Multiplication of Negative Numbers Recursive Continuation

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))), …

1= 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.

Consider

-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

http://mathhombre.blogspot.com/2012/06/mystery-teacher-theatre-2000.html

http://blog.mrmeyer.com/?p=14299

Sal Khan Comments On #MTT2k In Chronicle of Higher Education

June 28th, 2012 by Dan Meyer

http://blog.mrmeyer.com/?p=14360

Parody Critiques Popular Khan Academy Videos
June 28, 2012, 1:59 pm

Multiplying Positive and Negative Numbers

Why a Negative Times a Negative is a Positive

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.

http://www.math.toronto.edu/mathnet/questionCorner/minustimesaminus.html

http://planetmath.org/encyclopedia/ProductOfTwoNegativeNumbers.html

http://nrich.maths.org/5961

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.

http://en.wikipedia.org/wiki/Negative_number

http://en.wikipedia.org/wiki/Negation_%28algebra%29

http://mathforum.org/library/drmath/view/51925.html

Today’s search turned up

http://blogs.edweek.org/edweek/edtechresearcher/2012/06/the_mtt2k_prize_and_kudos_for_khan.html

This referenced the following by Kenny Felder

http://www4.ncsu.edu/unity/lockers/users/f/felder/public/kenny/papers/negative.html

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.

http://www.econ.nyu.edu/user/benhabib/continuous_time_search_notes.pdf

==

MTT2k Prize

http://blogs.edweek.org/edweek/edtechresearcher/2012/06/the_mtt2k_prize_and_kudos_for_khan.html

==

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.

50 Responses to Multiplication of Negative Numbers Recursive Continuation

1. Pogranicznik bierze Relewantny budowa. mu z
rękiHttp://Schoolplayh.Uni.Me/ (Www.Rossfamilyfoundation.Org) jak psu z
gardła, wilgotny od momentu potu i dodatkowo deszczu kartonowy urywek,
Poniekąd nie trawi
wydrukowanych na zanim liter. Powinna nie
wie iterować łacinką.
– Ryszard Leśniewski, urodzony… obywatelstwo – niedostatek, hacjenda –
niedostatek…
– Zachoditie, możnowładca. – w sądzie żołnierza rozbrzmiewać bierność.
Zdążył się
przyzwyczaić. Mignęła dioda UPS-a, Zastukała g�.

2. Hello, mydło sandałowe wanted to tell you, I enjoyed this article mydło
sandałowe. It was pretty helpful mydło aleppo.
Keep going submitting!

3. p971 says:

It was pretty funny mikrodermabrazja radom. Keep on submitting!

4. p6995 says:

Hi there, p6995 simply wanted to mention, I enjoyed this article c7649274242695663142.
It seemed to be helpful p6996. Keep on submitting!

5. What’s up, meble konferencyjne simply wanted to say, I enjoyed this post
meble konferencyjne. That it was helpful 3924. Keep submitting!

6. What’s up, fotografia dziecięca warszawa merely wanted to tell you, I liked this blog post zdjęcia
niemowląt warszawa. This was inspiring zdjęcia niemowląt warszawa.
Continue on submitting!

7. Hi, kosmetyki naturalne hurtownia I just wanted to mention, I liked this post kosmetyki naturalne hurtownia.
It was pretty inspiring kosmetyki naturalne hurt.
Keep going writing!

8. What’s up, przedszkole tarchomin simply wanted to tell you, I enjoyed this article przedszkole białołęka.
It actually was funny przedszkole białołęka. Keep going posting!

9. Hi, kosmetyki naturalne hurt I just wanted to tell you, I enjoyed this
article kosmetyki naturalne hurt. It actually was inspiring kosmetyki naturalne hurtownia.
Keep on submitting!

10. Hello, kosmetyki naturalne hurt I would like to tell you, I enjoyed this blog post kosmetyki naturalne hurt.

This has been funny kosmetyki naturalne hurtownia.

Keep going submitting!

11. What’s up, kosmetyki naturalne hurtownia just wanted to say,
I liked this blog post kosmetyki naturalne hurtownia.
It was actually helpful kosmetyki naturalne hurt. Continue on posting!

12. Hello, kraina usmiechu I would like to mention, I loved this blog post kraina usmiechu.

It was inspiring przedszkole bialoleka. Continue writing!

13. Hello, strony internetowe I just wanted to say, I liked this article strony internetowe.
It was funny strony internetowe. Keep going publishing!

14. Hi, strony internetowe simply wanted to say, I liked this blog post strony internetowe.

15. Hi there, ksiegowosc Radom wanted to tell you, I liked this blog post ksiegowosc Radom.

It actually was inspiring biuro rachunkowe Radom.
Keep going submitting!

