Negative times negative makes positive

[Genady]

Does anybody know/remember how this rule of number multiplication (see the title) is/was explained to little children?

[studiot]

4 hours ago, Genady said:

Does anybody know/remember how this rule of number multiplication (see the title) is/was explained to little children?

Do they have to be little ?

 

I seem to remember it was presented as a rule rather than withe proof/justification.

Here is a typical treatment with limited justification followed by

Learn the rule it is easier.

 

 

[exchemist]

What I do remember is the story of the teacher saying that, while two negatives make a positive, two positives still make a positive.

To which a voice at the back of the class remarked: "Yeah. Right".  

[joigus]

My recollections coincide with @studiot's. Rule.

Trying to think about it afresh, if you accept that 1+(-1)=0, which seems far more intuitive, as well as (-1)*0=0 and (-1)*1=-1, the distributive property forces you to admit that,

0=(-1)*0=(-1)*(1+(-1))=(-1)*1+(-1)*(-1)=-1+(-1)*(-1)

so (-1)*(-1) must be 1, which is the additive inverse of -1.

I'm not sure that would be very persuasive to children though...

6 minutes ago, exchemist said:

What I do remember is the story of the teacher saying that, while two negatives make a positive, two positives still make a positive.

To which a voice at the back of the class remarked: "Yeah. Right".  

I love this joke. It works in Spanish with "sí, sí" too. The intonation is essential.

[MigL]

It was a loooong time ago, but I seem to remember the teacher comparing to even and odd numbers, and how two odds make an even, while two evens also make an even.

[Genady]

"Little children" in my question are school kids which are taught this rule - 3rd grade?

As I don't remember any explanation given then to me, I think that there was not one. Like above. Just a rule. I wonder, is it still this way? Everywhere?

"Hand waving" explanations like in the book posted by @studiot and the odd-even metaphor recalled by @MigL help perhaps to "get" the rule.

How about a "Guess the Rule" game? Let them come up with a possible rule and find that other suggestions don't work?

Edited October 26 by Genady

[studiot]

5 hours ago, Genady said:

I wonder, is it still this way?

 

Thank you for posting this topic it made me stop and think.  +1

My offering was taken from a mid 20th century book.
Unfortunately my wife 'persuaded' me to let go some more modern elementary maths texts in favour of Music theory and Elementary Pharmaceutical Chemistry etc discarded by the younger generation.

However I have one modern book left, but it was really aimed at adults. "Maths made easy for Science , Engineering and Business'.  This presents analternative approach.

I have yet to dig out my older texts such as Hall and Knight.

Here also is another approach but requires an appeal to symmetry

 

 

 

 

[sethoflagos]

17 hours ago, Genady said:

Does anybody know/remember how this rule of number multiplication (see the title) is/was explained to little children?

As far as I remember:

• Learn multiplication tables by rote
• Apply multiplication tables to negative multiplicands
• Logically extend multiplication tables backwards for negative multipliers
• Hope little minds latch on to the underlying symmetries

 

[Genady]

An alternative to appealing to symmetries approach could be to uncover what these number operations do to number line:

- adding number shifts (translates) the line

- multiplying by positive number stretches / compresses the line

- multiplying by -1 flips / inverts the line.

When the line is flipped / inverted, 2 -> -2, 3 -> -3, ..., and correspondingly, -2 -> 2, -3 -> 3, ... .

[studiot]

8 minutes ago, Genady said:

An alternative to appealing to symmetries approach could be to uncover what these number operations do to number line:

 

- multiplying by -1 flips / inverts the line.

Exactly the first approach in my last post.

😀

[Genady]

3 minutes ago, studiot said:

Exactly the first approach in my last post.

😀

Right. It would be a part of the big picture if other operations were presented in this way, too.

[studiot]

2 minutes ago, Genady said:

Right. It would be a part of the big picture if other operations were presented in this way, too.

Yes they are also presented this way in the book, but you didn't ask about them.

Edited October 26 by studiot

[Genady]

1 hour ago, studiot said:

Yes they are also presented this way in the book, but you didn't ask about them.

