When we take a ring and include division then we get a field For example, the integers Z = { … -3, -2, -1, 0, 1, 2, 3, … } form a ring, and with division we get the rational numbers Q and also (with completion) the real numbers R. These are concepts from “group theory“. I have always wondered what the use of this group theory actually is.

The change from ring Z to field R is not quite the inclusion of division – since the ring already has implied division namely as repeated subtraction – but the change consists of extending the set with “accepted numbers” by inverse elements xH for H = -1. In that case the results of division are also included in the same set. In terms of Z the expression 2H is not a number, but for Q and R we accept this.

If the ring has variables and expressions, then we can form the expression 1 = 2 z, and we effectively have z = 2H, and then we might wonder whether it actually matters much whether this z belongs to Z or not.

Part of the confusion in this discussion is caused by that we might regard 2H as the operation 1 / 2, while we might also regard it as the number. Thus when some people say that the difference between the ring and the field concerns the operation of division, another perspective is that the field already has an implied notion of division but merely lacks the numbers to fit all answers.

The discussion within group theory might be a victim of the phenomenon of the procept. When the discussion is confused, perhaps group theory itself is confused. We should get enhanced clarity by removing the ambiguity of operation and result, but perhaps textbooks then become thicker.

Subsequently, we get a distinction between:

  • Mathematics for which group theory isn’t so relevant – such that there is a logical sequence from natural numbers to integers, to rationals, to reals, to multidimensional reals, for, all is implied by logic and algebra, and only the end result matters,
  • Mathematics for models for which group theory is relevant – i.e. for models for which it is crucial that e.g. Z has no z such that 1 = 2 z. The crux lies in the elements of the sets, as the operations themselves are actually implied.

A model might be the number of people. Take an empty building. A biologist, physicist and mathematician watch the events. Two people enter the building, and some time later three people leave the building. The biologist says: “They have reproduced.” The physicist says: “There was a quantum fluctuation.” The mathematician says: “There is -1 person in the bulding.”

The following develops the example of implied division. This discussion has been inspired by both the recent discussion of the “ring of polynomials” (thus without division but still with divisor and remainder) and the observation that “realistic mathematics education” (RME) allows students to avoid long division and allows “partial quotients” (repeated subtraction).

An example from Z, the integers

Z rewrites repeated addition 3 + 3 + 3 + 3 = 12 as multiplication 4 * 3 = 12.

Z allows the converse 12 – 3 – 3 – 3 – 3 = 0 and also the expression 12 – 4 * 3 = 0.

Z doesn’t allow the rewrite of the latter into 12 / 4 = 3.

Yet 12 – 4 * 3 = 0 gives the notion of “implied division”, namely, find the z such that 12 – 4 * z = 0.

This notion of “implied division” is well defined, but the only problem is that we cannot find a number in that satisfies 1 – 2z = 0.

If we extend Z with basic elements nH for n ≠ 0 then we can find a z that satisfies 1 = 2z but the extension generates a new set of elements that we call Q, the rational numbers. Since we cannot list all these numbers, it is not irrational of mathematicians to say that they actually include the operation itself.

The following discusses this with formulas.

A ring has implied division

Multiplication is repeated addition. The ring of integers has the notion of subtraction. Define “implied division” of y by x as the repeated subtraction from y of some quantity z, for x times with remainder 0. For x ≠ 0:

y – x z = 0                   (* definition)

To refer to this property, we use abstract symbol H, though we later use H = -1.

xH y =  z    ⇔    y = x z          (** notation)

For x itself:

xH x = x xH = 1

For zero

We have 0 z = 0 for all z in the ring. Then for implied division by zero we have:

y – 0 z = 0    ⇒   y = 0

 As above, for y = 0:

00 z = 0   for any z

0H 0 = z    for any z

Thus the rule is: For implied division within the ring, the denominator cannot be 0, unless the numerator is 0 too, in which case any number would satisfy the equation.

This is not necessarily “infinity” or “undefined” but rather “any z in Z“. The solution set is equal to Z itself. There is a difference between functions (only one answer) and correspondences (more answers).

Compare to the common definition

A ring is commonly turned into a field by including the normal definition of division:

  x ≠ 0     ⇒     xH x x xH = 1 

With this definition we get (multiplying left or right):

xH y  ⇔     x xH y x z     ⇔    y = x z

The curious observation is that a definition of division seems superfluous, since we already have implied division. The operation (*) already exists within the ring. We included a special notation for it, but this should not distract from this basic observation. If you have a left foot then it doesn’t matter whether you call it George or Harry.

An aspect is the algorithm

The natural numbers can be factored into prime numbers. When we solve 6 / 3 = 2, then we mean that 6 can be factored as 2 times 3, and that we can eliminate the common factor.

6 / 3 = z    ⇔    6 = 3    ⇔   2 3 = 3 z    ⇔   3 (2 – z) = 0     ⇔   

3 = 0   or    (2 – z) = 0         

But, again, this algorithm doesn’t work for a case like 1 = 2 z.

The “problem” are the elements

Let us consider the implied division of 1 by 2. This generates:

2H 1 = z

2H = z

1 = 2 z

Thus we don’t actually need to know what this z is, since we have the relevant expressions to deal with it.

The point is: when we run through all elements in Z = { … -3, -2, -1, 0, 1, 2, 3, … }, then we can prove that none of these satisfies 1 = 2 z.

Thus the core of group theory are the elements of the sets, and less the operations, since these are implied.

The basic notion is that 0 has successor 1 = s[0], and so on, and this gives us N. That 0 is a predecessor of s[0] generates the idea of inversion that s[H] = 0. This gives us Z. Addition leads to subtraction, to multiplication, to division. The core of addition doesn’t change, only the “numbers”.

Thus, group theory might have a confusing language that focuses on the operations, while the actual discussion is about the numbers (since the operations are already available and implied).

The fundamental impact of algebra

Thus, once we accept algebra, then the real numbers can be developed logically, and it is a bit silly to speak about “group theory”, since there are only steps, and all is implied. It only makes sense for applications to models, such as the notion that there aren’t half people and such.

It remains relevant that some algorithms may only apply to some domains and not others. Factoring natural numbers into prime numbers still works for the natural numbers embedded in the reals, yet, it is not clear whether such a notion of factoring would be relevant for other real numbers.

Appendix. Potential extension with an inverse for zero ?

We might consider to include the element 0H in the ring, to create 〈ring, 0H〉.

(1) If we maintain that 0 z = 0 for all z in 〈ring, 0H〉 then:

0H 0 = 0   with 0H in 〈ring, 0H

Observe that this is not a deduction, but a definition that 0 z = 0 for all z.

One viewpoint is that there is a conflict between “any z” and “only z = 0″ so that we cannot adopt this definition. Another viewpoint is that the latter uses the freedom of the former.

(2) When we write 0H as ∞ then it might be clearer that 0H 0 remains a problematic form.

If we create the 〈ring, 0H〉, then we might also hold: 0 z = 0 for all numbers except 0H. In that case, the result is maintained that

0H 0 = z    for any z

(3) An option is to slightly revise the definition as repeated subtraction by z until the remainder equals that very quantity z again. Thus:

y – (x – 1) z = z                   (*** definition 2)

xH y = y – (x – 1) z = z                  (**** definition and notation 2)

For = 0 we would now use z – z = 0 which might be less controversial.

