A simple method to approximate square root without calculator
I thought of this when I am in Secondary School which I love to compete with calculator for speed and accuracy. I encounter a problem when to calculate square root of 5, there is no built-in mechanism for my brain to square root. So how?
Then I remember that
(X+1/X)^2=2+X^2+1/X^2
or (X+1/2X)^2=1+X^2+1/X^2
Since 5=2^2+1, therefore we first approximate square root of 5 by 2, to add one in the square of the result, we need 2+1/4.
However, although the square of 2+1/4 is closer to 5 than 2, but there is an excess of square of 1/4 or 1/16 from the real square root. That is an error of 6.25%.
To eliminate this excess value, we remember that (X-1/X)^2=X^2-2+1/X^2
or (X-1/2X)^2=X^2-1+1/X^2
So we can try 2+1/4-(1/16/(2+1/4)/2), which would give us the excess of (1/16/(2+1/4)/2) squared, i.e. an excess of 1/72 squared or 1/5104. We are thus now making error of only 0.02% from the true value of square root of 5. That is good enough for most purpose.
The general methodology is like this: Suppose we need to find the square root of X^2+q.
Then we first approximate it by nearest square root, i.e. X, and remember we need to make the difference of q.
Thus the next step is adding q/2X to X for suppling this difference, giving an excess of q^2/4X^2.
Yet another step is treating the answer X+q/2X as X in the first step and difference as the q then we try to eliminate the difference by X-q/2X.
We could repeat the process however many times until the answer reach the accuracy we needed.
0. It is the HKEAA, not the Educational Bureau. (learn more before you write)
1. As mentioned by Mr Wai, the markers are teachers but not civil servants. They just work according to marking schemes and if the marking scheme was wrong, then they wo’t be held responsible.
2. I don’t think anyone can say “my English is so good ever since I learnt it” (英文我一向是一流). Saying so means you are so over-confident. (By the way, if you got a D, I don’t think your English is that good)
3. And if there is no definite marking scheme, many more problems would arise–remember this is a public examination which screens those who are not qualified for university and upper forms.
4. “任何考生仍然沒有權利去參閱他/她的考卷” if everyone wants to read his/her papers, the administrative work would be very time-consuming. Then the fee goes up. And people like you will view this with skepticism, saying the HKEAA discourages you from checking your papers.