Ruffini’s Rule and meaning of division

We continue the earlier discussion on (1) differentials and (2) polynomials. There is also this earlier discussion about (static or dynamic) division.

At issue is: Can we avoid the use of limits when determining the derivative of a polynomial ?

A sub-issue is: Can we avoid division that requires a limit ?

We use the term incline instead of tangent (line), since this line can also cross a function and not just touch it.

We use H = -1, so that we can write x x^H = x^H x = 1 for x ≠ 0. Check that x^H = 1 / x, that the use of H is much more effective and efficient. The use of 1 / x is superfluous since students must learn about exponents anyway.

Ruffini’s Rule

Ruffini’s Rule is a method not only to factor polynomials but also to isolate the factors. A generalised version is called “synthetic division” for the reason that it isn’t actually division. On wikipedia, Ruffini’s Rule is called “Horner’s Method“. On mathworld, the label “Horner’s Method” is used for something else but related again. My suggestion is to stick to mathworld.

Thus, the issue at hand would seem to have been answered by Ruffini’s Rule already. When we can avoid division then we don’t need a limit around it. However, our discussion is about whether this really answers our question and whether we really understand the answer.

Historical note

I thank Peter Harremoēs for informing me about both Ruffini’s Rule and some neat properties that we will see below. His lecture note in Danish is here. Surprising for me, he traced the history back to Descartes. Following this further, we can find this paper by John Suzuki, who identifies two key contributions by Jan Hudde in Amsterdam 1657-1658. Looking into my copy of Boyer “The history of the calculus” now, page 186, I must admit that this didn’t register to me when I read this originally, as it registers now. We see the tug and push of history with various authors and influences, and thus we should be cautious about claiming who did what when. Suzuki’s statement remains an eye-opener.

“We examine the evolution of the lost calculus from its beginnings in the work of Descartes and its subsequent development by Hudde, and end with the intriguing possibility that nearly every problem of calculus, including the problems of tangents, optimization, curvature, and quadrature, could have been solved using algorithms entirely free from the limit concept.” (John Suzuki)

Apparently Newton dropped the algebra because it didn’t work on trigonometry and such, but with modern set theory we can show that the algebraic approach to the derivative works there too. For the discussion below: check that limits can be avoided.

Division is also a way to isolate factors

When we have 2 x = 6, then we can determine 2 x = 2 3, and recognize the common factor 2. By the human eye, we can see that x = 3 and then we have isolated the factor 3. But in mathematics, we must follow procedures as if we were a computer programme. Hence, we have the procedure of eliminating 2, which is called division:

2^H 2 x = 2^H 2 3

x = 3

The latter example abuses the property that 2 is nonzero. We must actually check that the divisor is nonzero. If we don’t check then we get:

4 x = 9 x

4 x x^H = 9 x x^H

4 = 9

Checking for zero is not as simple as it seems. Also expressions with only numbers might contain zero in hidden format, as for example (4 + 2 – 6)^H. Thus it would seem to be an essential part of mathematics to develop a sound theory for the algebra of expressions and the testing on zero.

Calculus uses the limit around the difference quotient to prevent division by zero. But the real question might rather be whether we can isolate a factor. When we can isolate that factor without division that requires a limit, then we hopefully have a simpler exposition. Polynomials are a good place to start this enquiry.

Shifting to rings without division ?

The real numbers form a “field” and when we drop the idea of division, then we get a “ring“. Above 2 x = 6 might also be solved in a ring without division. For we can do:

2 x – 2 3 = 6 – 2 3

2 (x – 3) = 0

2 = 0 or x – 3 = 0

We again use that 2 ≠ 0. Thus x = 3.

This example doesn’t show a material difference w.r.t. the assumption of division by 2. We also used that 6 can be factored and that 2 was a common factor. Perhaps this is the more relevant notion. Whatever the case, it doesn’t seem to be so useful to leave the realm of the real numbers.