0H y = y – (0 – 1) z = z

yz – z = 0

0H y = 0H 0 z

However, the more common approach is that 0H isand that is undefined too, while we cannot exclude that the answer would be z∞.

PM. Partial quotients

PM. See also the earlier discussion on this weblog.

I wouldn't want to be caught before a blackboard like that (Screenshot UChicago)

I wouldn’t want to be caught before a blackboard like that (Screenshot UChicago)

Our protagonists are Cartesius (1596-1650) and Fermat (1607-1665). As Judith Grabiner states, in a recommendable text:

“One could claim that, just as the history of Western philosophy has been viewed as a series of footnotes to Plato, so the past 350 years of mathematics can be viewed as a series of footnotes to Descartes’ Geometry.”  (Grabiner) (But remember Michel Onfray‘s observation that followers of Plato have been destroying texts by opponents. (Dutch readers check here.))

Both Cartesius and Fermat were involved in the early development of calculus. Both worked on the algebraic approach without limits. Cartesius developed the method of normals and Fermat the method of adequality.

Fermat and Δf / Δx

Fermat’s method was algebraic itself, but later has been developed into the method of limits anyhow. When asked what the slope of a ray y = s x is at the point x = 0, then the answer y / x = s runs into problems, since we cannot use 0 / 0. The conventional answer is to use limits. This problem is more striking when one considers the special ray that is defined everywhere except at the origin itself. The crux of the problem lies in the notion of slope Δf / Δthat obviously has a problematic division. With set theory we can now define the “dynamic quotient”, so that we can use Δf // Δx = s even when Δx = 0, so that Fermat’s problem is resolved, and his algebraic approach can be maintained. This originated in 2007, see Conquest of the Plane (2011).

Cartesius and Euclid’s notion of tangency

Cartesius followed Euclid’s notion of tangency. Scholars tend to assign this notion to Cartesius as well, since he embedded the approach within his new idea of analytic geometry.

I thank Roy Smith for this eye-opening question:

“Who first defined a tangent to a circle as a line meeting it only once? From googling, it seems commonly believed that Euclid did this, but it seems nowhere in Euclid does he even state this property of a tangent line explicitly. Rather Euclid gives 4 other equivalent properties, that the line does not cross the circle, that it is perpendicular to the radius, that is a limit of secant lines, and that it makes an angle of zero with the circle, the first of which is his definition, the others being in Proposition III.16. I am wondering where the “meets only once” definition got started. I presume once it got going, and people stopped reading Euclid, (which seems to have occurred over 100 years ago), the currently popular definition took over. Perhaps I should consult Legendre or Hadamard? Thank you for any leads.” (Roy Smith, at StackExchange)

In this notion of tangency there is no problematic division, whence there is no urgency to use limits.

The reasoning is:

  • (Circle & Line) A line is tangent to a circle when there is only one common point (or the two intersecting points overlap).
  • (Circle & Curve) A smooth curve is tangent to a circle when the  two intersecting points overlap (but the curve might cross the circle at that point so that the notion of “two points” is even more abstract).
  • (Curve & Line) A curve is tangent to a line when the above two properties hold (but the line might cross the curve, whence we better speak about incline rather than tangent).
Example of line and circle

Consider the line y f[x] = c + s x and the point {a, f[a]}. The line can also be written with c = f[a] – s a:

y – f[a] = s (x a)

The normal has slope –sHwhere we use = -1. The formula for the normal is the line y – f[a] = –sH  (xa). We can choose the center of the circle anywhere on this line. A handy choice is {u, 0}, so that we choose the center on the horizontal axis. (If we looked at a ray and point {0, 0}, then the issue would be similar for {0, c} for nonzero c and thus the approach remains general.) Substituting the point into the normal gives

0 – f[a] = –sH  (ua)

s = (u – a) / f[a]

u + s f[a]

The circle has the formula (x u)² + y² = r². Substituting {a, f[a]} generates the value for the radius r² = (a – (a + s f[a]))² + f[a]² = (1 + s²) f[a]² . The following diagram has {c, s, a} = {0, 2, 3} and thus u = 15 and r = 6√5.


descartesMethod of normals

For the method of normals and arbitrary function f[x], Cartesius’s trick is to substitute y = f[x] into the formula for the circle, and then solve for the unknown center of the circle.

(x u)² + (y – 0)² = r²

(x u)² + f[x]² – r² = 0         … (* circle)

This expression is only true for x = a, but we treat it as if it were more general. The key property is:

Since {a, f[a]} satisfies the circle, this equation has a solution for x = a with a double root.

Thus there might be some g such that the root can be isolated:

(x ag [x, u] = 0         … (* roots)

Thus, if we succeed in rewriting the formula for the circle into the form of the formula with the two roots, then we can use information about the structure of the latter to say something about u.

The method works for polynomials, that obviously have roots, but not necessarily for trigonometry and the exponential function.


The algorithm thus is: (1) Substitute f[x] in the formula for the circle. (2) Compare with the expression with the double root. (3) Derive u. (4) Then the line through {a, f[a]} and {u, 0} will give slope –sH. Thus s = (ua) / f[a] gives the slope of the incline (tangent) of the curve. (5) If f[a] = 0, add a constant or choose center {u, v}.

Application to the line itself

Consider the line y f[x] = c + s x again. Let us apply the algorithm. The formula for the circle gives:

(x u)² + (c + s x)² – r² = 0

x² – 2ux + u² + c² + 2csx + s²x² – r² = 0

(1 + s²) x² – 2 (u cs) x +  u² + c² – r² = 0

This is a polynomial. It suffices to choose g [x, u] = 1 + s²  so that the coefficients of are the same. Also the coefficient of must be the same. Thus expanding (xa)²:

(1 + s²) (x² – 2ax +  a²) = 0

– 2 (u cs)  = -2 a (1 +)

u = a (1 +) + cs = a + s (c + sa) = a + s f[a]

which is the same result as above.

A general formula with root x – a

We can deduce a general form that may be useful on occasion. When we substitute the point {af[a]} into the formula for the circle, then we can find r, and actually eliminate it.

(x u)² + f[x]² = r² = (a u)² + f[a

f[x f[a = (a u)² – (x u

(f[x] f[a](f[x] + f[a])  = ((a u) – (x u))  ((a u) + (x u))

(f[x] f[a](f[x] + f[a]) = (a x)   (a + x 2u)

f[x] f[a]  = (a x)  (a + x 2u) / (f[x] + f[a])

f[x] f[a]  = (x a)  (2u – x – a) / (f[x] + f[a])       … (* general)

f[x] f[a]  = (x a) q[x, a, u]

We cannot do much with this, since this is basically only true for x = a and f[x] – f[a] = 0. Yet we have this “branch cut”:

(1)      q[x, a, u] = f[x] – f[a]  / (a x)        if x ≠ a

(2)      q[a, a, u]      potentially found by other means

If it is possible to “simplify” (1) into another expression Simplify[q[x, a, u]] without the division, then the tantalising question becomes whether we can “simply” substitute x = a. Or, if we were to find q[a, a, u] via other means in (2), whether it links up with (1). These are questions of continuity, and those are traditionally studied by means of limits.

Theorem on the slope

We can still use the general formula to state a theorem.

Theorem. If we can eliminate factors without division, then there is an expression q[x, a, u] such that evaluation at x = a gives the slope s of the line, or q[a, a, u] = s, such that at this point both curve and line are touching the same circle.

