Question

What is the fastest method to calculate a non-perfect square root without using a calculator?

Answer 1

Probably something like newton's method.

Answer 2

Suppose you are trying to find the square root of \(c\).
This is basically the same as finding the roots of \(f(x) = x^2-c\).

Answer 3

Wouldn't you still then have to use a calculator?

Answer 4

The derivative of this is: \(
f'(x) = 2x
\).
Newton's method says start with some guess \(x_0\), the following number will be closer to the square root: \[
x_1 = x_0-\frac{f(x_0)}{f'(x_0)} = x_0 - \frac{x_0^2-c}{2x_0}
\]

Answer 5

You can keep going until you think you're close enough.

Answer 6

A second degree taylor polynomial would be fairly close usually.

Answer 7

But, this is an algebra question.
(not calculus, so offering differentiation to solve it??)

Answer 8

You can estimate c between two perfect squares.
(I have seen that technique when I was helping my relative with math lessons on connections academy)

Answer 9

This is for my student who is 11 years old....so maybe something easier? Or is estimate and calculate and estimate the only real way?

Answer 10

For example, suppose I want the square root of \(13\).
I guess \(4\).\[
x_1 = 4 - \frac{4^2-13}{2(4)} = 4 - \frac{3}{8} = \frac{29}{8} = 3.625
\]Then I can do this again: \[
x_2 = 3.625 - \frac{3.625^2-13}{2(3.625)}
\]

Answer 11

for finding perfect square of a no. which wasn't be a perfect square, you need to know minimum of square root of first 20 natural no.
let you need to find sqrt 80.\[\sqrt{80}\]than first calculate LCM of this,now simply wright\[\sqrt{80}=\sqrt{2*2*2*2*5}\]now wright this as \[2*2\sqrt{5}\]now simply put the value of \[\sqrt{5}\] and you get required answer.

Answer 12

Hmmm, wio that is not too bad.

Answer 13

Next iteration I get: \[
x_2 = \frac{1673}{464} \approx 3.6056
\]Note that \[
\sqrt{13} \approx 3.60555
\]

Answer 14

Another method, that could be a bit easier, is to memorize the square root of prime numbers up to 25 or so, then to factor as newtonson suggests.

Answer 15

This is the method that engineers used before calculators.

Answer 16

You still really need a calculator. I will review newtonson

Answer 17

Except, they had a reference table for square roots of primes.

Answer 18

Typically math classes that don't allow calculators will allow you to keep the square root of a prime, and if not, then they expect you to have memorized the square root of the prime.

Answer 19

Thanks everybody.