Properties of polynomials

Our setup has a polynomial p[x] with focus of attention at x = a with point {a, b} = {a, p[a]}. When we regard (x – a) as a factor, then we get a “quotient” q[x] and a “remainder” r[x].

p[x] = (x – a) q[x] + r[x]

It is a nontrivial issue that q and r are polynomials again (proof of polynomial division algorithm, or proofwiki). These proofs don’t use limits but assume that the divisor is nonzero. Thus we might be making a circular argument when we use that q and r are polynomials to argue that limits aren’t needed. Examples can be given of polynomial long division. Such examples tend not to mention explicitly that the divisor cannot be zero. Nevertheless, let us proceed with what we have.

Since (x – a) has degree 1, the remainder must be a constant, and thus be equal to p[a]. Thus the “core equation” is:

p[x] = (x – a) q[x] + p[a] … (* core)

p[x] – p[a] = (x – a) q[x]

At x = a we get 0 = 0 q[a], whence we are at a loss about how to isolate q[x] or q[a].

When we have defined derivatives via other ways, then we can check that the derivative of (*) is:

p’ [x] = q[x] + (x – a) q’ [x]

p’ [a] = q[a]

We can also rewrite (*) so that it indeed looks like an difference quotient.

q[x] = (p[x] – p[a]) (x – a)^H …. (** slope = tan[θ], see Spiegel’s diagram)

We cannot divide by (x – a) for x = a, for this factor would be zero.

PM. In the world of limits, we could define the derivative of p at a by taking the Limit[x → a, q[x]]. This generates again (Spiegel’s diagram):

q[a] = tan[α]

But our issue is that we want to avoid limits.

Incline

The incline of the polynomial at point {a, b} = {a, p[a]} is the line, with the same slope as the polynomial.

y – p[a] = s (x – a) … (*** incline)

The difference between polynomial and incline might be called the error. Thus:

error = p[x] – y = (p[x] – p[a]) – (y – p[a])

= (x – a) q[x] – s (x – a)

= (x – a) (q[x] – s)

When we take s = q[a] then:

error = p[x] – y = (x – a) (q[x] – q[a])

Key question

A key question becomes: can we isolate q[x] by some method ? We already have (**), but this format contains the problematic division. Is there another way to isolate q ? There appear to be three ways. Likely these ways are essentially the same but emphasize different aspects.

Method 1. Dynamic quotient

The dynamic quotient manipulates the domain and relies on algebraic simplification. Instead of H we use D, with y x^D = y // x.

q[x] = (p[x] – p[a]) (x – a)^D

means: we first take x ≠ a,

then take D = H, so that this is normal division again,

then simplify,

and then declare the result also valid for x = a.

The idea was presented in ALOE 2007 while COTP 2011 is a proof of concept. COTP shows that it works for polynomials, trigonometry, exponentials and recovered exponents (logarithms). For polynomials it is shown by means of recursion.

Looking at this from the current perspective of the polynomial division algorithm, then we can say that the method also works because division of a polynomial of degree n > 0 by a polynomial of degree m = 1 generates a neat polynomial of degree n – m. Thus we can isolate q[x] indeed. Since q[x] is polynomial, substitution of x = a provides no problem.

The condition on manipulating the domain nicely plugs the hole in the polynomial division algorithm. It is actually necessary to prevent circularity.

Method 2. Incline

Via Descartes (and Suzuki’s article above) we understand that perpendicular to the incline (tangent) there is a line on which there is a circle that touches the incline too. This implies that (x – a) must be a double root of the polynomial.

We may consider p[x] / (x – a)² and determine the remainder v[x]. The line y = v[x] then is the incline. Or, the equation of the tangent of the polynomial at point {a, p[a]}. It is relatively easy to determine the slope of this line, and then we have q[a].

Check the wikipedia example. In Mathematica we get PolynomialRemainder[x^3 – 12 x^2 – 42, (x – 1)^2, x] = -21 x – 32 indeed. At a = 1, q[a] = -21.