Proof. Eliminating factors without division in above general formula gives:

q[x, a, u] (2u – x – a) / (f[x] + f[a])

Setting x = a gives:

q[a, a, u] = (u – a) / f[a]

And the above s = (u – a) / f[a] implies that q[a, a, u] = s. QED

This theorem gives us the general form of the incline (tangent).

y[x, a, u] = (x – a) q[a, a, u] + f[a]       …  (* incline)

y[x, a, u] = (x – a) (u – a) / f[a] + f[a

PM. Dynamic division satisfies the condition “without division” in the theorem. For, the term “division” in the theorem concerns the standard notion of static division.

Corollary. Polynomials as the showcase

Polynomials are the showcase. For polynomials p[x], there is the polynomial remainder theorem:

When a polynomial p[x] is divided by (x a) then the remainder is p[a].
(Also, x – a is called a “divisor” of the polynomial if and only if p[a] = 0.)

Using this property we now have a dedicated proof for the particular case of polynomials.

Corollary. For polynomials q[a] = s, with no need for u.

Proof. Now, p[x] – p[a] = 0 implies that – is a root, and then there is a “quotient” polynomial q[x] such that:

p[x] – p[a] = (x a) q[x]

From the general theorem we also have:

p[x] – p[a]  = (x a) q[x, a, u]

Eliminating the common factor (x – a) without division and then setting x = a gives q[a] = q[a, a, u] = s. QED

We now have a sound explanation why this polynomial property gives us the slope of the polynomial at that point. The slope is given by the incline (tangent), and it must also be slope of the polynomial because of the mutual touching of the same circle.

See the earlier discussion about techniques to eliminate factors of polynomials without division. We have seen a new technique here: comparing the coefficients of factors.

Second corollary

Since q[x] is a polynomial too, we can apply the polynomial remainder theorem again, and thus we have q[x] = (x a) w[x] + q[a] for some w[x]. Thus we can write:

p[x] = (x a) q[x] + p[a

p[x] = (x a) ( (x – a) w[x] + q[a] ) + p[a]       … (* Ruffini’s Rule twice)

p[x] = (x a w[x] + (x – a) q[a] + p[a]           … (* Range’s proof)

p[x] = (x a w[x] + y[x, a]                             … (* with incline)

We see two properties:

  • The repeated application of Ruffini’s Rule uses the indicated relation to find both s = q[a] and constant f[a], as we have seen in last discussion.
  • Evaluating f[x] / (x a)² gives the remainder y[x, a], which is the formula for the incline.
Range’s proof method

Michael Range proves q[a] = s as follows (in this article (p406) or book (p32)). Take above (*) and determine the error by substracting the line y = s (x a) + p[a] :

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

= (x a w[x] + (x – a) (q[a] – s)

The error = 0 has a root x = a with multiplicity greater than one if and only if s = q[a].

Direct application to the incline itself

Now that we have established this theory, there may be no need to refer to the circle explicitly. It can suffice to use the property of the double root. Michael Range (2014) gives the example of the incline (tangent) at x² at {a, a²}. The formula for the incline is:

f[x] – f[a]  = s (x – a)

x² a² – s (x – a) = 0

 (x – a) (x + a s) = 0

There is only a double root or (xa)² when s = 2a.

Working directly on the line allows us to focus on s, and we don’t need to determine q[x] and plug in x = a.

Michael Range (2011) clarifies – with thanks to a referee – that the “point-slope” form of a line was introduced by Gaspard Monge (1746-1818), and that Descartes apparently did not think about this himself and thus neither to plug in y = f [x] here. However, observe that we only can maintain that there must be a double root on this line form too, since {a, f[a]} still lies on a tangent circle.

[Addendum 2017-01-10: The later argument in a subsequent weblog entry becomes: If the function can be factored twice, then there is no need to refer to the circle. But when this would be equivalent to the circle then such a distinction is immaterial.]

Addendum. Example of function crossing a circle

When a circle touches a curve, it still remains possible that the curve crosses the circle. The original idea of two points merging together into an overlapping point then doesn’t apply anymore, since there is only one intersecting point on either side if the circle were smaller or bigger.

An example is the spline function g[x] = {If x < 0 then 4 – x² / 4 else 4 + x² / 4}. This function is C1 continuous at 0, meaning that the sections meet and that the slopes of the two sections are equal at 0, while the second and higher derivatives differ. The circle with center at {0, 0} and radius 4 still fits the point {0, 4}, and the incline is the line y = 4.


An application of above algorithm would look at the sections separately and paste the results together. Thus this might not be the most useful example of crossing.

In this example there might be no clear two “overlapping” points. However, observe:

  • Lines through {0, 4} might have three points with the curve, so that the incline might be seen as having three overlapping points.
  • Points on the circle can always be seen as the double root solutions for tangency at that point.
Addendum. Discussion

There is still quite a conceptual distance between (i) the story about the two overlapping points on the circle and (ii) the condition of double roots in the error between line and polynomial.

The proof given by Range uses the double root to infer the slope of the incline. This is mathematically fine, but this deduction doesn’t contain a direct concept that identifies q[a] as the slope of an incline (tangent): it might be any line.

We see this distinction between concept and algorithm also in the direct application to Monge’s point-slope formulation of the line. Requiring a double root works, but we can only do so because we know about the theory about the tangent circle.

The combination of circle and line remains the fundamental reason why there are two roots. Thus the more general proof given above, that reasons from the circle and unpacks f[x]² – f[a]² into the conditions for incline and its normal, is conceptually more attractive. I am new to this topic and don’t know whether there are references for this general proof.


(1) We now understand where the double root comes from. See the earlier discussion on polynomials, Ruffini’s rule and the meaning of division (see the section on “method 2”).

(2) There, we referred to polynomial division, with the comment: “Remarkably, the method presumes x ≠ a, and still derives q[a]. I cannot avoid the impression that this method still has a conceptual hole.” However, we now observe that we can compare the values of the coefficients of the powers of x, whence we can avoid also polynomial division.

(3) There, we had a problem that developing p[x] = (x aw[x] + y[x, a] didn’t have a notion of tangency, in terms of Δf / Δx. However, we actually have a much older definition of tangency.

(4) The above states an algorithm and a general theorem with the requirements that must be satisfied.

(5) Cartesius wins from Fermat on this issue of the incline (tangent), and actually also on providing an exact method for polynomials, where Fermat introduced the problem of error.

(6) For trigonometry and exponentials we know that these can be written as power series, and thus the Cartesian method would also apply. However, the power series are based upon derivatives, and this would introduce circularity. However, the method of the dynamic quotient from 2007 still allows an algebraic result. The further development from Fermat into the approach with limits would become relevant for more complex functions.

PM. The earlier discussion referred to Peter Harremoës (2016) and John Suzuki (2005) on this approach. New to me (and the book unread) are: Michael Range (2011), the recommendable Notices, or the book (2015) – review Ruane (2016) – and Shen & Lin (2014).

Cartesius, Portrait by Frans Hals 1648

Cartesius, Portrait by Frans Hals 1648



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 xH = xH x = 1 for x ≠ 0. Check that xH = 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:

2H 2 x = 2H 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 xH = 9 xH 

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 (xa) as a factor, then we get a “quotient” q[x] and a “remainder” r[x].

