It may look as if we are over it, but the negative numbers
still represent a puzzle for many.
Ms. Ruth McNeill[1] found it very
counter-intuitive to consider that a negative number times a negative number
comes out to a positive number. According to the source we have consulted, this lady
claimed that multiplying intensifies something, and thus two negative numbers
multiplied together should properly produce a very negative result.
All in human
knowledge is obviously convention, for even the most guaranteed of the
findings, say a person is definitely with worms in their digestive system, and
we should therefore remove those, if we see a roundworm coming out of their
body, depends on a set of premises.
So we have a body?
We obviously have a
body for as long as our senses tell us that there is a difference between us
and the others that is determined by our skin, for instance.
Trivially, if we find
out that our eye has a matrix that is not evolved enough, and an animal of some
sort has a better matrix for their eye, and we decide to use their matrix, and,
in doing that, we see no separation between our bodies and the bodies of
others, then we will start believing that the worm is not inside of our digestive
system: It is inside of our common body instead.
If things get worse,
and our new eye matrix reveals that we are actually also connected to the land
around us with no distinction, then the worm is another being fighting for its
life in the only environment that is available to it.
In this case, our
worm may become our next Tasmanian Devil, like it may need to be preserved,
and perhaps it is then OK if a few pieces of us die in order for it to keep
on existing.
If even physical
things, which we take for granted, are actually products of our imagination,
all founded on a set of premises, or assumptions, imagine the rest… .
Human knowledge is
fragile.
Inside of the World
of Mathematics, we have the negative numbers, which could be seen as the
reflection, in the mirror, of the non-negative and non-null numbers.
We should be able to
explain all basic mathematical operations with the negative and non-null
numbers in the same way that we explain the basic mathematical operations with the
non-negative and non-null numbers.
It could be possible
that Ms. Ruth were right and all our conventions, those that have to do with
the negative numbers, were actually not logical. If that were true, however,
our explanations would have to be odd, unnatural or counter-intuitive, as she
said they were.
When we add two
non-negative and non-null numbers, we think of two sets of items being merged, then of us counting the items, and coming up with a figure in the end. Some of us, however, think of the ruler and then
imagine a particle moving from the starting point, the 0, over as many units
as those we have in the first number, then over as many units as those we have
in the second number, and finally stopping. Those would go to the ruler and
take note of the number that is found under the particle as they finish doing that.
Well, if we put the
ruler in front of a mirror, then we will see its reflection in it. If we now
position the ruler in a way to have its zero connected to the surface of the
mirror, we will see another ruler in the virtual world that we have just created. As in Physics, we translate inside of the
mirror into a '-' sign. Now, if we forget the '-' sign of the amounts involved
and perform the requested mathematical operations in the way we described before, but
look at the mirror, at the reflection of the result, we will have our actual
result. We will then have both quantities together; therefore a value that
represents that, but, this time, this value will appear to the right side of a
minus sign.
Now, let’s think of
our bank accounts. Let’s think of the minus sign as a synonym for debt. We
would then increase our debt as we increase the quantity of numbers with negative
signs in our sum, right? Because it would still be a debt, we would have to
have the minus sign in front of the result of the sum.
If we had a debt of,
say, two dollars, and we wanted to make all our seven friends owe as much as we
do, therefore we would like to copy our situation for each one of them or we
would like to multiply it, we would have minus fourteen dollars, since that
would be two dollars in debt seven times or 7 x (-2).
If our friends then
decided that in fact they owe that money but left their seats, and they were sitting
at our table before, and they all have put the money they owed us over their
seat before they left, we then would have -7x(-2)=14.
We could also think
of the situation as being our seven friends who have paid their debts with us,
therefore our 7 x (-(-2)), that is, our seven friends that now can have that
debt taken away (with a minus) or 14 dollars entering our pocket since it is
two for each.
Besides, if we have a
positive balance in our bank account, say AU$ 4, and we then acquire a debt of
AU$ 10, we know that it is missing AU$ 6 for us to be OK with our bank,
that is, we have a -6 result.
If we divide a debt
of, say, AU$ 10 by two people who do not exist, we are probably trying to lure
the Taxation Office because we actually have a profit that we cannot explain in
our result with our business, let’s say. That means that we actually have a
non-negative balance to deal with. A 10-dollar debt that has never existed
divided by two people that have never existed ((-10)/(-2)) can only mean an excess of 10
dollars, therefore a +10 or a 10. In this case, we actually have an excess of 5 for each person.
Another situation that could explain the process of division involving two negative quantities is that in which we have the quantity of people we had before owing us minus two. They
all owed AU$ 10, therefore had a debt of AU$ 10 (-10 x 2). Now they have paid
their debt, so that we have (-10)x(-2), therefore we have a non-negative 20 as an entry in our bank statement… .
On the other hand, we
could have AU$10 to explain to the Taxation Office. We would then come up with
two people who owed us money, but two people who do not exist, in order to even
out all and not pay tax. We would then have a 10-dollar profit to divide by 2
people who do not exist (10/(-2)). Well, the result, in this case, can only be
that each one owed us AU$5, therefore each one got a -5 for their account… .
If we had a debt of
AU$ 10 to divide by two people that actually exist ((-10)/2), then we know
that each one would be getting a debt of 5, that is, a -5.
It then seems to us
that Ms. Ruth McNeill could be more accepting of the mathematical rules if she
had access to our explanations.
If we could go back
in the past, is it not? Only if we could… .
[1]
Ruth
McNeill, “A Reflection on When I Loved Math and How I Stopped.” Journal of
Mathematical Behavior, vol. 7 (1988) pp. 45-50, as mentioned at http://homepages.math.uic.edu/~saunders/MTHT591_F12_crypto/supplements/4gelfand.pdf (Bonnie
Saunders, September 2012, as seen by us on the 10th of July of 2013)
____________________________________________________________
____________________________________________________________
____________________________________________________________