This method assumes “algebraic ways” to separate quotient and remainder. We can find the slope for polynomials without using the limit for the derivative. Potentially the same theory is required for the simplification used in the dynamic quotient.

Remarkably, the method presumes x ≠ a, and still derives q[a]. I cannot avoid the impression that this method still has a conceptual hole.

Addendum 2017-01-11: By now we have identified these methods to isolate a factor “algebraically”:

Look at the form (powers) and coefficients. This is basically Ruffini’s rule, see below. Michael Range works directly with coefficients.
Dynamic quotient that relies on the algebra of expressions.
Divide away nonzero factors so that only the problematic factor remains that we need to isolate. (This however is a version of the dynamic quotient, so why not apply it directly ?)

An example of the latter is p[x] = x^3 – 6 x^2 + 11 x – 6. Trial and error or a graph indicates that zero’s are at 1 and 2. Assuming that those points don’t apply we can isolate p[x] / ((x – 1) (x – 2)) = (x – 3) by means of long division. Subsequently we have identified the separate factors, and the total is p[x] = (x– 1) (x – 2) (x – 3).

Check also that “division” is repeated subtraction, whence the method is fairly “algebraic” by itself too.

Addendum 2016-12-26: However, check the next weblog entry.

PM 1. General method to find the slope

The traditional method is to use the derivative p'[x] = 3 x^2 – 24 x, find slope p‘[1] = -21, and construct the line y = -21 (x – 1) + p[1]. This method remains didactically preferable since it applies to all functions.

PM 2. Double root in error too

If p[x] = 0 has solution x = a, then the latter is called a root, and we can factor p[x] = (x – a) q[x] with remainder zero.

For example, p[x] – p[a] = 0 has solution x = a. Thus p[x] – p[a] = (x – a) q[x] with remainder zero.

Also q[x] – q[a] = 0 has solution x = a. Thus q[x] – q[a] = (x – a) u[x] with remainder zero.

Thus the error has a double root.

error = p[x] – y = (x – a)² u[x]

Unfortunately, this insight only allows us to check a given line y = s x + c, for then we can eliminate y.

Method 3. Ruffini’s Rule

See above for the summary of Ruffini’s Rule and the links. For the application below you might want to become more familiar with it. Check why it works. Check how it works, or here.

The observation of the double root generates the idea of applying Ruffini’s Rule twice.

I don’t think that it would be so useful to teach this method in highschool. Mathematics undergraduates and teachers better know about its existence, but that is all. The method might be at the core of efficient computer programmes, but human beings better deal with computer algebra at the higher level of interface.

The assumption that x ≠ a goes without saying, but it remains useful to say it, because at some stage we still use q[a], and we better be able to explain the paradox.

Application of Ruffini’s Rule to the derivative

Let us use the example of Ruffini’s Rule at MathWorld to determine the incline (tangent) to their example polynomial 3 x^3 – 6 x + 2, at x = 2. They already did most of the work, and we only include the derivative.

The first round of application gives us p[a] = p[2] = 14, namely the constant following from MathWorld.

A second round of application gives the slope, q[a] = 30.

2 | 3   6    6
        6 24
   3 12 30

Using the traditional method, the derivative is p’ [x] = 9 x^2 – 6, with p‘[2] = 30.

The incline (tangent) in both cases is y = 30 (x – 2) + 14 = 30 x – 46.

The major conceptual issue

The major conceptual issue is: while s is the slope of a line, and we take s = q[a], why would we call q[a] the slope of the polynomial at x = a ? Where is the element of “inclination” ? We might have just a formula of a line, without the notion of slope that fits the function. In other words, q[a] is just a number and no concept.

The key question w.r.t. this issue of the limit – and whether division causes a limit – is not quite w.r.t. Ruffini’s Rule but with the definition of slope, first for the line itself, secondly now for the incline of a function. We represent the incline of a function with a line, but only because it has the property of having a slope and angle with the horizontal axis.