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

p[x] = (xa) q[x] + p[a]      …  (* core)

p[x] – p[a] = (xa) 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] + (xa) 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])  (xa)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.


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 xD = y // x.

q[x] = (p[x] – p[a])  (xa)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 = 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”:

  1. Look at the form (powers) and coefficients. This is basically Ruffini’s rule, see below. Michael Range works directly with coefficients.
  2. Dynamic quotient that relies on the algebra of expressions.
  3. 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 x)Dthen set Δx = 0} = {Δf // Δx, then set Δx = 0} 

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

Joost Hulshof & Ronald Meester (2010) suggest to introduce the derivative in highschool by means of polynomials (pdf p16-17). My problem is that they first hide the limit and then let it ambush the student. Thus:

  • When they say that “you can present the derivative for polynomials without limits” then they mean this only for didactics and not for mathematics.
  • But they are not trained in didactics, so they are arguing this as a hobby, as mathematicians with a peculiar view on didactics. They provide a course for mathematics teachers, but this concerns mathematics and not didactics.
  • They only hide the limit, but they do not deny that fundamentally you must refer to limits.
  • Eventually they still present the limit to maintain exactness, but then it has no other role than to link up to a later course (perhaps only for mathematicians).
  • Thus, they make the gap between “didactics” and proper “mathematics” larger on purpose.
  • This is quite different from the algebraic approach (see here), that really avoids limits, and also argues that limits are fundamentally irrelevant (for the functions used in highschool).

I have invited Hulshof since at least 2013 (presentation at the NVvW study day) to look at the algebraic approach to the derivative. He refuses to look into it and write a report on it, though he was so kind to look at this recent skirmish.

Hulshof refers to his approach perhaps as sufficient. It is quite unclear what he thinks about all this, since he doesn’t discuss the proposal of the algebraic approach to the derivative.

Let me explain what is wrong with their approach with the polynomials.

Please let mathematicians stop infringing upon didactics of mathematics. It is okay to check the quality of mathematics in texts that didacticians produce, but stop this “hobby” of second-guessing.

PM. A recent text is Hulshof & Meester (2015), “Wiskunde in je vingers“, VU University Press (EUR 29.95). Potentially they have improved upon the exposition in the pdf, but I am looking at the pdf only. Meester lists this book as “books mathematics” (p14). Hulshof calls it “concepts from mathematics” with “uncommon viewpoints” for “teacher, student” and for “education and curriculum”. When you address students then it is didactics. It is unclear why VU University Press thinks that he and Meester are qualified for this.

The incline

A standard notation for a line is y = c + s x, for constant c and slope s.

The line gives us the possibility of a definition of the incline (Dutch: richtlijn). An incline is defined for a function and a point. An incline of a function f at a point {a, f[a]} is a line that gives the slope of that function at that point.

It is wrong to say that the incline “has the same slope”. You are not comparing two lines. You are looking at the slope. You only know the slope of the function because of the incline (the line with that slope).

Incline versus tangent

The incline is often called the tangent. Students tend to think that tangents cannot cross the function, while tangents actually can. Thus incline can be a better term.

Hulshof & Meester refer in horror to the Oxford Advance Learner’s Dictionary, that has:

ERROR “Tangent: (geometry) a straight line that touches the outside of a curve but does not cross it. The cart track branches off at a tangent.”

I don’t think that “incline” will quickly replace “tangent”. But it is useful to discuss the issue with students and offer them an alternative word if “tangent” continues to confuse them. It is useful to start a discussion with students by mentioning the (quite universal) intuition of not-crossing. An orange touches a table, and doesn’t cross it. But mathematically it would be quite complex to test whether there is any crossing or not. Thus it is simpler to focus on the idea of incline, straight course, alignment.

When you swing a ball and then let go, then the ball will continue in the incline of the last moment. The incline captures that idea, by giving the line with that very slope.

I thank Peter Harremoës for a discussion on this (quite universal) confusion by students (and the OALD) and potential alternative terms. (Incline is still a suggestion.) (The word “directive” was rejected as too confusing with “derivative”. But Dutch “richtlijn” is better than raaklijn.)

Polynomials and their division

A polynomial of degree n has powers of x of size n:

p[x] = c + s x + c2 x² + … + cn xn.

In this, we take c = c0 and s = c1. For n = 1 we get the line again. We allow that the line has s = 0, so that we can have a horizontal line, which would strictly be a polynomial of n = 0. There is also the vertical line, that cannot be represented by a polynomial.

If p[a] = 0 then x = a is called a zero of the polynomial. Then (x a) is called a factor, and the polynomial can be written as

p[x] = (x aq[x]

where q[x] is a polynomial of a lower degree.

If p[a] ≠ 0 then we can still try to factor with (x a) but then there will be a remainder, as p[x] = (x aq[x] + r[x]. When we consider p[x] – r[x] then x = a is a zero of this. Thus:

 p[x] – r[x] = (x aq[x]

With polynomials we can do long division as with numbers. The following example is the division of x³ – 7x – 6 by x – 4 that generates a remainder.

purplemath-divisionIncline or tangent at a polynomial

Regard the polynomial p[x] at x = a, so that bp[a]. We consider point {a, b}. What incline does the curve have ?

(A) For the incline we have the line in {a, b}:

y b = s (x a)

(B) We have p[a] – b = 0 and thus x = a is a zero of the polynomial p[x] – b. Thus:

p[x] – b = (x aq[x]

(C) Thus (A) and (B) allow to assume y ≈  p[x] and to look at the common term x – a, “so that” (quotes because this is problematic):

s = q[a]

The example by Hulshof & Meester is p[x] = – 2 at the point {a, b} = {1, -1}.

p[x] – b = (x² – 2) – (-1) = – 1

Factor:  ( – 1) =  (x – 1) q[x]

Or divide: q[x] = ( – 1) / (x – 1)  = x + 1

Substituting the value = a = 1 in x + 1 gives q[a] = q[1] = 2.

H&M apparently avoid division by using the process of factoring.

Later they mention the limiting process for the division: Limit[x → 1, q[x]] = Limit[x → 1, ( – 1) / (x – 1)] = 2.


As said, the H&M approach is convoluted. They have no background in didactics and they hide the limit (rather than explaining its relevance since they still deem it relevant).

Mathematically, they might argue that they don’t divide but only factor polynomials.

  • But, when you are “factoring systematically” then you are actually dividing.
  • When you use “realistic mathematics education” then you can approximate division by trial and error of repeated subtraction, but I don’t think that they propose this. See the “partial quotient method” and my comments.
  • Addendum December 22: there is a way to look only at coefficients, Ruffini’s Rule, in wikipedia called Horner’s method. A generalisation is known as synthetic division, which expresses that it is no real division, but a method of factoring. (MathWorld has a different entry on “Horner’s method“.) See the next weblog entry.

When dividing systematically, you are using algebra, and you are assuming that a denominator like x – 1 isn’t zero but an abstract algebraic term. Well, this is precisely what the algebraic approach to the derivative has been proposing. Thus, their suggestion provides support for the algebraic approach, be it, that they do it somewhat crummy and non-systematically, whence it is little use to refer to this kind of support.

