Square root of a number without a calculator

Hey There

how are you

the answer is For the number 1111, we choose 33, which when squared, yields 99.

We thus rewrite our problem as follows: 11−−√=11×1−−−−−√=11×9/9−−−−−−−√=(11/9)×9−−−−−−−−√=11/9−−−−√×9–√=311/9−−−−√11=11×1=11×9/9=(11/9)×9=11/9×9=3⁢11/9

We have thus changed the nature of our problem to finding the square root of 11/911/9. Has progress been made? We maintain, yes.

Letâs expand 11/911/9 as a decimal. We then have 31.222222…−−−−−−−−−√3⁢1.222222⁢…

THis method can be used for any number It would be best to show a picture but I cant load one up so this script might help. Its a bit like long division.

Write 11 as 11,00,00,00,00 Note Ive separated each double zero by a comma.

Square root of 11 is between 3 and 4 so we first write 3.

Now multiply 3 by 3 to give 9 and subtract this from the 11 to give 2. now add the first pair of 00 to this to give 200. Multiply the 3 we got earlier by 2 to give 6. Now we add a digit to 6 so that when we multiply by this digit we get a number which is below but close to 200. This number is 3 because 63 * 3 = 189. Now we update our answer to 3.3. Now we proceed to the next decimal place. 200 - 189 = 11 . now we add another 00 to give 1100. Next multiply the

3.3 by 2 to give 66 ( we ignore the decimal point) . Now we need to add a digit to 66 so that when we multiply by this digit we get a number close to and less than 1100. So this digit is 1 because 661 * 1 = 661. So we again update our answer to 3.31. Repeating this process will give us the square root to any accuracy we need.

The answer correct to 3 decimal places is 3.317

answer: x=60