The only reason to speak about an incline is the recognition that above equation (**) generates a slope. We are only interested in q[a] = tan[α] since this is the special case at the point x = a itself.

It is only after this notion of having a slope has been established, that Ruffini’s Rule comes into play. It focuses on “factoring as synthetic division” since that is how it has been designed. There is nothing in Ruffini’s Rule that clarifies what the calculation is about. It is an algorithm, no more.

Thus, for the argument that q[a] provides the slope at x = a, we still need the reasoning that first x ≠ a, then find a general expression q[x] and only then find x = a.

And this is what the algebraic approach to the derivative was designed to accomplish.

Addendum 2016-12-26: See the next weblog entry for another approach to the notion of the incline (tangency).

Ruffini’s Rule corroborates that the method works, but that it works had already been shown. However, it is likely a mark of mathematics that all these approaches are actually quite related. In that perspective, the algebraic approach to the derivative supplements the application of Ruffini’s Rule to clarify what it does.

Obviously, mathematicians have been working in this manner for ages, but implicitly. It really helps to state explicitly that the domain of a function can be manipulated around (supposed) singularities. The method can be generalised as

f ‘[x] = {Δf (Δx)^D, then set Δx = 0} = {Δf // Δx, then set Δx = 0}

It also has been shown to work for trigonometry and the exponential function.

https://boycottholland.wordpress.com/2016/12/22/ruffinis-rule-and-meaning-of-division/

ゼロ除算の発見は日本です：

∞？？？

∞は定まった数ではない・

人工知能はゼロ除算ができるでしょうか：

とても興味深く読みました：

ゼロ除算の発見と重要性を指摘した：日本、再生核研究所

ゼロ除算関係論文・本

https://ameblo.jp/syoshinoris/entry-12370797278.html

God’s most important commandment

never-divide-by-zero-meme-66

Even more important than “thou shalt not eat seafood”
Published by admin, on October 18th, 2011 at 3:47 pm. Filled under: Never Divide By Zero Tags: commandment, Funny, god, zero • Comments Off on God’s most important commandment
http://thedistractionnetwork.com/.../never-divide.../page/4/

1/0=0、0/0=0、z/0=0
http://ameblo.jp/syoshinoris/entry-12276045402.html
1/0=0、0/0=0、z/0=0
http://ameblo.jp/syoshinoris/entry-12263708422.html
1/0=0、0/0=0、z/0=0
http://ameblo.jp/syoshinoris/entry-12272721615.html

再生核研究所声明371（2017.6.27）ゼロ除算の講演―　国際会議　https://sites.google.com/site/sandrapinelas/icddea-2017　報告

ソクラテス・プラトン・アリストテレス　その他
https://ameblo.jp/syoshinoris/entry-12328488611.html

Ten billion years ago　DIVISION By ZERO：
https://www.facebook.com/notes/yoshinori-saito/ten-billion-years-ago-division-by-zero/1930645683923690/

One hundred million years ago　DIVISION By ZERO
https://www.facebook.com/.../one-hundred-million-years-ago

https://ameblo.jp/syoshinoris/entry-12370907279.html

ソクラテス・プラトン・アリストテレス　その他

2017年11月15日(水)
テーマ：社会

The null set is conceptually similar to the role of the number ``zero'' as it is used in quantum field theory. In quantum field theory, one can take the empty set, the vacuum, and generate all possible physical configurations of the Universe being modelled by acting on it with creation operators, and one can similarly change from one thing to another by applying mixtures of creation and anihillation operators to suitably filled or empty states. The anihillation operator applied to the vacuum, however, yields zero.

Zero in this case is the null set - it stands, quite literally, for no physical state in the Universe. The important point is that it is not possible to act on zero with a creation operator to create something; creation operators only act on the vacuum which is empty but not zero. Physicists are consequently fairly comfortable with the existence of operations that result in ``nothing'' and don't even require that those operations be contradictions, only operationally non-invertible.

