4 ms·

Math Trick: Calculating the Square Root by Hand

48 points by aneesh 17y ago

windsurfer 17y ago
"What is the largest number that we can put in the units and multiply times the divisor and still be less than or equal to what we have?" Oh my goodness.
- zackattack 17y ago
  "Algebraically, what is d such that d × 4d ≤ 265?" made it much clearer. :)
- huhtenberg 17y ago
  Just read on, the process becomes obvious after the next iteration. It is very simple actually.
  - windsurfer 17y ago
    Oh yes. I just thought that sentence was fantastic!
dxjones 17y ago
I was taught this method of calculating a square root by hand when I was a child in public school. I have since been surprised by the number of people my own age who were never taught. Since many Canadian high schools only teach "pre-Calculus" now, it makes me wonder if anyone is learning the "long hand" method for square roots these days.
- lkozma 17y ago
  Same with me, learned this in 8th grade (although I forgot most of the details since). 1996, Romania.
cperciva 17y ago
Much faster algorithm, calling your input value X: 1. Move the decimal point 2N digits to the left, so that your value X is of the form a.bcdef... or ab.cdef... Remember the value N. 2. If X now has one digit before the decimal place, set Y = 5. Otherwise, set Y = 2. 3. Compute Y = Y * (300 - X * Y * Y) / 200 to as many decimal places as you want. Feel free to start by computing only a few decimal places. 4. Repeat step 2 until you're computing Y to the number of decimal places you want in your answer and it's not changing. 5. Compute Y = X * Y. 6. Move the decimal point in Y to the right by N-1 places.
- jpwagner 17y ago
  I'm sorry, but this is totally unclear (to me at least). For example what are you computing (in part 3) if you know what X and Y are? What is N? What is "the number of decimal places you want"? Can you explain in another way? Do you have an example? Why does this work?
  - jibiki 17y ago
    Part 3 makes perfect sense to me, it's an "x = x+1" style line. N is (((the number of digits of X), divided by 2), rounded down.) Edit: hmm, maybe I've got it wrong, the convergence is very slow. Anyways, here's how it works out for me: X = 105362 X = 10.5362 so N = 2 (half the number of digits to the right of the decimal point...) Y = 2 Y = 2*(300-10.5362*2*2) = 2.578552 Y = 2.578552*(300-10.5362*2.578552*2.578552) = 2.9646326 Y = 2.9646326*(300-10.5362*2.9646326*2.9646326) = 3.0742773 It should converge to around 3.24, at which point you would shift the decimal point back 2 digits to get 324.
    - jpwagner 17y ago
      I guess I'm confused by the fact that you are repeating step 2 which reassigns Y... Clear on N thanks. An example would make this easier...
      - jibiki 17y ago
alexitosrv 17y ago
Thinking in powers of two, aids to get faster estimates of square roots. When you are calculating just by hand, you can tolerate errors associated with this method. For example, the number of the explication is near to 2^16, so the square root is 2^8 = 256 approximately (with relative error of 0.7% and in less of a second :) Of course, it won't work when numbers are farther of powers of two.
- derefr 17y ago
  You just have to be familiar with how to scale numbers logarithmically. (That is, half-way between two units is 70% of the value, 1/10th of the way is 30% value, and so on.)
quizbiz 17y ago
dare I ask, why?
- pg 17y ago
  Among other things, knowing how to solve a problem with limited means helps you understand more precisely how hard it is.
- jrp 17y ago
  My dad and I were on a backpacking trip and while hiking we calculated how thin the atmosphere was compared to sea level. It feels more elegant sometimes to solve a problem simply (in paper, in your head) without bringing out the computers. See Dijkstra on sum/product puzzle: http://www.cs.utexas.edu/users/EWD/transcriptions/EWD06xx/EWD666.html http://www.cs.utexas.edu/users/EWD/transcriptions/EWD06xx/EW...
ableal 17y ago
For those intrigued by this kind of thing, this book (paper only, but the web page has more resources) may be interesting: Dead Reckoning: Calculating Without Instruments, by Ronald W. Doerfler (http://www.myreckonings.com/Dead_Reckoning/Dead_Reckoning.htm http://www.myreckonings.com/Dead_Reckoning/Dead_Reckoning.ht... ) More good stuff at http://myreckonings.com/wordpress/ http://myreckonings.com/wordpress/ . The nomography posts had caught my eye a while ago, and there may be more of those coming soon. (Disclosure: I recently exchanged a couple of emails with the author, after leaving a comment on the Heaviside post which was briefly featured here.)
jodrellblank 17y ago
We found longhand square root instructions with worked examples in my Grandma's recipe collection. :)