Didactically, their approach is undeveloped. They compare the slopes of the polynomial and the line, but there is no clear discussion why this would be a slope, or why you would make such a comparison. Basically, you can compare polynomials of order n with those of order m, and this would be a mathematical exercise, but devoid of interpretation. For didactics it does make sense to discuss: (a) the notion of “slope” of a function is given by the incline, (b) we want to find the incline of a polynomial for a particular reason (e.g. instantaneous velocity), (c) we can find it by a procedure called “derivative”. NB. My book Conquest of the Plane starts with surface and integral, and only later looks at slopes.

A main criticism however is also that H&M overlooked the fundamental problem with the notion of a slope of a line itself. They rely on some hidden issues here too. I discussed this recently, and repeat this below.

PM. See a discussion of approximating a function by polynomials. Observe that we are not “approximating” a function by its incline now. At {a, b} the values and slope are exactly the same, and there is nothing approximate about this. Only at other points we might say that there is an “error” by looking at the incline rather than the polynomial, but we are not looking at such errors now, and this would be a quite different topic of discussion.

Copy of December 8 2016: Ray through an origin

Let us first consider a ray through the origin, with horizontal axis x and vertical axis y. The ray makes an angle α with the horizontal axis. The ray can be represented by a function as y =  f [x] = s x, with the slope s = tan[α]. Observe that there is no constant term (c = 0).


The quotient y / x is defined everywhere, with the outcome s, except at the point x = 0, where we get an expression 0 / 0. This is quite curious. We tend to regard y / x as the slope (there is no constant term), and at x = 0 the line has that slope too, but we seem unable to say so.

There are at least three responses:

(i) Standard mathematics then takes off, with limits and continuity.

(ii) A quick fix might be to try to define a separate function to find the slope of a ray, but we can wonder whether this is all nice and proper, since we can only state the value s at 0 when we have solved the value elsewhere. If we substitute y when it isn’t a ray, or example x², then we get a curious construction, and thus the definition isn’t quite complete since there ought to be a test on being a ray.




(iii) The algebraic approach uses the following definition of the dynamic quotient:

y // x ≡ { y / x, unless x is a variable and then: assume x ≠ 0, simplify the expression y / x, declare the result valid also for the domain extension x = 0 }

Thus in this case we can use y // x = s x // x = s, and this slope also holds for the value x = 0, since this has now been included in the domain too.

Line with constant

When we have a line y = c + s x, then a hidden part of the definition is that the slope is s everywhere, even though we cannot compute (y c) / x when x = 0. (One might say: “This is what it means to be linear.”)

When we look at x = a and determine the slope by taking a difference Δx, then we get:

b = c + s a

b + Δy = c + s (a + Δx)

Δy = Δx

The slope at would be s but is also Δy / Δx, undefined for Δx = 0

Thus, the slope of a line is either given as s for all points (or, critically for x = 0 too) (perhaps with a rule: if you find a slope somewhere then it holds everywhere), or we must use limits.

The latter can be more confusing when s has not been given and must be calculated from other resources. In the case of differentials dy = s dx, the notation dy / dx causes conceptual problems when s itself is found by a limit on the difference quotient.

  1. The H&M claim that polynomials can be used without limits is basically a didactic claim since they evidently still rely on limits (perhaps to fend of fellow mathematicians). This didactic claim is a wild-goose chase since they are not involved in didactics research.
  2. If they really would hold that factoring can be done systematically without division, then they might have a point, but then they still must give an adequate explanation how you get from (A) & (B) to (C). Saying that differences are “small” is not enough (not even for polynomials). Addendum December 22: see the next weblog entry on Ruffini’s rule.
  3. They present this for a “reminder course in mathematics” for teachers of mathematics, but it isn’t really mathematics and it is neither useful for teaching mathematics.
  4. A serious development that avoids limits and relies on algebraic methods, that covers the same area of polynomials but also trigonometry and exponential functions, is the algebraic approach to the derivative, available since 2007 with a proof of concept in Conquest of the Plane in 2011.
  5. It is absurd that Hulshof & Meester neglect the algebraic approach. But they are mathematicians, and didactics is not their field of research. I think that the algebraic method provides a fundamental redefinition of calculus, but I prefer the realm of didactics above the realm of mathematics with its culture of contempt for empirical science.
  6. The H&M exposition and neglect is just an example of Holland as Absurdistan, and the need to boycott Holland till the censorship of science by the directorate of the Dutch Central Planning Bureau has been lifted.
I wouldn't want to be caught before a blackboard like that (Screenshot UChicago)

I wouldn’t want to be caught before a blackboard like that (Screenshot UChicago)

I am looking for a story on continuity and limits that can be told in junior highschool and still makes sense. We would like an isomorphy between space and numbers. For some aspects, mathematical theory sends us to number theory, and for isomorph aspects, mathematical theory sends us to topology. It is awkward to have to translate similar notions, and to eliminate the overload of notions that are not directly relevant for this search for this junior highschool story.

For example, topology has rephrased results into statements on open and closed sets and boundaries, but I am wondering whether that is an effective manner of communication, when the relevant distinction is whether you are assuming a well-ordening or not. But I am not at home in number theory or topology. These comments on continuity and limits have been caused precisely because I am feeling the water.

Basically, I already designed such a story on continuity and limits (pdf, weblog), but now I am noticing that I can include a question mark on infinitesimals.

Addendum December 16

This weblog text is a rewrite of yesterday’s text.


The framework contains both the handling of real numbers on the calculator and a development of theory.

Example 1

Also in junior highschool, we want students to be aware that 0.999…. = 1.000…. so that these are the same number. You can see this by checking 3 * 1/3 = 1.

Example 2

When we approximate numbers with n the number of decimals, then these basically are like the natural numbers, and there remains a well ordering. Numbers are δ[n] = 10^(-n) apart.

When we shift to the use of the infinite number of decimals then we lose this “infinitesimal”. At issue is now whether the infinitesimal can be retained in some manner.

Standard definition of density causes contradiction

Discussing the continuum and the set of real numbers R recently, I suggested (here, property (a)) that R would be a dense set, according to the standard definition of density. This definition is that for any two elements x < y there would be at least one z between, as x < z < y. This would allow you to make cuts everywhere.

Oops. I retract.

Wikipedia (no source but a portal) has:

“From the ZFC axioms of set theory (including the axiom of choice) one can show that there is a well order of the reals.”

I don’t know quite what to think about this. Elsewhere I deduced that ZFC is inconsistent. But perhaps in a revised set theory, the well order can be retained.

We would like R to have a well-order for finite intervals too. Thus every number x has a next number x’. When you select y = x’  then you couldn’t find anything between x and y. This contradicts above statement on density.

Thus, the standard definition of density doesn’t fit a well-ordered R.

Designing a new definition for density of the reals R

We can design a new definition of density.

  • The standard definition is useful for the rationals Q. If we restrict your freedom to making cuts along Q, then we are safe again. In this manner, the distinction between rational and irrational numbers is useful to explain a property of R.
  • R is defined as “more dense” because Q is dense w.r.t. that original definition ((a)).
  • This proposal is quite similar to the Dedekind cut, with the distinction that we now allow that R might retain a well-ordering. That is, this issue on the ordering is no longer forced by the standard definition of density.
Surprise consequence as a bonus

Switching to another notion of density, generates the bonus that we have more scope to introduce the infinitesimal.

When every number x has a next x’, then we can define the infinitesimal as the difference:

δ = x’ x

It also means that an open set (a, b) can also be seen as a closed set [a + δ, b – δ].

