Hello friends,
I am studying in 10th class. Actually I have a question and I’m unable to solve this question. My question is: How can we find the square root of a number by hand? How about cube roots? If anybody can solve my question I will grateful. Thanks in advance!

2. Originally Posted by daniel_dan
Hello friends,
I am studying in 10th class. Actually I have a question and I’m unable to solve this question. My question is: How can we find the square root of a number by hand?
I am going to give an an algorithmic approach. Meaning a step-by-step program to do by hand. Each time you do a step the better you get to the square root.

Let $x$ be some positive number. And we want to find $\sqrt{x}$. Let $a$ be your guess. Meaning what you think the square root it. You can be totally off, it will still work. But the better you guess the faster it will work.

Find $\frac{a^2 + x }{2a}$.
Call this $b$.

Now pretend that $b$ is your guess. (Which is now better).
Find,
$\frac{b^2+x}{2b}$.
Call this $c$.

Now this is $c$ is better so find $\frac{c^2+x}{2c}$. And keep on going .... The more you do it the closer you get.

For example, you want to find $\sqrt{2}$.

Make a guess $a=1$.

That means $b = \frac{1^2 + 2}{2(1)} =1.5$.

That means $c=\frac{1.5^2 + 2}{2(1.5)} = 1.41$.

You can see you start getting better and better.