# Did Newton Crack Square Roots For Us??? **URL:** \ **Category:** Factual Questions\ **Created:** [May 21, 2003, 4:19am UTC](https://boards.straightdope.com/t/did-newton-crack-square-roots-for-us/176499 "2003-05-21T04:19:39Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Jinx](https://avatars.discourse-cdn.com/v4/letter/j/c6cbf5/32.png) [@Jinx](https://boards.straightdope.com/u/Jinx)\ **Post date:** [May 21, 2003, 4:19am UTC](https://boards.straightdope.com/t/did-newton-crack-square-roots-for-us/176499/1 "2003-05-21T04:19:39Z")
Was Newton the first (only?) to develop a method for finding square roots? What about cube roots and other roots? For those unfamiliar with the technique for finding square roots by hand, it resembles a long division problem, but the technique is not quite identical, of course. - Jinx ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [May 21, 2003, 4:58am UTC](https://boards.straightdope.com/t/did-newton-crack-square-roots-for-us/176499/2 "2003-05-21T04:58:42Z")
I’m sure this isn’t what you’re talking about, but you can use Newton’s Method to get the root to any old equation, including x[sup]2[/sup] - 5 = 0. I understand one of the ancient mathematical cultures, like the Babylonians, used what was effectively Newton’s Method for finding square roots. ---
**Author:** ![raygirvan](https://avatars.discourse-cdn.com/v4/letter/r/ac91a4/32.png) [@raygirvan](https://boards.straightdope.com/u/raygirvan)\ **Post date:** [May 21, 2003, 12:00pm UTC](https://boards.straightdope.com/t/did-newton-crack-square-roots-for-us/176499/3 "2003-05-21T12:00:18Z")
And the Ancient Greeks used various geometrical methods to find square roots. For instance, if you have a line, you can easily construct the square of which that line is the diagonal. ---
**Author:** ![Paradigm](https://avatars.discourse-cdn.com/v4/letter/p/13edae/32.png) [@Paradigm](https://boards.straightdope.com/u/Paradigm)\ **Post date:** [May 21, 2003, 12:32pm UTC](https://boards.straightdope.com/t/did-newton-crack-square-roots-for-us/176499/4 "2003-05-21T12:32:01Z")
Do you mean the Newton-Raphson iteration method? ---
**Author:** ![Desmostylus](https://avatars.discourse-cdn.com/v4/letter/d/c57346/32.png) [@Desmostylus](https://boards.straightdope.com/u/Desmostylus)\ **Post date:** [May 21, 2003, 12:47pm UTC](https://boards.straightdope.com/t/did-newton-crack-square-roots-for-us/176499/5 "2003-05-21T12:47:57Z")
**Paradigm** , **Jinx** means this process: To find a = √b a) Guess a. b) Set a = (a + b/a)/2 c) Repeat step (b) until solved. “Solved” usually means that the last iteration didn’t change the value of a at the precision being used. Those are the steps involved in using NR to solve the problem. But that _particular_ procedure for that _particular_ problem predated calculus by millenia. ---
**Author:** ![Paradigm](https://avatars.discourse-cdn.com/v4/letter/p/13edae/32.png) [@Paradigm](https://boards.straightdope.com/u/Paradigm)\ **Post date:** [May 21, 2003, 1:19pm UTC](https://boards.straightdope.com/t/did-newton-crack-square-roots-for-us/176499/6 "2003-05-21T13:19:40Z")
Yep that’s what I thought (he never mentioned Raphson, but I have heard the NR method called Newton’s method, so I was just being sure), in which case the root you find is not actually the real root, just an accurate estimate (how accurate depends on how many iterations of that you choose). So in effect it’s not exactly ‘solved’ mathematically, but for most practical purposes it is. ---
**Author:** ![Paradigm](https://avatars.discourse-cdn.com/v4/letter/p/13edae/32.png) [@Paradigm](https://boards.straightdope.com/u/Paradigm)\ **Post date:** [May 21, 2003, 1:31pm UTC](https://boards.straightdope.com/t/did-newton-crack-square-roots-for-us/176499/7 "2003-05-21T13:31:55Z")
Oh, also it doesn’t always work… Try doing it with y=x³+9-9x² And take your first guess to be either 1 or 4.7034 I’ll admit it, I intentionally made that so it would fail, but i’m sure there are other examples where it wouldn’t. But it works fine if you just take a different initial starting point. Just for information, NR works by taking the point on the line and drawing its gradient to converge to the root, so you have to be careful to not miss any roots. Btw Desmostylus, your method only works for x^1/2 because you’ve effectively fiddled it. The real NR method is: For y=f(x) Guess a a-(f(a)/f’(a)) Repeat Yours works because d/dx(x^2) is (1/2)x^-1/2 ---
