# Re: [xmca] a minus times a plus

```Mike, according to our wonderful correspondents, the number
line is the favourite "visual/practical image". I believe them.

People have rightly suggested the use of this to present
mutliplication tables in the form of a number line, which,
to start with, stop at 0. Then you ask: "what if we extend
this ...?" This looks great to me.

My only point is that the teacher would use this
practical/visual approach to get kids filling in the blanks
correctly ... *and then* abstract the general rule "minus
times minus = plus". You have to learn this as a rule at
some point. Question is when and on what basis is it to be
justified and understood?

There is no way you can introduce that rule prior to
practical/intuitive manipulations. If some appropriate
intuitive image is used to present the base regularity, the
kids will be blissfully unaware that stepping across the
boundary into a new domain of numbers, which you have
presciently placed as a continuation of the starting doamin
is an unwarranted, arbitrary and unjustifable logical leap.
And why not! But until kids have left behind them the
intuitive justifications, they have not learnt mathematics,
which is after all the object of the exercise, isn't it? Or
is it?

The great thing about Ed's first contribution, I think, is
that he showed how foundations-of-mathematics thinking can
mirror the operation used for teaching: i.e., extension of a
domain. And this is a foundation on which people can reflect
and learn about mathematics, as well as carry out the
required opeations when asked.

My only point is that kids need to first practice the
operations in the narrow and then the wider domain, and then
abstract the rule, and then much later learn the logical
meaning of domain-extension. At bottom, the answer to "why
is - x - = + ?" is "because we say so". It is no more
provable than "1 + 1 = 2" and intuitive demonstrations which
purport to prove it are actually fooling people. But maybe
it is necessary to fool people first and explain why later,
if they're interested.

To return to the example that we were discussing the other
day: e to the i pi = -1. This is justified in university
maths classes using the domain extension rationale. It is
difficult to see any other way of approaching it. And that
presupposes familiarity with 4 domains of numbers. But it
was fantastic at the time to already have the *Argand
diagram*. The Argand Diagram looks like Cartesian x-y
coordinates, and is understood as a representation of
"complex numbers" in which the domain of numbers has been
extended to include the square root of minus one, and
represented as a "2 dimensional number". With the Argand
Diagram a lot of the symmetries of this great equation
become visually very compelling.

Mike Cole wrote:
```
```Andy-- Visual image PLEASE!!!
mike

```
On Mon, Apr 27, 2009 at 6:05 PM, Andy Blunden <ablunden@mira.net <mailto:ablunden@mira.net>> wrote:
```
From the worst ex-maths teacher in the world ...

Certainly I think Ed's explanation of "why" minus numbers behave the
way they do when included in operations that make intuitive
explanation impossible is right. I.e., you ask that regularities
that applied in the domain so far ought to be retained when the
domain is extended by adding a new group of numbers. There is no
meaning for "multiplying by a negative number" that can be reliably
deduced from intuitive definitions of "multiply" and "negative". So
the rule is that you can grasp the idea of "multiply" intuitively
through the idea of repeated addition, just as you grasped the idea
of addition by repeated counting. And you can grasp the idea of
negative numbers in some equally intuitive way (there are several
options), but not a way which can be fitted into the idea of

So you take Ed's advice and rely on some general rule or visual
image that worked before and require that it still work for negative
numbers. In that way you move out of the bounds of intuition into
mathematical thinking, guided no longer by plausible intuition, but
by a mathematical rule.

That still leaves open the question as to whether you can teach
general rules and mathematical reasoning to someone who has had no
practice in applying the rules whose jutification you claim to
achieve by this "rule extension" rationale that Ed exlained.

I was of a generation that learnt my times-tables by rote and had my
first lesson on real mathematics in my last years as an undergrad.
15 years later, and then 6 years later was asked to teach "modern
mathematics" to 13 year old kids who couldn't count and had no idea
of what "1/2" meant except a 1 a stroke and a 2. I was not a happy
chappy at the time. I blame Piaget and his "Genetic Epistemology"
and a whole lot of absurdity that went down in the early 1970s.

I say: learn to ride your bike, and then learn dynamics to make
sense of it afterwards.

Andy

Mike Cole wrote:

Great!! Thanks Ed and Eric and please, anyone else with other
ways of
explaining the underlying concepts.
Now, we appear to have x and y coordinates here. If I am using a
number line
that ranges along both x and y axes from (say) -10 to +10 its pretty
easy of visualize the relations involved. And there are games
that kids can
play that provide them with a lot of practice in getting a
strong sense
of how positive and negative positions along these lines work.

What might there be of a similar nature that would help kids and
old college
professors understand why -8*8=64 while -8*-8=64?

Might the problem of my grand daughter, doing geometry, saying,
"Well, duh,
grandpa, its just a fact!) arise from the fact (is it a fact?) that
they learn multiplication "facts" before they learn about
algebra  and
grokable explanations that involve even simple equations such as
y+a=0 are unintelligible have become so fossilized that the required
reorganization of understanding is blocked?

mike

On Mon, Apr 27, 2009 at 4:16 PM, Ed Wall <ewall@umich.edu
<mailto:ewall@umich.edu>> wrote:

Mike

It is simply (of course, it isn't simple by the way)
because, the
negative integers (and, if you wish, zero) were added to the
natural numbers
in a way that preserves (in a sense) their (the natural
numbers) usual
arithmetical regularities. It would be unfortunate if
something that was
true in the natural numbers was no longer true in the
integers, which is a
extension that includes them.  Perhaps the easiest way to
the negative x
positive business is as follows (and, of course, this can be
precise - smile):

3 x 1 = 3
2 x 1 = 2
1 x 1 = 1
0 x 1 = 0

so what, given regularity in the naturals + zero) do you
think happens
next? This thinking works for, of course, for negative times
negative. The
opaque proof is more or less as follows.

Negative numbers are solutions to natural number equations
of the form (I'm
simplifying all this a little)

x + a = 0    ('a' a natural number)

and likewise positive numbers  are solutions to natural
number equations of
the form

y = b          ('b' a natural number)

Multiplying these two equations in the usual fashion within
the natural
numbers gives

xy + ay = 0

or substituting for y

xy + ab = 0

so, by definition, xy is a negative number.

Notice how all this hinges on the structure of the natural
numbers (which
I've somewhat assumed in all this).

Ed

On Apr 27, 2009, at 6:47 PM, Mike Cole wrote:

Since we have some mathematically literate folks on xmca,
could someone

please post an explanation of why

multiplying a negative number by a positive numbers
yields a negative
number? What I would really love is an explanation
that is representable in a manner understandable to old
college professors
and young high school students alike.

mike
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```
-- ------------------------------------------------------------------------
```    Andy Blunden http://home.mira.net/~andy/
Hegel's Logic with a Foreword by Andy Blunden:
From Erythrós Press and Media <http://www.erythrospress.com/>.

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```
```
--
------------------------------------------------------------------------
Andy Blunden http://home.mira.net/~andy/
Hegel's Logic with a Foreword by Andy Blunden:
From Erythrós Press and Media <http://www.erythrospress.com/>.

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