I used to perform a lot of mental arithmetic when I was kid and even read some books about different techniques (Vedic mathematics, Trachtenberg system, etc.).

I used to perform a lot of mental arithmetic when I was kid and even read some books about different techniques (Vedic mathematics, Trachtenberg system, etc.). I don't remember the specific techniques from the books these days. Since I can also do multiplication and division in my head, I simply figure out which two consecutive integers when squared bound the number of interest. I then divide the difference between the number of interest and the lower integer by the sum of the two integers and add that fraction to the lower integer to get the square root. For example, if we want to calculate the square root of 1051, I know that 32^2 is 1024 and 33^2 is 1089. Then, 1051^0.5 = 32 + (1051-1024)/(32+33) = 32 + (27/65) ~ 32 + (26/65) + (1/65) ~ 32 + 0.4 + 0.02 ~ 32.42. The square of 32.42 is actually 1051.0564, so that's pretty close.