Do you know how to calculate a square root using paper and pencil?

[eric l]

The long division-like method is based on the formula

(a+:lol:² = a² + 2ab + b² = a*(a+2b) + b²

The aproximation based is based on

a² - b² = (a+:hihi:*(a-:ud:

I know this is not enough as an explication, but for explaining I would actually need a blackboard and a piece of chalk (or a whiteboard and a marker) but most of all constant response of the pupil(s). Which probably makes me an old style teacher.

By the way, do you know why long division is done the way it's done ? Because otherwise I would have to start by explaining that.

[ronthepon]

Long division I know, how it works.

[ronthepon]

For the past ~1 hour, I've been thinking about how it works.

 

My brain is in a mess, probably been full of gamma waves all the time.

 

Although I've figured out the basic workings, I need someone to guide me in my mad quest to figure out how this works.

 

Please, Please just let me have some directions! Anybody!

[eric l]

Let's take the long division type method first.

Just take any number, for the sake of the argument we take my birthday, written as yearmonthdate, so 19471107.

In sections of two digits starting wit hte decimal point this becomes 19 47 11 07.

This means actually (19*1000000) + (47*10000) + (11*100) + (7*1).

We take the square root of the first part. Root of 19 is 4, wich in this case means : root of 19000000 is 4000. Square of 4000 is 16000000, so we are actualy 3000000 short.

Remenber : our a in the formula is 4000, and a² is 16000000.

Now we are going to find the b in the (a + :doh:² formula. For that we include the second section.

The number we want to find the root of is now 19000000 + 470000, or (16000000 + 3000000) + 470000. Or 16000000 + 3470000. (Hence the sticking the next section of two digits to the residue of the first step).

16000000 is the a²; 3470000 must be the [b*(2a+B)]

Because of this (2a+B) we double the result we had so far and stick the following section behind. In our example 3470000 = b*(2a+B)

If we devide 347 by (2*4) we find 4. In our example 3470000 = 400* (8000 + 400). This multiplication gives us 3360000. We now have a residue of 3470000 - 3360000 or 110000.

Are you still with me ? The first two digits of our result are 44, representing 4400. The square of 4400 is 19360000, which is in fact 110000 short of 19470000, the number of which we calculated the root so far.

Now we can go for the next digit of the root. Again, this next digit is the b in the fromula (a + B)², but our "a" is now 4400.

To the "11" repsesenting 110000, we stick the next section of 2 digits (by chance this is again 11)? The number is now 1111, representing actualy 111100.

This number equals the [b*(2a+B)] in the calculation of our next digit. The "2a" part of it is 2*4400. Dividing 111100 by 8800 leaves us with the number 1 for b. [b*(2a+B)] is now [10*(8800+10)] or 88100. This is 23000 short of 111100, so our new residue is 23000.

The next section of 2 digits is 07, representing 07. We add this to the residue and find 23007. This now to be the product of [b*(2*4410+B)]. The value for b now is 2; the residue is 5363.

The root we end up with is 4412. The square of 4412 is 19465744, which is 5363 short of 19471107, the number we wanted to calculate the root of.

We could go on to calculate the decimals, but I rather stop here.

Please make the calculation for yourself, the notation we use on the European continent is slightly different from the notaton in the US. The indication "Northern Hemisphere" for your location does not give me a clue as to which notation I should use.

If you want, I try to explain the other method a next time.

[CraigD]

Although I've figured out the basic workings, I need someone to guide me in my mad quest to figure out how this [long division-like square root calculation] works.

 

Please, Please just let me have some directions! Anybody!

Eric 1 gives a good rendering of the usual explanation, which includes the idea of residues.

 

If you’re trying to reason out how it (or most algorithms) work, a good direction is to try putting specific examples in general algebraic terms. For example, the square of a 2-digit number is given by:

[math](a +b)^2 = a^2 +2ab +b^2 = a^2 +(2a +b)b[/math]

A specific example is:

[math]256 = 10^2 + (20 +6)6 = (10 +6)^2 = 16^2[/math]

 

For a 3-digit result:

[math](a+b+c)^2 = a^2 +2ab +2ac +b^2 +2bc +c^2 = a^2 +(2a +b)b +(2a +2b +c)c[/math]

A specific example is:

[math]103041 = 300^2 +(600 +20)20 + (600 +40 +1)1 = (300 +20 +1)^2 = 321^2[/math]

 

You likely can guess that this explanation can be extended to explain any number of digits. While it’s far from a complete explanation, hopefully it hints one.

[ronthepon]

Thanks a lot, I got it perfectly.