Question

How do you find the square root of a number without using a calculator?

Answer 1

Well, you can learn your perfect square roots, like the \[\sqrt{9}=3\] and so on. :p c:

Answer 2

Ok thanks

Answer 3

But do you know of way to figure the rounded answer of the square root?

Answer 4

theres a long division method, there is also a calculus method

Answer 5

the calculus method might require a calculator for speed, the long division method is pretty much pencil to paper

Answer 6

You can use the following to find \(\sqrt S\):
$$
x_{n+1}=\cfrac{1}{2}\left(x_n+\cfrac{S}{x_n}\right)\\
x_0=\sqrt{1}=1
$$
For example, for \(S=10\)
$$
x_{n+1}=\cfrac{1}{2}\left(x_n+\cfrac{10}{x_n}\right)\\
x_0=1\\
x_1=\cfrac{1}{2}\left(1+\cfrac{10}{1}\right)=5.5\\
x_2=\cfrac{1}{2}\left(5.5+\cfrac{10}{5.5}\right)=3.66\\
x_3=\cfrac{1}{2}\left(3.66+\cfrac{10}{3.66}\right)=3.2\\
x_4=3.16\\
x_5=3.16\\
x_6=3.16\\
\cdots
$$
We see here that \(\sqrt {10}\approx 3.16\)

See - https://en.wikipedia.org/wiki/Methods_of_computing_square_roots

As alluded to earlier, even though this can be done manually, it would be very tedious for arbitrary values of \(S\).

Does this make sense?