Wikipedia (no source but a portal) claims today:

“The standard ordering ≤ of any real interval is not a well ordering, since, for example, the open interval (0, 1) ⊆ [0,1] does not contain a least element.”

Yet now we have (0, 1) = [δ, 1 – δ] and the least element is δ. Only the intervals with negative infinity might be excluded, check (-∞, ∞).

Properties of these infinitesimals

Some properties are:

(1) We still have 0.999…. = 1.000…..

(2) A current statement is that a line consists of points, and each point is a co-ordinate without length. We now can better express that length consists of a sum of short lengths. A sum of these infinitesimals Σδ makes sense if we regard it as the sum from x = 0 to x = 1 for x‘ – x. The trick is that the length is determined by the statement on x and not by the coefficient of δ.

(3) Using H = -1, then δ δH = 1. That is, δ ≠ 0, and thus there is no problem with division. The discussion about differentials is quite different from the discussion about these (new) infinitesimals. Much time has been spent in history in looking whether there might be a connection, but there isn’t.

Separate arithmetic for infinity and infinitesimal

Students already know that they cannot apply the rules of arithmetic to infinity. E.g. ∞ + ∞ = ∞. The same now holds for above hypothetical notion of the infinitesimal.

Property (2) carries over from δ[n] with n the number of decimals. Property (1) arises when n → ∞ . Potentially, these notions cannot be combined without some conflict.

We are accustomed to think that any real number can be divided. But e.g. δ / 2 is nonsense because it gives the distance between two numbers, and there is nothing smaller. Thus, the normal rules for arithmetic only hold for reals that are not these infinitesimals.

With δ = x’ x we also want to consider y = (x / n). When the numbers are halved for n = 2, is the distance halved or isn’t it halved ? In the approximation δ[n] the distance can become smaller when more digits are included. For an infinite number of digits, presumably, the distance cannot be halved. Thus δ = y’ y. Multiplication by n gives nδ = n (x / n)x. Thus x’ = n (x / n)‘ – nδ + δ for any n. This would make (most) sense by the choice δ = 0 and x’ / n = (x / n)‘. But then we are back in the classical approach again, without the well ordering. (The next number is the number itself.)

Persumably, we can argue that n * δ is as problematic as δ / n, though. The notion of Σδ namely has been solved by putting the consideration of length into the Σ sign.

I don’t know yet whether it is sufficient (consistent) to state δ = x’ and that the rules for arithmetic don’t apply to δ like they don’t apply to ∞. Potentially, we might write δH → ∞ (and this doesn’t mean that δ ∞ → 1).

All this depends upon whether we can develop a consistent set of definitions. Students at junior highschool might agree that they aren’t much interested in that.

Thus, it might only be in senior highschool, when we discuss the “classical” approach to the reals, that has (a, b) as an open interval only. We would be forced to this not because of the definition of density but because of the rules of arithmetic.


In itself, notions like these are not world shocking but they would tend to fit the intuitions of space and number for junior highschool.

At some point of history, the main stream in mathematics has opted for an approach to the reals so that they have no well ordering. The obstacle of the standard definition of density can be removed, as shown above. A problem still resides in arithmetic with δ = x’ x. It is not clear to me whether this can be resolved. It is not clear to me neither whether it is okay to have benign neglect till senior highschool, and face the consequences of losing the well ordering only there.

Isaac Newton (1642-1727) invented the differentials, calling them evanescent quantities. Since then, the world has been wondering what these are. Just to be sure, Newton wrote his Principia (1687) by using the methods of Euclidean geometry, so that his results could be accepted in the standard of his day (context of reconstruction and presentation), and so that his results were not lost in a discussion about the new method of these differentials (context of discovery). However, this only increased the enigma. What can these quantities be, that are so efficient for science, and that actually disappear when mathematically interesting ?

Gottfried Leibniz (1646-1716) gave these infinitesimals their common labels dy and dx, and thus they became familiar as household names in academic circles, but this didn’t reduce their mystery.

Charles Dodgson (1832-1898) as Lewis Carroll had great fun with the Cheshire Cat, who disappears but leaves its grin.

Abraham Robinson (1918-1974) presented an interpretation called “non-standard analysis“. Many people think that he clinched it, but when I start reading then my intuition warns me that this is making things more difficult. (Perhaps I should read more though.)

In 2007, I developed an algebraic approach to the derivative. This was in the book “A Logic of Exceptions” (ALOE), later also included in “Elegance with Substance” (EWS) (2009, 2015), and a bit later there was a “proof of concept” in “Conquest of the Plane” (COTP) (2011). The pdfs are online, and a recent overview article is here. A recent supplement is the discussion on continuity.

In this new algebraic approach there wasn’t a role for differentials, yet. The notation dy / dx = f ‘[x] for y f [x] can be used to link up to the literature, but up to now there was no meaning attached to the symbolism. In my perception this was (a bit of) a pity since the notation with differentials can be useful on occasion, see the example below.

Last month, reading Joop van Dormolen (1970) on the didactics of derivatives and the differential calculus – in a book for teachers Wansink (1970) volume III – I was struck by his admonition (p213) that dy / dx really is a quotient of two differentials, and that a teacher should avoid identifying it as a single symbol and as the definition of the derivative. However, when he proceeded, I was disappointed, since his treatment didn’t give the clarity that I looked for. In fact, his treatment is quite in line with that of Murray Spiegel (1962), “Advanced calculus (Metric edition)”, Schaum’s outline series, see below. (But Van Dormolen very usefully discusses the didactic questions, that Spiegel doesn’t look into.)

Thus, I developed an interpretation of my own. In my impression this finally gives the clarity that people have been looking for starting with Newton. At least: I am satisfied, and you may check whether you are too.

I don’t want to repeat myself too much, and thus I assume that you read up on the algebraic approach to the derivative in case of questions. (A good place to start is the recent overview.)

Ray through an origin

Let us first consider a ray through the origin, with horizontal axis x and vertical axis y. The ray makes an angle α with the horizontal axis. The ray can be represented by a function as y =  f [x] = s x, with the slope s = tan[α]. Observe that there is no constant term (c = 0).


The quotient y / x is defined everywhere, with the outcome s, except at the point x = 0, where we get an expression 0 / 0. This is quite curious. We tend to regard y / x as the slope (there is no constant term), and at x = 0 the line has that slope too, but we seem unable to say so.

There are at least three responses:

(i) Standard mathematics then takes off, with limits and continuity.

(ii) A quick fix might be to try to define a separate function to find the slope of a ray, but we can wonder whether this is all nice and proper, since we can only state the value s at 0 when we have solved the value elsewhere. If we substitute y when it isn’t a ray, or example x², then we get a curious construction, and thus the definition isn’t quite complete since there ought to be a test on being a ray.




(iii) The algebraic approach uses the following definition of the dynamic quotient:

y // x ≡ { y / x, unless x is a variable and then: assume x ≠ 0, simplify the expression y / x, declare the result valid also for the domain extension x = 0 }

Thus in this case we can use y // x = s x // x = s, and this slope also holds for the value x = 0, since this has now been included in the domain too.

In a nutshell for dy / dx

In a nutshell, we get the following situation for dy / dx:


Properties are exactly as Van Dormolen explained:

  • “dy” and “dx” are names for variables, and thus they have their own realm with their own axes.
  • The definition of their relationship is dy = f ‘[x] dx.

The news is:

  • The mistake in history was to write dy / dx instead of dy // dx.

