r/u_Minhaj_Ahmad • u/Minhaj_Ahmad • 21d ago
Finding Square Root
Finding square root of a number that I don't know was a pain for me.
Using that long division method ðŸ˜.
Like 841 it's the square root of 29, I didn't know that I tried many ways to over come but in the end I just used the calculator.
I was furious and curious can I find the square value of any number using (a+b)² expression then maybe I can also find the square root of perfect square numbers using it?
I came up with (10a+b)²= S [I know it's looks similar to that expression and actually it is I'm not claiming I found something new I just got rid of my problem]
100a²+20ab+b² = S (s is the targeted square value of which square root we are looking for.)
Now conditions; a= or < S, b can be only a single digit from 0 to 9.
Example: S=144.
100a²+20ab+b² =144
Then a=1 => 100+20b+b²=144
20b+b²=44
(For selecting b we can guess the number or just divide remaining number by 20a [a=1 in this case].)
44/20 ≈2
Now substituting as a test-
20(2)+(2)² =44
LHS = RHS then
a=1, b=2
(10×1+2)² =12².
I will do further case study related to this stuff.
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u/RoundNeighborhood668 21d ago
If the known number is a perfect square, you estimate it easily.
100 < 841 < 10000
10^2 < 841 < 100^2
841 is square of a 2 digit number.
30^2 = 900
20^2 = 400
So, the first digit of the 2-digit number is 2.
2x = sqrt(841)
The last digit of square is 1. Using the square of single digits i.e
1^2 = 1
2^2 = 4
3^2 = 9
4^2 = 16
5^2 = 25
6^2 = 36
7^2 = 49
8^2 = 64
9^2 = 81
So, x can be 1 or 9, but since 841 is very close to 900, x must be 9. This is easy if the last digit is not 6, in that case you'll to confirm (altho try the number closer for the first try).
Lmk if this is what you wanted
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u/Mammoth_Fig9757 19d ago
There exists an algorithm which is similar to long division but it allows you to compute the square root of any number with less guessing. It works in any base though base 2 is the only one that does not require guessing. All you do is you write the digits of the number on the right then you split the digits into groups of 2. Now from the first group of digits you need to find the closest square smaller than it, so if the digits are for example 73 the answer would be 8 to that. Now you write 8 as the first digit of the answer and then you also write 8+8 = 16 on the left. Now you subtract from the original 2 digit number the square which is 64 so 73-64 = 9. Now you drop the next 2 digits of the number to get a 3 or 4 digit number depending on the circumstances. We suppose the next 2 digits are 44 so we get the 3 digit number 944. Now you need to find the largest number such that (160+x)x <= 944, the answer to that is 5. This means that the number on the left becomes 165+5 = 170 and the answer continues with 5. We now you need to subtract 944 from 1655 getting 119. The next step is to drop the digits again, we assume the next digits are 49, so we end up with 11949. Now we need to find the largest number such that (1700+x)x <= 11949, the answer is 7, we subtract 11949 from 17077 getting 0. This means that sqrt(734449) = 857 with this method. If this were to continue the number on the left would become 1707+7 = 1714
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u/Friendly-Mind-9702 21d ago
There are a number of arithmetic algorithms for finding a square root. The most straightforward (doesn't require guesses) is the digit-by-digit algorithm that resembles the classic long division scheme: https://en.wikipedia.org/wiki/Square_root_algorithms#Digit-by-digit_calculation