Finding accurate estimates of square roots.

[Craiger1987]

Hi all, this is my first post here and I wanted to run something by people who are more knowledgeable than I. In the past couple days I've decided that I need to teach myself more advanced mathmatics as it is of great importance to my studies in physics, astronomy and biology. In school we never went past algebra 1 and geometry and that has proven to be a hindrance. So I decided to brush up on the basic math that I learned in school and then move on to more advanced maths. I decided to make up a bunch of problems to solve while waiting for my books to arrive and I ran into my old nemesis, finding square roots of large numbers. I was taught prime factorization, which didn't work for this number, and the Babylonian Method, which can yield accurate results depending on the accuracy of your initial estimate and how many times you're willing to run the equation. What I wanted was an equation that used simple math and a known number instead of an assumed estimate (that must be high in the case of the Babylonian Method) that didn't have to be high or low that would yeald an equally/greater accurate estimate of the square as does the BM, and would only have to be run once. I realize this simple mathmatics compared to what is normally discussed here, so I apologize, and also please forgive me if my terminology isn't quite correct. What I came up with was 

√S≈((S/2)/(e/2)+e)/2

in this S is the number we are trying to find the sqaure of and e is the closest perfect square regardless of weather it is high or low. 

For exaple, using 1,863 the number I was initially trying to find the sqaure of, we know that the closest perfect sqaure is 43. So it would look like this

√1,863≈((1863/2)/(43/2)+43)/2

√1,863≈((931.5)/(21.5)+43)/2

√1,863≈(43.325+43)/2

√1,863≈ 86.325/2

√1,863≈ 43.163

Actual square of 1863=43.1624

so the estimate it yielded was very close. I've run a bunch of numbers through it both large and small and so far it appears to give better estimates than the BM (depending on the accuracy of your initial high estimate and number of times you run it through the equation) and without the requirement that the estimate is high, and also we don't have to make an initial estimate ourselves as the starting estimate is fixed for us by the nature of the equation. 

What do you guys think? Is there something I've missed or perhaps is there an easier equation like it that I am not familiar with?

Thanks!

 

Sorry left out a couple words in my haste lol. E is the closest perfect square root. 

[mathematic]

It looks like you have a slightly more complicated version of the usual method.  Let S be the number you want to get the square root of and x the square root.  Let x_0 be your first guess (in your example 43) and calculate x_(n+1)=(x_n+S/x_n)/2.  The x_n's will converge to x.

Edited August 25, 2017 by mathematic
need latex - can't make it work.

[studiot]

 

Hello Craiger and welcome.

First of all, full marks for wanting to use your brain. +1

4 hours ago, Craiger1987 said:

I realize this simple mathmatics compared to what is normally discussed here, so I apologize, and also please forgive me if my terminology isn't quite correct.

Please note that square roots are very important so square root algorithms have been discussed for hundreds of years and there are probably hundred of methods about.

So no it's not too simple at all.

 

You may remember or may never have done logs but here is one method.

if y = aⁿ then if we take logs of both sides (rememebr you don't change an equation if you do the same thing to both sides)

log y = n log (a)

So if we take the log of 1863 and divide it by 2 (multiply it by 0.5) we have the log of the square root of a.

So taking the antilog of this give us the square root.

This method works for other roots besides square roots, including awkward fractional powers used in some Physics equations.

Another method that is simple is the 'bracketing method'

You can work on one place digit at a time, to any desired dgeree of accuracy.

It works like this

(40)² = 1600
(50)² = 2500

So the root is bwtween 40 and 50 since (40)² is less than 1869 and (50)² is greater than 1869
So the first digit is a 4

But 1869 is closer to (40)² than to (50)² so try numbers below 45
(42)² = 1764
(43)² = 1849
(44)² = 1936

So (43)² is less than 1869
but  (44)² is greater than 1869

So the root lies between 43 and 44 so the second digit is a 3.

You can continue this process for as many digits as you like, without having to work out complicated formulae.
And you know that when you arrive you have the correct answer.
Self checking processes are the best.