**Author:** ![Desmostylus](https://avatars.discourse-cdn.com/v4/letter/d/c57346/32.png) [@Desmostylus](https://boards.straightdope.com/u/Desmostylus)\ **Post date:** [May 21, 2003, 1:51pm UTC](https://boards.straightdope.com/t/did-newton-crack-square-roots-for-us/176499/8 "2003-05-21T13:51:32Z")
**Paradigm** , I explained a simple point to you, i.e.: “Do you mean the Newton-Raphson iteration method?” You are a new member. Welcome aboard. You shouldn’t assume from the outset that you know more than any other given member. This sort of stuff: > [@](#): > > _Originally posted by Paradigm_ > **Btw Desmostylus, your method only works for x^1/2 because you’ve effectively fiddled it.** doesn’t fly here. ---
**Author:** ![Paradigm](https://avatars.discourse-cdn.com/v4/letter/p/13edae/32.png) [@Paradigm](https://boards.straightdope.com/u/Paradigm)\ **Post date:** [May 21, 2003, 2:01pm UTC](https://boards.straightdope.com/t/did-newton-crack-square-roots-for-us/176499/9 "2003-05-21T14:01:54Z")
Sorry man, I didn’t mean any disrespect ☹ All I meant was it was different to the version I was familiar with, and your method only worked for x^1/2 because it was just a manipulation of that method (which came first I honestly don’t know). All I meant by ‘fiddle’ is a manipulation of something that makes it work in specific circumstances. I don’t assume (to assume makes an ass out of u and me ;)) to be smarter than anyone, intelligence is an abstract concept, of course. Plus i’m only 18 and i’m sure there are many more experienced, wiser, and simply more intelligent people out there 😉 Maybe I should put a disclaimer at the bottom of my posts: ‘This post only opinion of poster, may not represent real fact’ 😉 Also, thx for the welcome, I am indeed new… ---
**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\ **Post date:** [May 21, 2003, 2:15pm UTC](https://boards.straightdope.com/t/did-newton-crack-square-roots-for-us/176499/10 "2003-05-21T14:15:35Z")
Guys, I’m not certain **Jinx** means what you think he means: > [@](#): > > \*Originally posted by Jinx \* > **For those unfamiliar with the technique for finding square roots by hand, it resembles a long division problem, but the technique is not quite identical, of course.** I’ve seen a method for computing square roots by hand which resembles long division, in that the digits of the square root are determined one at a time. It’s similar but not quite the same as Newton’s method of applying a-\>(a+b/a)/2 repeatedly. For one thing, the “long-division-like” method approaches the square root exclusively from below (just as long division approaches the quotient from below), whereas Newton’s method approaches the square root from both sides. Unfortunately I don’t know the origin of the “long-division-like” method, but I wouldn’t be surprised to learn that it predates Newton. (I’ve never heard it referred to as “Newton’s method”, either.) ---
**Author:** ![sailor](https://avatars.discourse-cdn.com/v4/letter/s/a587f6/32.png) [@sailor](https://boards.straightdope.com/u/sailor)\ **Post date:** [May 21, 2003, 2:33pm UTC](https://boards.straightdope.com/t/did-newton-crack-square-roots-for-us/176499/11 "2003-05-21T14:33:47Z")
> [@](#): > > \*Originally posted by Desmostylus \* > \*\ ***Paradigm** , **Jinx** means this process: > > To find a = √b > > a) Guess a. > > b) Set a = (a + b/a)/2 > > c) Repeat step (b) until solved. > > “Solved” usually means that the last iteration didn’t change the value of a at the precision being used. > > Those are the steps involved in using NR to solve the problem. But that _particular_ procedure for that _particular_ problem predated calculus by millenia. \*\* I do not think he does. Finding the square root by iterative successive approximations is not the same as the method which “resembles a long division problem”. They are two different methods. BTW, in my implementation of the recursive algorithm I have never heard or used the “guess a number” to begin with. I used (1+ N/2) or some such. ---
**Author:** ![Engywook](https://avatars.discourse-cdn.com/v4/letter/e/bc8723/32.png) [@Engywook](https://boards.straightdope.com/u/Engywook)\ **Post date:** [May 21, 2003, 2:50pm UTC](https://boards.straightdope.com/t/did-newton-crack-square-roots-for-us/176499/12 "2003-05-21T14:50:21Z")
> [@](#): > > \*Originally posted by raygirvan \* > \*\*And the Ancient Greeks used various geometrical methods to find square roots. For instance, if you have a line, you can easily construct the square of which that line is the diagonal. \*\* I’ve a feeling I’ll be smacking my head in a mere moments, but how does this method give you the square root of anything? Let _y_ be length of diagonal. Let _x_ be length of side of square. 2x[sup]2[/sup] = y[sup]2[/sup] From this, I don’t see a way to get to the square root of either _x_ or _y_ being equal to anything but something times the square root of _y_ or _x_. ---
**Author:** ![Desmostylus](https://avatars.discourse-cdn.com/v4/letter/d/c57346/32.png) [@Desmostylus](https://boards.straightdope.com/u/Desmostylus)\ **Post date:** [May 21, 2003, 2:58pm UTC](https://boards.straightdope.com/t/did-newton-crack-square-roots-for-us/176499/13 "2003-05-21T14:58:04Z")
> [@](#): > > \*Originally posted by sailor \* > \*\*BTW, in my implementation of the recursive algorithm I have never heard or used the “guess a number” to begin with. I used (1+ N/2) or some such. \*\* And what exactly would you call your starting point? Estimate? Starting Point, maybe. Guesstimate? :rolleyes: ---
