On Sat, Sep 1, 2012 at 1:58 AM, Jonathan Crabtree <firstname.lastname@example.org> wrote: > Arithmetic has been flawed ever since Euclid edited the Elements around 300 BCE. > > The reason is the incorrect definition of multiplication included in Book VII Def. XV > > "Def. 15. A number is said to multiply a number when the latter is added as many times as there are units in the former." > ... > I am grateful to Professor David Joyce for changing the definition on the above page from the previous definition, > > "Def. 15. A number is said to multiply a number when that which is multiplied is added to ITSELF (my caps) as many times as there are units in the other." > ... > David updated Euclid's definition of multiplication on his wonderful Euclid website after an exchange of emails in which I pointed out that Euclid's definition of multiplication was wrong. > > I wrote about this here > http://mathforum.org/kb/message.jspa?messageID=7626442 > > I'm a bit of a heretic suggesting arithmetic has been built on shaky foundations, yet Euclid can be forgiven as he wrote prior to the acceptance of both zero as a number and and negative numbers. He would have learnt much from Diophantus and more from Brahmagupta of Indian fame. > > I have finally found another mathematician who has stated that mk = m added to itself k-1 times. > > His name is S. F. Lacroix, whio wrote about multiplcation, "... by adding the number to itself as many times, wanting one, as is to be repeated. > > For instance, by the following addition, > > 16 > 16 > 16 > 16 > - -- > 64 > > the number 16 is repeated four times, and added to itself three times." ie k-1 times > > So the non Euclidean definition I have created is, > > "mk = the combination of m either added to or subtracted from zero k times" > > >From this correct definition of multiplication, all rational arithmetic becomes easy, including integer and fraction multiplication. The number lines can now be used for fractions as well as multiplying negatives. > > The whole Devlin MIRA debate has been a red herring. > > Until now, multiplication itself has never been correctly defined for the rational numbers. > > Jonathan Crabtree > 1st September 2012 > > P.S. Think of a photocopying machine. You have one letter and you need four altogether. So what button do you press on the multiplication machine? Three! ie k-1 Press the four button and you end up with five. >
Any of these repeated-addition definitions for product ab does not work in a field, especially for the set of all real numbers but even for the set of all rational numbers, unless we restrict the multiplication such that a or b is an integer.
It seems to me that this fact is what motivated Devlin to imply that at the very least we ought to come up with a general enough definition of multiplication that if not replaces then at least supplements these repeated-something "definitions" of binary multiplication.
That is, with such a general enough definition students would from the beginning have a mental model of binary multiplication that holds up on any set on which absolute value is defined. And this includes the natural numbers all the way up through the real numbers.
His idea was to think of positive multiplication as scaling.
That is, to visualize this idea, we could imagine that at point 0 on the number line we have a typical measuring tape machine such that we could pull out measuring tape attached to a spring such that if we let go, all the measuring tape gets pulled back into the machine.
For product xb, with can think of pulling the tape out to length b, and then pulling it out further from there by a scaling factor of x to a point represented by the product xb.
Of course we have to explain what we mean by scaling, that we think of length b as a unit, but this is good thing: It creates a mental foundation for proportional reasoning very early on. (Using the number line as model regardless has the same effect, in my view. It helps the mind see the scaling inherent in the moving away from 0 when we are multiplying positive elements.)
Likewise we could think of positive division as the inverse of this:
For division a/b for a = xb, with can think of starting at point a that is a product obtained from multiplying xb = a, and then letting the tape machine pull back the tape by an inverse scaling factor of b to arrive back at point x.
Side note: Yes, I think that it's a good idea to teach that a/b can always be viewed as (xb)/b for some x since dividend a always equals xb for some x, and I think that it's a good idea to teach that this x is where we end up on the number line (or more generally in the set of elements that is the domain of our variables) when we divide a or xb by b.
Back to the point: What about negative numbers or 0? Even with them, the model I just gave above on the positive numbers still works - just apply the rules of signs to get there such that these rules would come into play after we taught positive multiplication and positive division.
But Devlin I think had another beef with these repeated-something "definitions" of binary multiplication: The term "binary" should give way the beef. That is, for multiplication on a set on which absolute value is defined, for product xb, the term "binary" suggests that we have two elements x and b such that on the number line we should be able to use one of the elements x to get from the other element point b to the product point xb in just one step. That is, repeated-something "definitions" give us no mental model at all for how to get from b to xb in just one step.
But even so, if we want a repeated-something model to include in our mix of models, I still prefer a somewhat broader alternative I've laid out here a few times here at math forum - and yes, it address the k-1 problem. Here it is in action, explaining the right side in the identity p/q = p(1/q) for integer p and non-zero integer q - note that it allows using the generalized associative law for addition on a sum of n elements, x_1 + ... + x_n:
Multiplication p(1/q) means to add together p instances of (1/q). (Yes, we can use the rules of signs to make this fit.)
I like this language of adding together a number of instances of something - it does not require a particular ordering or grouping of the elements. (Yes, I think that what is meant with the term "instances" can be easily explained and understood.)