So, what is multiplication?

Hurkyl

Please try to be far more specific than you have been. I have to guess at the fine details of what you mean.


Long division is an exact operation on decimal numerals. Every computable operation on real numbers will have to deal with precision issues of some sort. Even addition.

When applied to decimals that represent rational numbers (because at some point a sequence of digits repeats forever), a slight modification allows long division to terminate after a finite number of steps.


Incidentally, when applied to rational numbers represented as a quotient of integers, the division algorithm and multiplication algorithms are pretty much identical.

apeiron

OK, forget I mentioned real numbers at one point as irrational numbers are another example of how the simplistic notion of construction or addition breaks down in practice.
Limits have to be introduced as a further action.

In fact forget the whole question because you clearly are not interested in actually addressing it, just talking around it forever.

Hurkyl

Eh? There were two and a half pages addressing the original question. You brought up a new claim -- that division is somehow working backwards from a destination to a construction, but yet you are not thinking in terms of division being an inverse of multiplication, and that this is somehow a fundamental difference between division and other arithmetic operations, rather than just being one of many ways to view division.

If "addressing" your point means unquestioningly buying into your assertion, then yes, I am uninterested in "addressing" it.

Studiot

I gave you an answer that is absolutely precise, but you chose to do exactly what you are accusing others of - you ignored it.

apeiron

Eh? You said...

I don't know about precise, but that's the feyest attempt at an insult I've seen in a long time.

apeiron

The OP said...

So that was what I was throwing out as a question. Your answer is that division is simply inverse multiplication. But that does not deal with the OP comment that division is not repeated subtraction.

Again, if you have nothing useful to say on the matter, just leave it to someone else.

Studiot

No insults were intended so I'm sorry if you feel insulted.

However the fact remains that up to the late 1960s boys in their first year in an English grammar school would be taught the construction, from Euclid, that I was referring to.

In those days such a construction was used by engineers and draughtsmen and a version appeared on many boxwood scales of that time.

You should also remember this is the pure maths section of the forum. In pure maths we are allowed the luxury, as was Euclid, of perfect constructions. Remember also there are very specific mathmatical rules governing perfect constructions.
But I assume you already know all this?

There is a further twist to your question. You have not specified what x is, but it cannot be any random real number, it can only be an integer. This makes the construction basic.

apeiron

I seem to have lost my boxwood scales and skipped 1960s grammars, so you might actually have to state what it is you are referring to here. Does it have a name? Can you provide a link? Or would that be too shockingly precise?

Studiot

Construction to divide a line into n equal parts.

Draw the line (in your case equal to the random real number to be divided or mark any line at a random point if you like to create a random real number)

Draw an auxiliary line at a convenient angle (30[tex]^{o}[/tex] is generally convenient) and crossing the original line at one end.

With compasses set to any convenient length mark off n steps along the auxiliary line, commencing at the intersection with the original line.

Join the mark representing the last step to the other end or marked point of the original line.
Through each mark along the auxiliary line draw a line parallel to this third line to intersect the original line.

You have now divided the original line perfectly into n equal parts.

apeiron

Thanks for that. But are you saying it is a geometric representation of either inverse multiplication or repeated subtraction as arithmetic operations?

Rather I think what it shows is a process in which "divide by x" is handled by creating a model of x outside the numberline and then morphing it to fit between two points on the numberline.

So a "whole" is constructed by additive steps, but then the whole is shrunk to fit. Which is a continuous transformation rather than as a series of discrete steps.

The other three operations are constructing a whole from the parts (additive actions). Whereas division starts with the whole and asks for a reduction to a set of parts.

So we can start by trying repeated subtraction with an example like 7/3. We can subtract twice then get down to having to divide the remainder 1 into 3 parts. We are now dealing with a "whole" unit - and subtraction depends on working with multiples of this unit. As does addition and multiplication.

The answer is to shrink the numberline in scale - morph the 1 to 10 to create an internal decimal division of .1 to 1. Then pick up with the subtraction at this new scale. And morph again if we need to get into hundredths or thousandths.

So division does seem deeply different in this light. The other three operations are straightforwardly constructive - operations that are discretely additive. But division involves the extra step of a morphing of a constructed co-ordinate space. Something quite different in nature is required to go from the whole to its parts. Even if in the end it is no big deal because you have multiplication as a "look-up table" of inverse operations and decimals as a standard way to fractalise the dimensionality of the numberline. (Base 10 is just a choice, not something derived axiomatically, is it? God created the integers but not the decimals? )

Studiot

I didn't say anything of the sort.

I said before that I answered a specific question you made, asking for a proceedure to "divide a given random number into x equal parts" and thought you also implied that you did not think this could be done.

As it happens this proceedure was documented centuries before we had anything like modern arithmetic so it preceeded such theory and cannot therefore be said to be derived from it in any way.

I am rather suprised you have not heard of it since in another thread you claimed a classical education within the British/Irish system.

I fully agree with Hurkyl that there are also arithmetic algorithms fully developed to handle the question. Obviously these came later in the hsitory of mathematics.


go well

apeiron

Then you misunderstood the question. It was about the OP point that "division seems different" and so about the precise nature of that difference in number theory.

They were teaching new maths by the time I came along I guess.

Studiot

I can only understand what is written.

Conforms exactly to the question I answered and later paraphrased.

apeiron

OK, I understand. You can't answer the larger question that was posed. You are not interested in how a procedure works, only that it works.

Studiot

You do seem to like putting (incorrect) words into the mouths of others.

I have been following this thread since near its inception, and even posted way back although my comment at that time has not been addressed.

disregardthat

This is purely due to the choice of representation by decimals. Division is a wholly constructive operation, but it is simply the case that not all rational and real numbers can be written as a finite decimal expansion in base 10. There isn't anything imprecise about the result of an operation that iteratively gives you the base 10 digits of a given real or rational number, it gives you exactly what you want.

You talked about the necessity to introduce limits when it comes to real numbers - as a sort of flaw, but it is in this sense real numbers are defined (or can be defined) - as limits.

Actually, there are simple constructive methods to give you the full prime factorization of any given integer. It is not necessary to guess at any point. Even the most naive (ineffective) ones will not involve any guesswork.

apeiron

Yes, division can be a wholly constructive operation (namely, repeated subtraction) but only because a further "natural" step has been taken in breaking the symmetry of the number line by choosing a base 10 numbering system.

So it seems to me that an extra geometrical argument has been introduced at this point. Whereas the numberline is a linear additive concept, we are now laying over the top of it a geometric expansion which gives us "counting in orders of magnitude and decimal scale".

Now of course I am sure people will say they see no issue here because all the points on the numberline exist. So 1.3, or pi, are just as natural as entities as 1,2,3.

But that is what I am musing about. Some extra constraint appears needed to break the naive symmetry of the numberline. The challenge was to connect something that is essentially discrete (a string of points) with what also had to be essentially continuous (a line) and breaking the scale of counting in this way, using a base as a further constraint, seems like the way it has been done.

So anyway, the answer for me now goes clearly beyond the original question about the nature of division and is clearly part of all the conversations about irrational numbers and infinities.

The numberline is founded on the notion of "one-ness". And that is a symmetric or single-scale concept. But as soon as you introduce an asymmetry, a symmetry-breaking constraint - such as any base system starting even from base 2 - then there is something new. A connection is forged between the original point-like discreteness and the continuity implied by a numberline. Scale is broken geometrically over all scales. Allowing then measurement down to the "finest grain".