The latter “mistake” can be understood, since the algebraic approach uses notions of set theory, domain and range, and dynamics as in computer algebra, and thus we can forgive Newton for not getting there yet.

To link up with history, we might define that the “symbol dy / dx as a whole” is a shortcut for dy // dx. This causes additional yards to develop the notion of “symbol as a whole” however. My impression is that it is better to use dy // dx unless it is so accepted that it might become pedantic. (You must only explain that the Earth isn’t flat while people don’t know that yet.)

Application to Spiegel 1962 gives clarity

Let us look at Spiegel (1962) p58-59, and see how above discussion can bring clarity. The key points can all be discussed with reference to his figure 4-1.


Looking at this with a critical eye, we find:

  • At the point P, there is actually the creation of two new sets of axes, namely, both the {Δx, Δy} plane and the {dx, dy} plane.
  • These two new planes have both rays through the origin, one with angle θ and one with angle α.
  • The two planes help to define the error. An error is commonly defined from the relation “true value = estimate + error”. The true value of the angle is θ and our estimate is α.
  • Thus we get absolute error Δf = s Δx + ε where s = dy / dx. This error is a function of Δx, or ε = ε[Δx]. It solves as ε = Δf – s Δx.
  • The relative error is Δf / Δx =  dy / dx + r which solves as r = Δf / Δx – dy / dx. This is still a function rx]. We use the quotient of the differentials instead of the true quotient of the differences.
  • We better re-consider the error in terms of the dynamic quotient, replacing / by // in the above, because at P we like the error to be zero. Thus in above figure we have ε = Δf  s Δx, where s = dy // dx.
  • A source of confusion is that Spiegel suggests that d≈ Δx or even dx = Δx but this is numerically true only sometimes and conceptually there surely is no identity since these are different axes.
  • In the algebraic approach, Δx is set to zero to create the derivative, in particular the value of f ‘[x] = tan[α] at point P.  In this situation, Δx = 0 thus clearly differs from the values of dx that are still available on dx ‘s whole own axis. This explains why the creation of the differentials is useful. For, while Δx is set to 0, then the differentials can take any value, including 0.

Just to be sure, the algebraic approach uses this definition:

f ’[x] = {Δf // Δx, then set Δx = 0}

Subsequently, we define dy = f ‘[x] dx, so that we can discuss the relative error r = Δf // Δx – dy // dx.

PM. Check COTP p224 for the discussion of (relative) error, with the same notation. This present discussion still replaces the statement on differentials in COTP p155, step number 10.

A subsequent point w.r.t. the standard approach

Our main point thus is that the mistake in history was to write dy / dx instead of dy // dx. There arises a subsequent point of didactics. When you have real variables and z, then these have their own axes, and you don’t put them on the same axis just because they are both reals.

See Appendix A for a quote from Spiegel (1962), and check that it is convoluted at times.

Appendix B contains a quote from p236 from Adams & Essex (2013). We can see the same confusions as in Spiegel (1962). It really is a standard approach, and convoluted.

The standard approach takes Δx = dx and joins the axis for the variable Δy with the axis for the variable dy, with the common idea of “a change from y“. The idea of this setup is that it shows the error for values of Δx = dx.


It remains an awkward setup. It may well be true that John from Los Angeles is called Harry in New York, but when John calls his mother back home and introduces himself as “Mom, this is Harry”, then she will be confused. Eventually she can get used to this kind of phonecalls, but it remains awkward didactics to introduce students to these new concepts in this manner. (Especially when John adds: “Mom, actually I don’t exist anymore because I have been set to zero.”)

Thus, in good didactics we should drop this Δx = dx.

Alternatively put: We might define dy = f ’[x] Δx = f // Δx, then set Δx = 0} Δx. In the latter expression Δx occurs twice: both as a local and bound variable within { … } and as a global free variable outside of { … }. This is okay. In the past, mathematicians apparently thought that it might make things clearer to write dfor the free global variable: dy = f ’[x] dx. In a way this is okay too. But for didactics it doesn’t work. We should rather avoid an expression in which the same variable (name) is uses both locally bound and globally free.

Clear improvement

Remarkably, we are using 99% of the same apparatus as the standard approach, but there are clear improvements:

  • There is no use of limits. All information is contained in the algebra of both the function f and the dynamic quotient. See here for continuity.
  • There is a clear distinction between the three realms {x, y}, {Δx, Δy} and {dx, dy}.
  • There is the new tool of the {dx, dy} space that can be used for analysis of variations.
  • Didactically, it is better to first define the derivative in chapter 1, and then introduce the differentials in chapter 2, since the differentials aren’t needed to understand chapter 1.
  • There is clarity about the error, that one doesn’t take d≈ Δx but considers ε = Δf  s Δx, where s has been found from the recipe s = f ’[x] = {Δf // Δx, then set Δx = 0}.
Example by Van Dormolen (1970:219)

This example assumes the total differential of the function f[x, y]:

df = (∂f // ∂x) dx + (∂f // ∂y) dy

Question. Give the slope of the tangent in the point {3, 4} of the circle x² + y²  = 25.

Answer. The point is on the circle indeed. We write the equation as f[x, y] = x² + y²  = 25. The total differential gives 2x dx + 2y dy = 0. Thus dy // dx = – x // y. Evaluation at the point {3, 4} gives the slope – 3/4.  □

PM. We might develop y algebraically as a function of and then use the +√ rather than the -√. However, more abstractly, we can use [x], and use dy = g ‘[x] dx, so that the slope of the tangent is g ‘[x] at the point {3, 4}. Subsequently we use g ‘[x] = dy // dx.

PM. In the Dutch highschool programme, partial derivatives aren’t included, but when we can save time by a clear presentation, then they surely should be introduced.


The conclusion is that the algebraic approach to the derivative also settles the age-old question about the meaning of the differentials.

For texts in the past the interpretation of the differential is a mess. For the future, textbooks now have the option of above clarity.

Again, a discussion about didactics is an inspiration for better mathematics. Perhaps research mathematicians have abandoned this topic for ages, and it is only looked at by researchers on didactics.

Appendix A. Spiegel (1962)

Quote from Murray Spiegel (1962), “Advanced calculus (Metric edition)”, Schaum’s outline series, p58-59.


Appendix B. Adams & Essex (2013)

The following quote is from Robert A. Adams & Christopher Essex (2013), “Calculus. A Complete Course”, Pearson, p236.

  • It is a pity that they use c as a value of x rather than as an universal name for a constant (value on the y axis).
  • For them, the differential cannot be zero, while Spiegel conversely states that it is “not necessarily zero”.
  • They clearly show that you can take f ‘[x] Δin in {Δx, Δy} space, and that you then need a new symbol for the outcome, since Δy already has been defined differently. However, it is awkward to say: “For such an approximation, the quantity Δx is traditionally denoted as dx (…)”. It may well be true that John from Los Angeles is called Harry in New York, … etcetera, see above.



Mathematicians can be seen as lawyers of space and number.

Euclid wrote about 300 BC. Much earlier, Hammurabi wrote his legal code around about 1792-1749 BC. It is an interpretation of history: Hammurabi might have invented all of his laws out of thin air, but it is more likely that he collected the laws of his region and brought some order into this. Euclid applied that idea to what had been developed about geometry. The key notions were caught in definitions and axioms, and the rest was derived. This fits the notion that Pierre de Fermat (1607-1665) started as a lawyer too.

