How can I find square roots without a calculator?

Answer 1

This algorithm doesn't require repeated multiplications or divisions.

Say we want to find the square root of 5400976

Separate the digits of the number into groups of two, starting at the decimal point.

05'40'09'76

Start with the leftmost "pair" separated and with the number 1 to the right.

5'400976 ~ 1

Subtract the number on the right from the separated part of the left number. Then add 2 to the number on the right.

4'400976 ~ 3

Repeat.

1'400976 ~ 5

Since we can no longer subtract the right number from the separated part of the number on the left, subtract 1 from the right number and tack on a 1 to the end of it. Move the separator over 2 digits.

140'0976 ~ 41

Repeat.

99'0976 ~ 43
56'0976 ~ 45
11'0976 ~ 47
1109'76 ~ 461
648'76 ~ 463
185'76 ~ 465
18576 ~ 4641
13935 ~ 4643
9292 ~ 4645
4647 ~ 4647
0 ~ 4649

If we arrive at zero on the left hand side, it means we have a perfect square. At this point, we can subtract 1 from the right number and then divide it by 2 to get the exact square root.

4649 - 1 = 4648
4648 / 2 = 2324

If the separator is still to the left of the decimal by so many digits on the left, then add half as many zeros to the right number after subtracting 1 and before dividing by 2.

If the separator is to the right of the decimal by so many digits on the left, then put the decimal point half as many digits in from the rightmost side of the right number after subtracting 1 and before dividing by 2.

If we don't have a perfect square, we can do this last step at any point.

Answer 2

The simplest way, though not necessarily the fastest, is through trial and error. For example, if you were trying to find the square root of two, you might start with an initial guess of 1.5. Squaring this gives you a number bigger than 2, so try 1.4 Squaring this gives you a number smaller than 2, so try 1.45. Squaring this gives you a number bigger than 2, so try 1.425, then see what its square is, etc. Though enough trial and error like this, you can (in theory) take the decimal out to as many places as you wish.

Another way is to use log tables, which was the common practice before calculators. If x is the number you want to take a square root of, notice that if y = x^(1/2), then
Log[base x] (y) = 1/2
Log[base 10](y) / log[base 10](x) = 1/2
Log[base 10](y) = (1/2)[ log[base 10](x) ]
So if you look up the log of x in your log table and take half of it, you can extrapolate and use the same table to find the corresponding value of y.

Answer 3

You can find the prime factors of the number and then any number you have 2 of, you can pull out of the radical sign... it's easier if I show you an example:

Square root of 36: (I know this is an easy one... but it's a good example)
First you would prime factor it:
16 = 2 x 2 x 3 x 3
Then you can circle the pairs: (2 x 2) and (3 x 3) Since you have a pair of 2's, you can take out a 2, and since you have a pair of 3's, you can take out a 3.... 2 x 3 = 6
so the square root of 36 is 6

Hope this helps and doesn't confuse you!

Answer 4

Unless it's a perfect square, it's hard without a calculator. You can get close. For instance the sqrt of 30 will be about 5 1/2 (halfway between 25 and 36.
Just see which two squares it's between and estimate. If you need more exact, you'll have to get a calculator. Windows has one in accessories.

Answer 5

start a whole number and multiply it by itself to get a number that is closest to the number you're trying to find the squre root of. if it is too high try the next one either higher or lower. then start doing the same thing but keep picking numbers in between the set of previous two numbers. for example, lets say i am trying to find the square root of 10. 3x3 is 9 and 4x4 is 12. so now i try 3.5x3.5 and get 12.25. so i try 3.25x3.25 and get 10.5625. now i try 3.125x3.125 and get 9.765625 so now i try 3.1875x3.1875 and get 10.16015625. each time i get closer and closer to 10. keep up this process and you'll eventually get very, very close to 10, but never 10 exactly (not even a calculator can do that).

Answer 6

Here is a link that shows how to calculate square roots by hand.

http://www.nist.gov/dads/HTML/squareRoot.html

This is the method I learned from my dad when I was in elementary school. They didn't teach it in school.

Answer 7

use newton's method of approximation. researh for it. you can extract any roots of any number accurate to any digit you want without using a calculator!

Answer 8

See the following wesite, it's too hard to explain:

http://www.nist.gov/dads/HTML/squareRoot.html