Expert: Josh - 1/4/2005

Question
I cannot remember how to do the arithmetic to find the sq root of a number, e.g. I know the sq root of 625 is 25; but how to do the arithmetic to get 25. Can you help me?
   I am so embarrassed with myself because this is simple elementary maths...DK

Answer
Hi Dave,

I am very sorry for this late reply.
I thought I had entered the right date for my vacation on the system, but apparently questions are still getting through.

A brilliant hindu mathematician -- Srinivasa Ramanujan -- solved many unusual and difficult mathematical problems around the 1720's. I'm sure one of the tricks that he touched on later opened up the field of "Vedic mathematics". [you may wish to check up on that using a search engine]

I am not aware of any approach taught in mainstream mathematics, other than the trial-and-error approach to finding the square roots of numbers. Most people acquire familiarity with these combinations through experience and become gradually more aware of the magnitude of square numbers. Of course, there are rules which eliminate many impossibilities. For instance, the product of two even number cannot yield an odd number and so forth.

One may also conduct a binary search.
e.g., to find x such that x*x=1369
You know that 10*10=100 and 100*100=10000.
This is somewhere in between.
So, you can start somewhere in the middle, say, with x=50.
Immediately, you see that 50*50=2500 which is larger than 1369. So, you may try to halve x from 50 to 25.
Now, 25*25=625, which is smaller than the target 1369.
So, we increase this to the mid-value between say, 25 and 50 (let's pretend we pick 37).
This process is repeated until we finally converge to the answer. In this case, it turns out that we hit the right answer at the third attempt. i.e., x indeed equals 37.

You see, this is a much faster way than performing an exhaustive, linear search. Ramping up from x=10 to x=11 to x=12,..., and so on, until we reach 37. This method is what programmers and scientists tend to use.

Hope this answers your question.

Sorry once again for the late reply. I didn't know you could reach me while I was away.

Cheers,
Josh