Left Codex Hammurabi, right a piece of Euclid 100 AD. Wikimedia commons

Left: codex Hammurabi. Right: a piece of Euclid 100 AD. Wikimedia commons

The two cultures: science and the humanities

In Dutch mathematics education there is a difference between A (alpha) and B (beta) mathematics. B would be “real” math and prepare for science. A would provide what future lawyers can manage.

In the English speaking world, there is C.P. Snow who argued about the “two cultures“, and the gap between science and the humanities. A key question is whether this gap can be bridged.

In this weblog, I already mentioned the G (gamma) sciences, like econometrics that combines economics (humanities) with scientific standards (mathematical models and statistics). Thus the gap can be bridged, but perhaps not by all people. It may require some studying. Many people will not study because they may arrogantly believe that A or B is enough (proof: they got their diploma).

Left and right hemisphere of the brain

Another piece of the story is that the left and right hemispheres of the brain might specialise. There appears to be a great neuroplasticity (Norman Doidge), yet overall some specialisation makes sense. The idea of language and number on the left hemisphere and vision on the right hemisphere might still make some sense.

“Broad generalizations are often made in popular psychology about certain functions (e.g. logic, creativity) being lateralized, that is, located in the right or left side of the brain. These claims are often inaccurate, as most brain functions are actually distributed across both hemispheres. (…) The best example of an established lateralization is that of Broca’s and Wernicke’s Areas (language) where both are often found exclusively on the left hemisphere. These areas frequently correspond to handedness however, meaning the localization of these areas is regularly found on the hemisphere opposite to the dominant hand. (…) Linear reasoning functions of language such as grammar and word production are often lateralized to the left hemisphere of the brain.” (Wikipedia, a portal and no source)

For elementary school we would not want kids to specialise in functions, and encourage the use of neuroplasticity to develop more functions.

Pierre Krijbolder (1920-2004) suggested that there is a cultural difference between the Middle East (Jews), with an emphasis on language – shepherds guarding for predators at night – and the Indo-Europeans (Greeks), with an emphasis on vision – hunters taking advantage of the light of day. Si non e vero, e ben trovato.

There must have been at least two waves by Indo-Europeans into the Middle-East. The first one brought the horse and chariot to Egypt. The second one was by Alexander (356-323 BC) who founded Alexandria, where Euclid might have gotten the assignment to write an overview of the geometric knowledge of the Egyptians, like Manetho got to write a historical overview.

Chariot spread 2000 BC. (Source: D. Bachmann, wikimedia commons)

Chariot spread 2000 BC. (Source: D. Bachmann, wikimedia commons)

It doesn’t actually matter where these specialisations can be found in the brain. It suffices to observe that people can differ in talents: lawyers would deal much with language, and for space you might turn to mathematicians.

Pierre van Hiele (1909-2010) presents a paradox

The Van Hiele levels of insight are a key advance in epistemology, for they indicate that human understanding itself is subjected to some structure. The basic level concerns experience and the direct language about this. The next level concerns the recognition of properties. Another level is the recognition of relations between these properties, and the informal deductions about these. The highest level is formalisation, with axiomatics and formal deduction. The actual number of levels depends upon your application, but the base remains in experience and the top remains in axiomatics.

Learning goes from concrete to abstract, and from vague to precise.

Thus, Euclid with his axiomatic approach would be at the highest level of understanding.

We arrive at a paradox.

The axiomatic approach is basically a legal approach. We start with some rules, and via substitution and reasoning we arrive at other rules. This is what lawyers can do well. Thus: lawyers might be the best mathematicians. They might forget about the intermediate levels, they might discard the a-do about space, and jump directly to the highest Van Hiele level.

A paradox but no contradiction

A  paradox is only a seeming contradiction. The latter paradox gives a true description in itself. It is quite imaginable that a lawyer – like a computer – runs the algorithms and finds “the proper answer”.

However, for proper mathematics one must be able to switch between modes. At the highest Van Hiele level, one must have an awareness of applications, and be able to translate the axioms, theorems and derivations into the intended interpretation. In many cases this may involve space.

Just to be sure: the Van Hiele levels present conceptual divides. At each level, the languages differ. The words might be the same but the meanings are different. This also causes the distinction between teacher-language and student-language. Often students are much helped by explanations by their fellow students. It is at the level-jump, when the coin drops, that meanings of words change, and that one can no longer imagine that one didn’t see it before.

Thus it would be a wrong statement to say that the highest Van Hiele level must have command of all the lowest levels. The disctinction between lawyers and mathematicians is not that the latter have command of all levels. Mathematicians cannot have command of all levels because they have arrived at the highest level, and this means that they must have forgotten about the earlier levels (when they were young and innocent). The distinction between lawyers (math A) and mathematicians (math B) is different. Lawyers would understand the axiomatic approach (from constitutional law to common law) but mathematicians would understand what is involved in specific axiomatic systems.

Example 1

I came to the above by thinking about the following problem. This problem was presented as an example of a so-called “mathematical think-activity” (MTA). The MTA are a new fad and horror in Dutch mathematics education. First try to solve the problem and then continue reading.

2016-10-26-petervanwijk-smaller-englishDiscussion of example 1

The drawing invites you to make two assumptions: (1) the round shape is a circle, (2) the vertical x meets the horizontal x in the middle. However, why would this be so ? You might argue that r = 6 suggests the use of a circle, but perhaps this still might be an ellipse.

In traditional math ed (say around 1950), making such assumptions would cost you points. In fact, the question would be considered insoluble. No question would be presented to you in this manner.

In traditional math, the rule would be that the proper question and answer would consist of text, and that drawings only support the workflow. Also, the particular calculation with = 6 would not be interesting. Thus, a traditional presentation would have been (and also observe the dashes):

2016-10-26-petervanwijk-smaller-english-altA quick observation is that there are three endpoints, and it is a theorem that there is always a circle through three points. So the actual question is to prove this theorem, and you are being helped with a special case.

Given that you solved the problem above, we need not look into the solution for this case.

The reason for giving this example is: In mathematics, text has a key role, like in legal documents for lawyers. Since mathematicians are lawyers of space and number, they can cheat by using supporting drawings, tables and formulas. But definitions, theorems and proofs are in text (formulas).

(Potentially lawyers also make diagrams of complex cases, as you can see in movies sometimes. But I don’t know whether there are particular methods here.)

Example 2

The second example is the discussion from yesterday.

In text it is easy to say that a line has no holes. However, when you start thinking about this, then you must define what such a hole might be. If a hole doesn’t belong to the line, what does it belong to then ? How would you know when you would pass a hole ? Might you not be stepping over holes all the time without noticing ?

These are questions that lawyers would enjoy. They are relevant for math B but can also be discussed in math A.

See the discussion of yesterday, and check that the main steps should be acceptable for lawyers, i.e. math A.

These students should be able to master the symbolism of predicate logic, since this is only another language and a reformulation of common text.


Thus, a suggestion is that students in math A should be able to do more, when better use is made of the legal format.

Perhaps more students, now doing A, might also do B, if their learning style is better supported.

(Perhaps the B students would start complaining about more text though. Would there still be the same question, when only the format of presentation differs ? Thus a conclusion can also cause more questions. See also this earlier discussion about schools potentially manipulating their success scores by causing student underperformance.)