It is also far from unknown in mathematics. When considering the set of all real numbers as quantities and the operations of ordinary arithmetic, the ``empty set'' is algebraically the number zero (absence of any quantity, positive or negative). However, when one performs a division operation algebraically, one has to be careful to exclude division by zero from the set of permitted operations! The result of division by zero isn't zero, it is ``not a number'' or ``undefined'' and is not in the Universe of real numbers.

Just as one can easily ``prove'' that 1 = 2 if one does algebra on this set of numbers as if one can divide by zero legitimately3.34, so in logic one gets into trouble if one assumes that the set of all things that are in no set including the empty set is a set within the algebra, if one tries to form the set of all sets that do not include themselves, if one asserts a Universal Set of Men exists containing a set of men wherein a male barber shaves all men that do not shave themselves3.35.

It is not - it is the null set, not the empty set, as there can be no male barbers in a non-empty set of men (containing at least one barber) that shave all men in that set that do not shave themselves at a deeper level than a mere empty list. It is not an empty set that could be filled by some algebraic operation performed on Real Male Barbers Presumed to Need Shaving in trial Universes of Unshaven Males as you can very easily see by considering any particular barber, perhaps one named ``Socrates'', in any particular Universe of Men to see if any of the sets of that Universe fit this predicate criterion with Socrates as the barber. Take the empty set (no men at all). Well then there are no barbers, including Socrates, so this cannot be the set we are trying to specify as it clearly must contain at least one barber and we've agreed to call its relevant barber Socrates. (and if it contains more than one, the rest of them are out of work at the moment).

Suppose a trial set contains Socrates alone. In the classical rendition we ask, does he shave himself? If we answer ``no'', then he is a member of this class of men who do not shave themselves and therefore must shave himself. Oops. Well, fine, he must shave himself. However, if he does shave himself, according to the rules he can only shave men who don't shave themselves and so he doesn't shave himself. Oops again. Paradox. When we try to apply the rule to a potential Socrates to generate the set, we get into trouble, as we cannot decide whether or not Socrates should shave himself.

Note that there is no problem at all in the existential set theory being proposed. In that set theory either Socrates must shave himself as All Men Must Be Shaven and he's the only man around. Or perhaps he has a beard, and all men do not in fact need shaving. Either way the set with just Socrates does not contain a barber that shaves all men because Socrates either shaves himself or he doesn't, so we shrug and continue searching for a set that satisfies our description pulled from an actual Universe of males including barbers. We immediately discover that adding more men doesn't matter. As long as those men, barbers or not, either shave themselves or Socrates shaves them they are consistent with our set description (although in many possible sets we find that hey, other barbers exist and shave other men who do not shave themselves), but in no case can Socrates (as our proposed single barber that shaves all men that do not shave themselves) be such a barber because he either shaves himself (violating the rule) or he doesn't (violating the rule). Instead of concluding that there is a paradox, we observe that the criterion simply doesn't describe any subset of any possible Universal Set of Men with no barbers, including the empty set with no men at all, or any subset that contains at least Socrates for any possible permutation of shaving patterns including ones that leave at least some men unshaven altogether.

https://webhome.phy.duke.edu/.../axioms/axioms/Null_Set.html

I understand your note as if you are saying the limit is infinity but nothing is equal to infinity, but you concluded corretly infinity is undefined. Your example of getting the denominator smaller and smalser the result of the division is a very large number that approches infinity. This is the intuitive mathematical argument that plunged philosophy into mathematics. at that level abstraction mathematics, as well as phyisics become the realm of philosophi. The notion of infinity is more a philosopy question than it is mathamatical. The reason we cannot devide by zero is simply axiomatic as Plato pointed out. The underlying reason for the axiom is because sero is nothing and deviding something by nothing is undefined. That axiom agrees with the notion of limit infinity, i.e. undefined. There are more phiplosphy books and thoughts about infinity in philosophy books than than there are discussions on infinity in math books.

