how to find square root of big number
Answers
Answer:
I'm going to take a different tack than the earlier answers.
First, most numbers don't have whole number square roots, so I'm going to talk about getting close to a square root - without finding its factors, and even without a calculator, if you can manage a little mental arithmetic (or manual arithmetic). Secondly, if it is a problem that has a whole number square root, this gives you a way to quickly guess at it.
Step 1: mentally ignore an even number of digits to the right, so you only look at 1 or 2 digits to the left:
6 _ _
Step 2: Find the largest integer whose square is smaller than this new number (in this case, 22=4 and 32=9, so it's 2.
This is the first digit of the square root.
Step 3: It will have as many additional digits as half the number of digits you "left off" above (which was two: " _ _"), so in this case it has one more digit.
Step 4: Now if the original is the square of a whole-number, you can see the second digit must be either 4 or 6, since those are the only end-digits that produce a 6 in their square (the squares of 0 to 9 end in 0,1,4,9,6,5,6,9,4,1). So the square root in that case must be either 24 or 26. If you then know the trick for squaring numbers ending in 5 (252=20×30+25=625), it's immediately clear it must be 26, which is easy to find the factors of, so this also speeds up the identification of the factors of the original square number.
But as I was suggesting, most numbers you encounter don't have whole-number square roots. So one way to get close is to guess at it and then improve the guess by averaging the guess with the original number divided by the guess.
Since we know the first digit is 2, the number is at least 20 and below 30. So let's start by guessing "25".
Then we improve the guess by averaging 25 and 67625=27125. The average is 26150 (halfway between 25 and 27 plus half the leftover fraction). However, these second-guess averages are always a little too big for the square root, so while 26150=26.02 is a good next guess, it might help to round down to 26 ... whereupon we find when we try to calculate the next guess that we have hit it exactly. In cases where it's not a whole number we can hit exactly, a couple of rounds of such dividing and averaging can get really close quite rapidly.
Let's try this on a bigger problem which we're told has a whole-number square root: 375769
Leave off an even number of digits on the right, to leave one or two digits:
3 7 _ _ _ _
The first digit must be 6 (62=36). We left off 4 digits, so the square root has two more digits.
Since 62 is quite close to 37, guess "600".
we want to average 600 and 375769600 ... but we don't have to get it exactly, so in that second term let's round off the numerator to the nearest 100: 375800600=37586=626.<something>. Now average with the 600 we started with to get our second guess: 613.<and a bit> ... but this number will be slightly too high.
Clearly the last digit must be either 3 or 7 (since they are the only digits whose square ends in 9).
so we need a number a bit smaller than 613.<and a bit> that either ends with 3 or 7. The obvious one to try is 613 ... which is the required number. (If 613 wasn't the square root, it would either be 607 or we made an error somewhere. With practice this sort of thing can be very rapid.)
Note also that there are only 22 three digit whole-number squares. It's surprisingly quick to learn them all (there are several patterns that can be exploited to speed that up); this will speed up identifying square roots and can provide numerous shortcuts to calculations like the one we just did -- and to other calculations as well (e.g. 17×19=182−1=323).
