Recently, the subject of how to teach or justify rules for multiplication of negative numbers has come out. The following is an approach based on the Peano Axioms.
Natural numbers are 0, S(0), S(S(0))), …
S(0) = 1
S(1) = 1′ = 2
As a technicality we assume that we first define and prove order properties of natural numbers and then define and prove the recursion theorem that lets us do definition by induction and form functions by recursion.
We define addition of natural numbers recursively by
x+0 = x (Right Zero Identity)
x+y’ = (x+y)’ (Right Successor Identity)
Multiplication of natural numbers
x*0 = 0
x*y’ = x*y + y
We can define the predecessor by P(x) or ‘x as the inverse of the successor S() function.
P(1) = 0.
‘1 = 0.
P(2) = 1
‘2 = 1
P(S(x)) = x
For 0, P(0) is not defined when P is restricted to natural numbers.
We can then define negative numbers as a pair of a sign and a size (+,n) and (-,n) where n is a natural number. We can call these signed number pairs, integers.
We then define the functions P and S on the signed number pairs. This requires an appropriate extension or interpretation of the recursion theorem.
(We could also start out with axioms that included two directions for Successor and Predecessor but restricted to integers as signed numbers similar to the Peano Axioms for unsigned natural numbers.)
A key identity is
‘(-n) = -(n’)
Which we can also write as
P(-n) = – S(n)
Having defined S and P for sign and size pairs, we can then define addition recursively for negative numbers as well.
x + 0 = x
x + ‘y = ‘(x+y)
This parallels the addition definition but using the predecessor.
We can then define subtraction and adding negative numbers.
x – 0 = x
x- y’ = ‘(x-y)
x’-y = (x-y)’
And so on. (Consistency has to be addressed in the mathematical structure of definitions and theorems.)
We can do the same thing for multiplication
x*0 = 0
x*(‘y) = x*y – x
‘x*y = x*y – y
So we extend the domain of these recursive functions from natural numbers to positive sign natural numbers to negative sign natural numbers. We can also view these as new recursive definitions from scratch on ordered pairs of a sign and a size.
A slightly more complicated issue is iterated use of minus or negative signs. This can be defined recursively itself.
-(- a) = a
-(-(-a)) = -a
Let f(n,a) be the use of minus signs n times before a.
f(0,a) = a
f(1,a) = -a
f(2,a) = a
f(n’,a) = – f(n,a)
Alternatively, we can define an odd number of negative signs as negative and an even number as positive or no negative sign as context indicates.
This approach to negative signs and multiplication does use recursion. The use of recursion in this way can be described as multiplication as repeated addition.
-5 * 3 = -5 -5 -5 = -15
This can be considered as repeated subtraction.
Use of the distributive law as a way to justify the extensions or “prove” them is old. This topic traces to the 19th century and the work of De Morgan, Peacock and others. I saw some reference that I will try to find indicating they were involved in this or something related.
One can define multiplication with negative signs, one or two by a rule of signs. This is defining the function multiplication on the signed integers. Similarly one defines other functions like addition on the signed integers.
When we take a recursive function and use its recursion equations to define the function on the signed integers, we get a different approach.
One can take one approach as the definition and prove the other as a theorem. This can be done either way.
I developed the above discussion primarily after reading the “snarky” criticism thread at the first Dan Meyer dy/dan on the video criticism of a Khan Academy video.
Given the continued discussion of this topic recently, I decided to go ahead and publish the above approach now instead of in a later volume of my ongoing series of books on Peano Axiom based arithmetic.
For other functions, we can try to proceed as follows. Suppose we have some recursive equations to define a function on the natural numbers. We try to form equations of the form:
f(0,0) has some known value
f(x,’y) = A(x,y,f(x,y))
f(‘x,y) = B(x,y,f(x,y))
f(x’,y) = C(x,y,f(x,y))
f(x,y’) = D(x,y,f(x,y))
This lets us extend the function f(x,y) recursively on the signed integers. We have to consider consistency on the paths of (x,y) values that reach the same point.
We can call this recursive continuation or analytic recursive continuation (ARC) or even analytic continuation. Analytic continuation is usually interpreted to mean a continuation by power series. However, it can also mean using a formula in a wider domain of inputs. So this method can be called analytic continuation. Calling it recursive continuation gives a distinct name and removes the context of power series or reals or complex numbers.
Analytic Recursive Continuation (ARC) as a term has the advantage that analytic continuation can be done along an arc in the plane for power series. Thus it fits into analytic continuation.
So we can talk about the recursive continuation of the addition function and multiplication function from positive natural numbers to the signed integers. This includes combinations of one positive and one negative input or two negative inputs.
If we think of order of signed integers as being done with functions that indicate order, and if those are set up recursively, then we can use this approach to extend order from positive numbers to signed integers. This then can explain why we have the order relations we have for signed integers.
The above is a sketch of this approach and not meant to be other than a quick draft. There may be redundancies or consistency issues to be worked out and some of that may have been done in other references already. Please bring any references to my attention (with or without snarkiness) in the comments.
== John Golden, David Coffey, MTT2k, Dan Meyer, Khan Academy, and Sal Khan related links (and see further below)
Mystery Teacher Theatre 2000
June 20th, 2012 by Dan Meyer
Sal Khan Comments On #MTT2k In Chronicle of Higher Education
June 28th, 2012 by Dan Meyer
Parody Critiques Popular Khan Academy Videos
June 28, 2012, 1:59 pm
By Angela Chen
Post parody Khan Academy videos (links from Justin Reich piece)
Also Reich indicates
“Khan’s work is similar to a 2009 video from James Tanton.”
== Some References on this.
My internet search on this topic earlier found the following references.
It was not until the 19th century when British mathematicians like De Morgan, Peacock, and others, began to investigate the ‘laws of arithmetic’ in terms of logical definitions that the problem of negative numbers was finally sorted out.
Today’s search turned up
This referenced the following by Kenny Felder
The Felder piece has a section “The Third Answer: Progress from the Positive Numbers” which can be considered related to recursion.
Searching on the term “recursive continuation” which I altered from analytic continuation to be distinctive, some hits come up showing some but not much use of it. One example is by Jess Benhabib.
Snarky comments welcome.