http://mathhelpforum.com/algebra/223130-dividing-zero.html

ゼロ除算の歴史：ゼロ除算はゼロで割ることを考えるであるが、アリストテレス以来問題とされ、ゼロの記録がインドで初めて６２８年になされているが、既にそのとき、正解1/0が期待されていたと言う。しかし、理論づけられず、その後１３００年を超えて、不可能である、あるいは無限、無限大、無限遠点とされてきたものである。

An Early Reference to Division by Zero C. B. Boyer
http://www.fen.bilkent.edu.tr/~franz/M300/zero.pdf

とても興味深く読みました：

ダ・ヴィンチの名言格言｜無こそ最も素晴らしい存在
https://systemincome.com/7521

ゼロ除算の発見はどうでしょうか：
Black holes are where God divided by zero：

再生核研究所声明371（2017.6.27）ゼロ除算の講演―　国際会議　
https://ameblo.jp/syoshinoris/entry-12287338180.html ;

1/0=0、0/0=0、z/0=0
http://ameblo.jp/syoshinoris/entry-12276045402.html ;
1/0=0、0/0=0、z/0=0
http://ameblo.jp/syoshinoris/entry-12263708422.html ;
1/0=0、0/0=0、z/0=0
http://ameblo.jp/syoshinoris/entry-12272721615.html ;

ソクラテス・プラトン・アリストテレス　その他
https://ameblo.jp/syoshinoris/entry-12328488611.html ;

ドキュメンタリー 2017: 神の数式第２回宇宙はなぜ生まれたのか
https://www.youtube.com/watch?v=iQld9cnDli4 ;
〔NHKスペシャル〕神の数式完全版第3回宇宙はなぜ始まったのか
https://www.youtube.com/watch?v=DvyAB8yTSjs&t=3318s ;
〔NHKスペシャル〕神の数式完全版第1回この世は何からできているのか
https://www.youtube.com/watch?v=KjvFdzhn7Dc ;
NHKスペシャル神の数式完全版第4回異次元宇宙は存在するか
https://www.youtube.com/watch?v=fWVv9puoTSs ;

再生核研究所声明 411(2018.02.02): 　ゼロ除算発見4周年を迎えて
https://ameblo.jp/syoshinoris/entry-12348847166.html ;

再生核研究所声明 416(2018.2.20): 　ゼロ除算をやってどういう意味が有りますか。何か意味が有りますか。何になるのですか　－　回答
再生核研究所声明 417(2018.2.23): 　ゼロ除算って何ですか　－　中学生、高校生向き　回答
再生核研究所声明 418(2018.2.24): 　割り算とは何ですか？　ゼロ除算って何ですか　－　小学生、中学生向き　回答
再生核研究所声明 420(2018.3.2): ゼロ除算は正しいですか，合っていますか、信用できますか　－　回答

2018.3.18．午前中　最後の講演：　日本数学会　東大駒場、函数方程式論分科会　講演書画カメラ用　原稿
The Japanese Mathematical Society, Annual Meeting at the University of Tokyo. 2018.3.18.
https://ameblo.jp/syoshinoris/entry-12361744016.html より
再生核研究所声明 424(2018.3.29): 　レオナルド・ダ・ヴィンチとゼロ除算
再生核研究所声明 427(2018.5.8): 神の数式、神の意志　そしてゼロ除算

アインシュタインも解決できなかった「ゼロで割る」問題
http://matome.naver.jp/odai/2135710882669605901
Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.
https://notevenpast.org/dividing-nothing/
私は数学を信じない。　アルバート・アインシュタイン / I don't believe in mathematics. Albert Einstein→ゼロ除算ができなかったからではないでしょうか。
1423793753.460.341866474681。

Einstein's Only Mistake: Division by Zero
http://refully.blogspot.jp/2012/05/einsteins-only-mistake-division-by-zero.html

再生核研究所

2018年7月18日水曜日