48 points by aneesh 17y ago

"What is the largest number that we can put in the units and multiply times the divisor and still be less than or equal to what we have?" Oh my goodness.

zackattack 17y ago
"Algebraically, what is d such that d × 4d ≤ 265?" made it much clearer. :)
huhtenberg 17y ago
Just read on, the process becomes obvious after the next iteration. It is very simple actually.
- windsurfer 17y ago
  Oh yes. I just thought that sentence was fantastic!

dxjones 17y ago

I was taught this method of calculating a square root by hand when I was a child in public school. I have since been surprised by the number of people my own age who were never taught. Since many Canadian high schools only teach "pre-Calculus" now, it makes me wonder if anyone is learning the "long hand" method for square roots these days.

lkozma 17y ago
Same with me, learned this in 8th grade (although I forgot most of the details since). 1996, Romania.

cperciva 17y ago

Much faster algorithm, calling your input value X: 1. Move the decimal point 2N digits to the left, so that your value X is of the form a.bcdef... or ab.cdef... Remember the value N. 2. If X now has one digit before the decimal place, set Y = 5. Otherwise, set Y = 2. 3. Compute Y = Y * (300 - X * Y * Y) / 200 to as many decimal places as you want. Feel free to start by computing only a few decimal places. 4. Repeat step 2 until you're computing Y to the number of decimal places you want in your answer and it's not changing. 5. Compute Y = X * Y. 6. Move the decimal point in Y to the right by N-1 places.

jpwagner 17y ago
I'm sorry, but this is totally unclear (to me at least). For example what are you computing (in part 3) if you know what X and Y are? What is N? What is "the number of decimal places you want"? Can you explain in another way? Do you have an example? Why does this work?
- jibiki 17y ago
  Part 3 makes perfect sense to me, it's an "x = x+1" style line. N is (((the number of digits of X), divided by 2), rounded down.) Edit: hmm, maybe I've got it wrong, the convergence is very slow. Anyways, here's how it works out for me: X = 105362 X = 10.5362 so N = 2 (half the number of digits to the right of the decimal point...) Y = 2 Y = 2*(300-10.5362*2*2) = 2.578552 Y = 2.578552*(300-10.5362*2.578552*2.578552) = 2.9646326 Y = 2.9646326*(300-10.5362*2.9646326*2.9646326) = 3.0742773 It should converge to around 3.24, at which point you would shift the decimal point back 2 digits to get 324.
  - jpwagner 17y ago
    I guess I'm confused by the fact that you are repeating step 2 which reassigns Y... Clear on N thanks. An example would make this easier...
    - jibiki 17y ago

alexitosrv 17y ago

Thinking in powers of two, aids to get faster estimates of square roots. When you are calculating just by hand, you can tolerate errors associated with this method. For example, the number of the explication is near to 2^16, so the square root is 2^8 = 256 approximately (with relative error of 0.7% and in less of a second :) Of course, it won't work when numbers are farther of powers of two.

derefr 17y ago
You just have to be familiar with how to scale numbers logarithmically. (That is, half-way between two units is 70% of the value, 1/10th of the way is 30% value, and so on.)

quizbiz 17y ago

dare I ask, why?

pg 17y ago
Among other things, knowing how to solve a problem with limited means helps you understand more precisely how hard it is.
jrp 17y ago
My dad and I were on a backpacking trip and while hiking we calculated how thin the atmosphere was compared to sea level. It feels more elegant sometimes to solve a problem simply (in paper, in your head) without bringing out the computers. See Dijkstra on sum/product puzzle: http://www.cs.utexas.edu/users/EWD/transcriptions/EWD06xx/EWD666.html http://www.cs.utexas.edu/users/EWD/transcriptions/EWD06xx/EW...

ableal 17y ago

For those intrigued by this kind of thing, this book (paper only, but the web page has more resources) may be interesting: Dead Reckoning: Calculating Without Instruments, by Ronald W. Doerfler (http://www.myreckonings.com/Dead_Reckoning/Dead_Reckoning.htm http://www.myreckonings.com/Dead_Reckoning/Dead_Reckoning.ht... ) More good stuff at http://myreckonings.com/wordpress/ http://myreckonings.com/wordpress/ . The nomography posts had caught my eye a while ago, and there may be more of those coming soon. (Disclosure: I recently exchanged a couple of emails with the author, after leaving a comment on the Heaviside post which was briefly featured here.)

jodrellblank 17y ago

We found longhand square root instructions with worked examples in my Grandma's recipe collection. :)