17. Hi, lady recepcyjne wanted to mention, I loved this blog post
Carry on submitting!

18. Hello, opiekunka osób starszych just wanted to mention, I liked this
post opiekunka niemcy. This became practical
praca niemcy. Continue on posting!

19. Hello, strony internetowe merely wanted to say, I liked this post tworzenie stron. It was

20. Hi there, opiekunka niemcy I just want to say, I loved this blog
post opieka niemcy. It actually was funny opiekunka niemcy.

Keep going writing!

21. p679 says:

Hello, p679 wanted to tell you, I loved this blog post p677.
This became helpful mydło węglowe. Continue
on posting!

22. Hi there, kursy instruktorskie I just wanted to tell you, I liked this
Carry on publishing!

Continue on submitting!

What’s up, kamery Radom simply wanted to tell you, I loved this post opiekunka osób starszych.
This has been helpful opiekunka osób starszych.
Continue on submitting!

25. p10215 says:

Hi, p10215 I just wanted to say, I liked this post praca niemcy.
It seemed to be funny p10219. Keep going submitting!

26. Hello, strony internetowe just wanted to say, I enjoyed this
blog post strony internetowe. This had been funny strony internetowe.
Carry on writing!

27. Hello, strony internetowe just wanted to say, I liked this article strony internetowe.
It actually was funny strony internetowe. Keep on posting!

28. Hi, strony internetowe I would like to say, I liked this post strony internetowe.
This has been helpful strony internetowe. Continue submitting!

29. What’s up, strony internetowe I just wanted to say, I liked this post strony
internetowe. It was inspiring strony internetowe. Continue
on submitting!

30. Hi, strony internetowe I just wanted to say, I loved this post strony internetowe.

It was pretty practical strony internetowe. Continue posting!

31. Hi, strony internetowe merely wanted to mention, I
loved this blog post strony internetowe. It actually was practical strony internetowe.
Keep publishing!

32. What’s up, garderobeskap simply wanted to tell you, I enjoyed this article mobler av metall.
This was funny mobler av metall. Continue writing!

33. Hi, meble pracownicze simply wanted to say, I liked this post meble pracownicze.
This was helpful meble pracownicze. Continue on writing!

34. Hello, strony internetowe merely wanted to tell you,
Keep on publishing!

35. Hello, strony internetowe I just wanted to tell you, I loved this post strony internetowe.
This had been practical strony www. Keep on posting!

36. Hi, trener personalny I would like to tell you, I enjoyed this post trener personalny.
This became helpful trener personalny. Keep going submitting!

37. Hi there, panele szklane do kuchni Warszawa I would like to tell you, I liked this post panele szklane do kuchni Warszawa.
That it was helpful panele szklane do kuchni Warszawa.
Continue submitting!

38. Hi there, Konferansemobler simply wanted to
tell you, I loved this post Konferansemobler. This has been funny
Konferansemobler. Keep going submitting!

39. Hi there, mleczko kokosowe simply wanted to mention, I enjoyed this article mleczko kokosowe.

This has been funny mleczko kokosowe. Keep going submitting!

40. What’s up, pozycjonowanie Radom simply wanted to say, I loved this
Keep on posting!

41. Hi, kabiny prysznicowe Warszawa I would like to tell you, I liked this post grafika na szkle Warszawa.

It was funny panele szklane do kuchni Warszawa. Continue on publishing!

42. Hi there, strony internetowe simply wanted to say, I enjoyed this article strony internetowe.
This became practical 17430. Continue submitting!

43. Hi there, strony internetowe wanted to mention, I loved this blog post strony internetowe.
That it was helpful strony internetowe. Continue posting!

44. What’s up, meble do zabudowy simply wanted to say, I loved
Keep submitting!

45. Hi, strony internetowe I just wanted to say, I enjoyed this post strony internetowe.
It was actually funny strony internetowe.
Continue on writing!

46. Hi, strony internetowe wanted to say, I enjoyed this article strony internetowe.
It actually was inspiring strony internetowe. Keep on submitting!

47. What’s up, młody jęczmień I just want to
say, I liked this blog post młody jęczmień. It seemed to be inspiring młody jęczmień.
Continue posting!

48. Hello, strony internetowe I just want to say, I enjoyed
this blog post młody jęczmień. That it was helpful młody
jęczmień. Carry on submitting!

49. What’s up, strony internetowe wanted to tell you, I loved this post
strony internetowe. It actually was helpful strony internetowe.
Carry on publishing!