this is my first post here and I wanted to run something by people who are more knowledgeable than I. In the past couple days I've decided that I need to teach myself more advanced mathematics as it is of great importance to my studies in physics, astronomy and biology. In school we never went past algebra 1 and geometry and that has proven to be a hindrance. So I decided to brush up on the basic math that I learned in school and then move on to more advanced maths. I decided to make up a bunch of problems to solve while waiting for my books to arrive and I ran into my old nemesis, finding square roots of large numbers. I was taught prime factorization, which didn't work for this number, and the Babylonian Method, which can yield accurate results depending on the accuracy of your initial estimate and how many times you're willing to run the equation. What I wanted was an equation that used simple math and a known number instead of an assumed estimate (that must be high in the case of the Babylonian Method) that didn't have to be high or low that would yield an equally/greater accurate estimate of the square as does the BM, and would only have to be run once. I realize this is simple mathematics compared to what is normally discussed here, so I apologize, and also please forgive me if my terminology isn't quite correct. What I came up with was
in this S is the number we are trying to find the square of and e is the closest perfect square root regardless of weather it is high or low.
For example, using 1,863 the number I was initially trying to find the square of, we know that the closest perfect square root is 43. So it would look like this
Actual square of 1863=43.1624
so the estimate it yielded was very close. I've run a bunch of numbers through it both large and small and so far it appears to give better estimates than the BM (depending on the accuracy of your initial high estimate and number of times you run it through the equation) and without the requirement that the estimate is high, and also we don't have to make an initial estimate ourselves as the starting estimate is fixed for us by the nature of the equation.
What do you guys think? Is there something I've missed or perhaps is there an easier equation like it that I am not familiar with?