Question

describe how you would estimate the square root of a number that is not a perfect square without using a calculator? i don't understand how to do this, can someone help me and please give me a couple examples please but don't make it to complicated. and yes i know that sounds like i'm stupid im not its just i sometimes need very specific steps or instructions.

Answer 1

Say you want to find the square root of 20. This is not a perfect square.

Answer 2

20 is between 16 and 25, which are perfect squares

so sqrt(20) is between sqrt(16) = 4 and sqrt(25) = 5

Answer 3

So one good guess is sqrt(20) = 4.5

this is close, but a bit off, it's a good guess though

Answer 4

now divide 20 by 4.5 to get 4.4444 (you can use long division to find this)

The result we get is not 4.5 like we want, so we have to average 4.5 and 4.4444 to get

(4.5 + 4.4444)/2 = 4.4722

Then try again

20/4.4722 = 4.47207

and we can see that we're getting closer

Answer 5

you repeat these steps to get better and better accuracy

you will never be able to get all of the decimal digits since they go on forever (without a pattern)

Answer 6

omg thank you so much you explained it just right for me to understand it thank you again i did not understand it either i mean i didn't know how to do this.

Answer 7

What @jim_thompson5910 just demonstrated is the way the Babylonians computed square roots long ago. It actually converges to the right answer very quickly!

Answer 8

wow what i was learning in the lesson i understanded but i didn't see any thing about having to do all those steps in my lesson

Answer 9

I wrote a little function to calculate the next value, and compared the results from each step with the square root of 20 out to quite a few digits:

\[\begin{array}{cc}
\text{estimate} & \text{error} \\
\hline \\
4.5 & 0.027864 \\
4.47222 & 0.0000862672 \\
4.47214 & 8.32028668185103*10^{-10} \\
4.47214 & 0. \\
\end{array}\]