low go up
fourth guess 86 (for ease of calculation)
6400 480 + 480 36
7396
too low go up: 93
8529
too high go down: 90
8100
too high go down: 88
7744
Etc.
It's not a great method, but it's what first came to mind. I do it sometimes in my head for a quick estimate.
7720 in binary is 1111000101000. This is 13 bits. Square root is at most half as long, so first guess is 10000000 in binary, or 128.
Square that, realize that it's too high, set the highest bit to 0 and next bit to 1.
Square 1000000 binary or 64, realize that it's too low. Keep the top bit, raise the next bit.
Square 1100000 binary or 96, too high, drop that bit back to zero, raise the next bit.
Square 1010000 binary or 80, too low, raise the next bit.
Square 1011000 binary or 88, too high, drop this bit and raise the next one.
Square 1010100 binary or 84, too low, raise the next bit.
Square 1010110 binary or 86, too low, raise the next bit.
Got 1010111 binary or 87.
I've never heard of anyone below the age of 30 learning in school how to do them manually.
(Of course, I learned how to calculate square roots on paper before I went to college, but only because I asked my father to show me how.)
nyah!