Square roots calculation without the use of a calculator

Question

How do you find a square root of a number when the square root isn't a whole number, without a calculator

Example: 35

Answer 1

There is a well known algorithm that is simillar to long division. Say you wish to find the square root of $x$, where $x = x_1x_2x_3...x_n$(this is not a product, but rather represents the digits of $x$, i.e if $x = 129, x_1 = 1, x_2 = 2, x_9 = 9$).

It is evalutated using the square root sign like you are doing long division. The number whose square root you will be taking(radicand) will be placed in the sign, and the answer will be written on the top. Just like long division, you will be computing 1 digit at a time.

The process consist of first, ordering the digits into pairs, then finding a number whose square is less than the first pair. This number will be the first digit of your answer. Subtract the numbers square from the first pair, and then bring down the next pair.

You will then bring down the first digit of the answer and then multiply it by 2, call this number $y$. Then you will find a number, say $z$ such that when you add it as the last digit of $y$ and then multiply it by $z$, you get a number less than or equal to the new number formed after bringing the previous pair down. Call this number $p$. Once you have done so, you will subtract $p$ from the pair, and $z$ will be the next digit of your answer.

Next you will bring down the entire answer(so far containing two digits) and multiply it by 2. You can then repeat the process above until you have reached your desired amount of digits.

You can read more about it here(in the section find square roots using an algorithm).

Answer 2

Here is a good page to learn:

http://www.basic-mathematics.com/square-root-algorithm.html

Also Newton made a great approximation.

So pick an approximate square root, $A$. In this case it would be $6$ since $35$ is close to $36$. Then calculate,

$$\frac{1}{2} (\frac{N}{A}+A)$$

Where $N$ is the number ($35$).

So the approximation would be $\frac{1}{2} (\frac{35}{6}+6)= 5.91\bar6 $. The actual answer is about $5.91607978309962$.