Well then. The only issue is that this book is aimed at adults as you said rather than at "little children" and thus does not answer the OP question. 

[StringJunky]

This is pretty intuitive, I think:

Rule: A negative sign reverses the positive sign to negative, and another negative sign reverses the sign again back to positive. I thought it was just like an axiom to make things consistent with the other mathematical rules.

Edited October 26 by StringJunky

[Genady]

1 hour ago, StringJunky said:

This is pretty intuitive, I think:

Rule: A negative sign reverses the positive sign to negative, and another negative sign reverses the sign again back to positive. I thought it was just like an axiom to make things consistent with the other mathematical rules.

What you refer to is, -(-2)=2. It is not intuitive that it has to do with multiplication, i.e., (-1)*(-2)=2.

[Eise]

Found this elsewhere:

[Genady]

11 minutes ago, Eise said:

Found this elsewhere:

Sorry, the link is broken. 

[Eise]

Huh? I do not see it on my tablet, but I posted in Chromium on my Linux notebook. And there I see it again. But when I 'fly over' I see 'unavailable'. Maybe local cache. Trying again:

[Genady]

... unless you're a fermion 🙂

Edited October 27 by Genady

[studiot]

21 hours ago, Genady said:

Well then. The only issue is that this book is aimed at adults as you said rather than at "little children" and thus does not answer the OP question. 

Well you have been disappointingly combative towards replies in this thread. I thought you wanted to discuss this excellent subject and I welcome the input and ideas from several others.

 

Out of interest here is what Richard Courant  the famous Mathematician and mathematical educator has to say on the subject:

Note carefully he claims that it is impossible to proove that (-1) x (-1) = +1.  I has to be defined that way.

[Genady]

6 minutes ago, studiot said:

It has to be defined that way.

Yes, this is exactly my point, with the emphasis on "has to".

[Genady]

2 hours ago, studiot said:

Well you have been disappointingly combative towards replies in this thread. ...

I welcome the input and ideas from several others.

Sorry for this impression.

I went back and expressed my sincere appreciation of the inputs.

[joigus]

2 hours ago, Genady said:

... unless you're a fermion 🙂

Good point. I was thinking of using reflections too, until I remembered fermions.

I don't think children would care too much about fermions tho...

2 hours ago, studiot said:

Note carefully he claims that it is impossible to proove that (-1) x (-1) = +1.  I has to be defined that way.

I agree. Any other choice would give you problems with the distributive property and/or other equally fundamental properties though.

After all, there must be a reason why we've been choosing that option and no other one has resulted in an interesting algebraic framing.

[Genady]

3 minutes ago, joigus said:

Good point. I was thinking of using reflections too, until I remembered fermions.

I don't think children would care too much about fermions tho...

+1, but I've used up my daily quota. Will try to remember tomorrow.

7 minutes ago, joigus said:

After all, there must be a reason why we've been choosing that option and no other one has resulted in an interesting algebraic framing.

This option gives integers a ring structure.

[studiot]

3 hours ago, joigus said:

I agree. Any other choice would give you problems with the distributive property and/or other equally fundamental properties though.

After all, there must be a reason why we've been choosing that option and no other one has resulted in an interesting algebraic framing.

 

3 hours ago, Genady said:

This option gives integers a ring structure.

 

Well it should be remembered that an algebraic ring structure has two differnt definitionss, dependin which side of the atlantic you are on.

 

But it should also be remembered that our construction of (formal) algebraic sturcutures are designed to relflect the convenient arithmetic structures we have found convenient for other purposes.

It would actually be OK to define (-1) x (-1) = -1.

That would still form a hemi group (or semigroup if you wish) under multiplication.

However american practice requires there to be a unique non zero identity for multiplication, which would not be the case with -1.

 

It is notable that Birkhoff and MacLane (american definitions) start off on page 1 with this subject and reach the crux of it by page 6  (so a bit much to post here as an extract) with 'the integers following 8 postulates.

Definitely  a bit much for 12 year olds I feel.

 

 

Edited October 27 by studiot