The concept of lines of numbers [linearly ordered systems of numbers] was invoked long before Wallis. In accord with the 17th century confusion about signed numbers during his times, Wallis proffered a pedagogically *horrible* interpretation ... the kind of thing that presently abounds among mathematically challenged educators who have not studied how/what students actually learn.
For students' who are first encountering "multiplication" of signed numbers in scholastic curricula, such "teachings" typically make no mathematical (or common) sense to the students, themselves. Their only recourse is to follow the path of "Yah, yah, whatever you say ... just tell me what you want me to do" ... commonly known as "rote."
See my forthcoming note on "MACS syllabus", in this thread ... about the common-sensibility of signed-numbers "multiplication."
- -------------------------------------------------- From: "Jonathan Crabtree" <email@example.com> Sent: Sunday, December 02, 2012 8:58 PM To: <firstname.lastname@example.org> Subject: Re: Some important demonstrations on negative numbers
>> >I have also provided a precise verbal mapping of the >> instructions that reveal the logic of why the >> products of both -ve x - and - x -ve are positive. >> >> Perhaps you can rationalize it - you can't prove it >> in the mathematical sense without making prior >> assumptions. > > Hi Joe > > Just letting you know I use the same verbal logic as John Wallis, credited > as the inventor of the number line. > > QUOTE > > Even the rule of signs was but a consequence of "the true notion of > (arithmetic] Multiplication [which] is ... to put the Multiplicand, or > thing Multiplied (whatever it be) so often, as are the Units in the > Multiplier." > >>From the latter definition Wallis argued consecutively that > > a) multiplication of a negative multiplicand and positive multiplier > involved no more than taking the multiplicand the specified number of > times, and thus getting a negative sum; > > b) multiplication of a positive multiplicand and a negative multiplier > involved nothing more than taking the multiplicand away the specified > number of times; and, finally, > > c) multiplication of a negative multiplicand and a negative multiplier > involved "taking away a Defect or Negative," which "is the same as to > supply it" ? and thus getting a positive." > > NOTE: My lettering for clarity. > > SOURCE: Symbols, Impossible Numbers, and Geometric Entanglements. by > Helena M. Pycior