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 *x ^{H} *for

*H*= -1. In that case the results of division are also included in the same set. In terms of

*Z*the expression 2

*is not a number, but for*

^{H}*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 = 2* ^{H}*, 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 2* ^{H}* 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 relevan**t*– 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 *z *in *Z *that satisfies 1 – 2*z* = 0.

If we extend *Z *with basic elements *n ^{H}* for

*n*≠ 0 then we can find a

*z*that satisfies 1 = 2

*z*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.

*x ^{H} y = *z ⇔

*y =*

*x*

*z*(** notation)

For *x *itself:

*x ^{H} x *=

*x x*= 1

^{H}##### 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:

0* – *0 *z* = 0 * *for any *z *

0* ^{H} *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 ⇒ * * * x ^{H} x *=

*x*

*x*= 1

^{H}

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

*x ^{H} y *=

*z*⇔

*x*

*x*=

^{H}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 *z * ⇔ 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:

2* ^{H} *1 =

*z*

*2 ^{H} *=

*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 0* ^{H} *in the ring, to create 〈ring, 0

*〉.*

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

0* ^{H} *0 = 0

*with 0*

*in 〈ring, 0*

^{H}*〉*

^{H}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 0* ^{H} as ∞ *then it might be clearer that 0

*0 remains a problematic form.*

^{H}If we create the 〈ring, 0* ^{H}*〉, then we might also hold: 0

*z*= 0 for all numbers except 0

*. In that case, the result is maintained that*

^{H}0* ^{H} *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)

*x ^{H} y *=

*y –*(

*x*– 1)

*z*=

*z*(**** definition and notation 2)

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

0* ^{H} y *=

*y –*(0 – 1)

*z*=

*z*

*y* = *z* – *z* = 0

0* ^{H} y *= 0

*0 =*

^{H}*z*

However, the more common approach is that 0* ^{H} *is

*∞*and 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.