**Author:** ![DaveW](https://avatars.discourse-cdn.com/v4/letter/d/9d8465/32.png) [@DaveW](https://boards.straightdope.com/u/DaveW)\ **Post date:** [May 21, 2003, 3:02pm UTC](https://boards.straightdope.com/t/did-newton-crack-square-roots-for-us/176499/14 "2003-05-21T15:02:24Z")
The square root method resembling long division can be found [here](http://www.mathpages.com/home/kmath146.htm). ---
**Author:** ![CalMeacham](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/calmeacham/32/35_2.png) [@CalMeacham](https://boards.straightdope.com/u/CalMeacham)\ **Post date:** [May 21, 2003, 3:04pm UTC](https://boards.straightdope.com/t/did-newton-crack-square-roots-for-us/176499/15 "2003-05-21T15:04:25Z")
Newton Cracks Roots, and I Don’t Care, Newton Cracks Roots, and I Don’t Care, Newton Cracks Roots, and I Don’t Care, I Can Solve Iteratively. ---
**Author:** ![Jinx](https://avatars.discourse-cdn.com/v4/letter/j/c6cbf5/32.png) [@Jinx](https://boards.straightdope.com/u/Jinx)\ **Post date:** [May 21, 2003, 3:07pm UTC](https://boards.straightdope.com/t/did-newton-crack-square-roots-for-us/176499/16 "2003-05-21T15:07:30Z")
> [@](#): > > \*Originally posted by Orbifold \* > **Guys, I’m not certain Jinx means what you think he means…I’ve seen a method for computing square roots by hand which resembles long division, in that the digits of the square root are determined one at a time. It’s similar but not quite the same as Newton’s method of applying a-\>(a+b/a)/2 repeatedly…** Yes, that is correct, Oribifold. If you want the square root of 18, for example, your first number in the solution will be the root of the closest square…I forget the exact method after this obvious step. I’ll see if I can locate it in a reference book. In any sense, I was taught it was developed by Newton. FYI: No, this is not the same as the Newton-Raphson method of iterating to find roots to solve polynomials, but one might stem from the other, possibly? I’ll look further into it, and post again ASAP… - Jinx ---
**Author:** ![sailor](https://avatars.discourse-cdn.com/v4/letter/s/a587f6/32.png) [@sailor](https://boards.straightdope.com/u/sailor)\ **Post date:** [May 21, 2003, 3:15pm UTC](https://boards.straightdope.com/t/did-newton-crack-square-roots-for-us/176499/17 "2003-05-21T15:15:38Z")
> [@](#): > > \*Originally posted by Desmostylus \* > \*\*And what exactly would you call your starting point? Estimate? Starting Point, maybe. Guesstimate? :rolleyes: \*\* I am not sure the meaning or purpose of your rolleyes. maybe my meaning wasn’t clear. i emant to say that in a practical implementation if the algorithm in a computer program “guess a number” (in other words, get a random number) is not the easiest or most effective way to do it and starting out doing it the way I said is easier and generally more effective. Does this deserve a rolleyes? ---
**Author:** ![Desmostylus](https://avatars.discourse-cdn.com/v4/letter/d/c57346/32.png) [@Desmostylus](https://boards.straightdope.com/u/Desmostylus)\ **Post date:** [May 21, 2003, 3:25pm UTC](https://boards.straightdope.com/t/did-newton-crack-square-roots-for-us/176499/18 "2003-05-21T15:25:31Z")
Newton’s method converges quadratically provided that the starting point (“guess”) is sufficiently close to the solution. _You_ inferred that “guess” meant “random number”. ---
**Author:** ![sailor](https://avatars.discourse-cdn.com/v4/letter/s/a587f6/32.png) [@sailor](https://boards.straightdope.com/u/sailor)\ **Post date:** [May 21, 2003, 3:30pm UTC](https://boards.straightdope.com/t/did-newton-crack-square-roots-for-us/176499/19 "2003-05-21T15:30:57Z")
I still don’t get the rolleyes but anyway, how do you implement in a real computer programing language the instruction “make a guess that is sufficiently close to the solution” in a manner that is simpler an more effective than what I proposed? ---
**Author:** ![Desmostylus](https://avatars.discourse-cdn.com/v4/letter/d/c57346/32.png) [@Desmostylus](https://boards.straightdope.com/u/Desmostylus)\ **Post date:** [May 21, 2003, 3:42pm UTC](https://boards.straightdope.com/t/did-newton-crack-square-roots-for-us/176499/20 "2003-05-21T15:42:44Z")
> [@](#): > > \*Originally posted by sailor \* > \*\*I still don’t get the rolleyes but anyway, how do you implement in a real computer programing language the instruction “make a guess that is sufficiently close to the solution” in a manner that is simpler an more effective than what I proposed? \*\* Right shift the exponent field. “sufficiently close to the solution” has a precise meaning in terms of the convergence of Newton’s method. Look it up. :rolleyes: [Next page](https://boards.straightdope.com/t/did-newton-crack-square-roots-for-us/176499.md?page=2)