 

[Craiger1987]

Thanks that helps a lot  we never got past geometry in school, which looking back is a shame, and for some reason I never really applied myself heavily in math class back then, I always focused my attention on science, chemistry etc not quite realizing just how much easier life would be once I got into the heavier stuff if I was proficienct with more advanced mathmatics. Didn't quite grasp the gravity of 'math is everything' back then lol. But, now that I'm going back over the base concepts I learned in school on my own I'm actually enjoying it quite a lot  and once I'm looking forward to working my way up to the more advanced maths. Thanks again and I'm sure I'll be here asking questions as I go. 

[studiot]

Talking of simplicity and the Babylonians,

you may like to know that computers use the babylonian method of multiplication.

 

That is they don't, they use the babylonian 'doubling and halving' method you may have heard of.
They do this electronically by using what are known as shift registers.

[Craiger1987]

That is really interesting  I'm gonna look that up and do some reading 

[studiot]

 

[mathspassion]

Respected All

Everybody knows this work done years ago the man who created this i think he also understand......

why all of u try to down creative man work....

1. if he has done something new appreciate his work.

2.after a long time I am writing comments BECAUSE I am here for appreciate the work not here for down someone.

 

Craiger1987  keep it up.....

[Strange]

1 hour ago, mathspassion said:

why all of u try to down creative man work....

I thought everyone was being very encouraging. It is pretty smart to rediscover an old method. Just as clever as discovering a totally new one.

[John Cuthber]

The algorithm here is, I think, the one they taught my dad many years ago.
http://www.homeschoolmath.net/teaching/square-root-algorithm.php

I never learned, or used it, but it's interesting to look at.

[mathematic]

There is a method, which I learned in elementary school (4th grade?) called "digit by digit".  It is one section of the following.

https://en.wikipedia.org/wiki/Methods_of_computing_square_roots

[studiot]

Since we are having a revival of old methods love in

Another fine method of multiplication is by quarter squares.
Tables for this and other purposes used to be published regualrly.

 

https://www.cut-the-knot.org/arithmetic/TablesForMultiplication.shtml

[OldChemE]

When I want to know a square root, I just pull my old slide rule from my desk  (it uses the Log method).

[DrKrettin]

Does anybody else think that an accurate estimate is an oxymoron?

[John Cuthber]

On 8/26/2017 at 11:32 PM, studiot said:

Tables for this and other purposes used to be published regualrly.

Did they change regularly?

3 minutes ago, DrKrettin said:

Does anybody else think that an accurate estimate is an oxymoron?

Not particularly, but I'm a chemist so I'm used to seeing the instruction that says "weigh accurately about 1.2 grams of the sample..."

Just now, John Cuthber said:

Did the multiplication tables  change regularly?

Not particularly, but I'm a chemist so I'm used to seeing the instruction that says "weigh accurately about 1.2 grams of the sample..."

 

[studiot]

Good morning John.

The history of the creation and development of mathematical tables is an interesting subject (to some) in its own right.

Most tables were originated as a skeleton list at easy to calculate points and extended by some form of interpolation.

The powerful imperative driving this was evidenced by the many famous names assocaited with this process, Napier, Newton to name just a couple.

Over the centuries as tables were extended and the gaps filled in and sometimes revised , major and minor revisions were published.

Better techniques became available for calculation.

The only 'tables' that I can think of that started exact and have remained so are those due to Cayley, which kicked off the modern development of logic.

Edited August 28, 2017 by studiot

[Craiger1987]

6 hours ago, DrKrettin said:

Does anybody else think that an accurate estimate is an oxymoron?

Lol, kind of sounds it doesn't it? Idk if it's the same in everything but at least in the work that I do we make a distinction between accuracy and precision. Accuracy is relative, if I want to be within say .5 of an exact point something that is .3 is considered accurate, but precision is exacting. 

[studiot]

4 hours ago, Craiger1987 said:

Lol, kind of sounds it doesn't it? Idk if it's the same in everything but at least in the work that I do we make a distinction between accuracy and precision. Accuracy is relative, if I want to be within say .5 of an exact point something that is .3 is considered accurate, but precision is exacting. 

Did you hear the one about the appprentice and the fruit gum?

This rather dopey apprentice was always chewing gum.

One day a piece got transferred to his micrometer and got stuck against the base plate.

Dopey didn't notice and went on using his micrometer.

Thenceforth he measured with very poor accuracy but very great precision.

 

[mathspassion]

hello appreciate someone work .................