Why does minus one times minus
one equal plus one or more
generally why when we take a
negative number and multiply it
by another negative number, do
we get a positive answer?
This is a question that has
plagued every student of
arithmetic. It disturbs us.
It disturbs us because it seems
to lie outside our familiar
experience. It doesn't seem to
fit. Let me explain by reviewing
the rules of arithmetic for
multiplying together 2 numbers.
And we should start by
multiplying 2 positive numbers.
Multiply positive 5.
By plus 3.
And we know the answer.
Is 15.
This. We're comfortable with it
matches our experience. When,
for example, were counting
counting money, so we can think
of 3 * 5 as representing three
piles, three separate piles.
In each pile, there being 51
pound coins. So in total when we
have them all together, we have
15. So 3 * 5 is 15
were quite happy with.
Next Let's see what happens when
we take a negative number.
Negative one for example, and
I'll put brackets around for
convenience. When we multiply
negative one by one, the answer
is minus one.
If we then multiply negative one
by two, the answer is minus 2.
And we can go on multiply
minus one by three, and the
answer is minus 3.
And you can see where developing
what really is a Times table for
minus one, but where convertible
with this, because again it
matches our experience. We can
think of it again in terms of
our bank account when dealing
with money. We can think of 1 *
1. As taking £1 out of our
account on one occasion only and
so our account is in deficit by
one pound. Two times minus one
we can think of as taking £1 out
of our account on two separate
occasions on what happens is our
account is in deficit by 2
pounds, and so on.
I multiplying a positive number
by a negative number, giving
rise to a negative answer is OK,
it fits. What then when we
multiply 2 negative numbers
together? Minus one times minus
one, but so the answer is
plus one. Why is
this so? Where on earth does it
come from? It didn't seem
to correspond to anything
in our familiar experience.
So what can you do?
Well, you could phone a friend.
That's if you've got a friend
who is a math teacher.
Or you could ask the math
teacher. And I recall doing
precisely that many, many years
ago. I asked him why does minus
one times minus one equals plus
one. And what he said was just
accept it. For now. You'll
understand it later on.
Very unsatisfactory, I
thought I ask a question but
I don't get an answer.
But
When you think of it, this
happens very often in life. A
question is posed, but the
answer is out of reach. For
example, when a small child asks
her parents what is a black hole
or where on earth where is
Infinity? The answer isn't
necessarily clear. In order
to appreciate the answer,
more information, more
knowledge is required.
So let's return to minus one
times minus one equal plus one.
What extra information is
required in order to
understand this?
It turns out that we need 2
extra bits of information, 2
rules of arithmetic.
And these rules are one the
rule of precedence.
What is presidents? Well,
presidents tells us.
Which operation to do first?
Well, next in any given
calculation. So if we look at an
example with positive numbers 3.
Times bracket 4 + 2. You can see
that. We've got multiple
multiplication to do, and we
have an addition to do. Which do
we do first? President says you
do what's in the brackets first.
4 + 2 is 6.
3 * 6.
Is 18. No problem.
So that's one.
Piece of information that we're
going to make yourself a second
that we're going to make use of.
Is the fact that multiplication
is distributive over addition?
Now what does that mean?
Multiplication is distributive
over addition. Well, that's best
appreciated again by an example.
And we can use the same example
that we've got here.
3 * 4 + 2.
It involves multiplication and
addition. If multiplication is
distributive over addition, it
means that this calculation is
equivalent to multiplying 3 by
4. And then adding three by
two 3 * 2.
And we can check
that this is so.
We've already worked out three.
Times bracket 4 + 2 using the
rule of presidents and the
answers 18. On the right hand
side with 3 * 4, which is 12.
Also we've got 3 * 2 which is
612 + 6 is 18, right outside
equals left outside.
So this fact that
multiplication is
distributive over addition
works for the numbers 3,
four and two.
As a second example, I'm going
to show it works when we have a
negative number. Say minus one.
Times 2 + 1.
Got two and one
both positive numbers.
And we multiply that bracket by
minus one. If multiplication is
distributive over addition, it
means that minus 1 * 2.
It means that the left hand side
is minus 1 * 2.
Plus minus 1 * 1.
There are precedents, the left
hand side we do what's in
the brackets first 2 plus one
is 3 times minus one.
Is minus 3.
On the right hand side, minus 1
* 2 is minus 2.
Minus 1 * 1 is minus one.
And you can see there on
the right hand side we have
minus two and minus one
added together, which is
minus three. So left hand
side equals right hand
side. I multiplication is
distributive over addition
for the numbers minus 1,
two and one.
Now the key.
For understanding why minus one
times minus one equals plus one.
Is that we insist that
multiplication is distributive
over addition for all numbers.
Whether negative or positive and
what we need to consider is
a particular calculation minus
one times. Same as we had up
there, but instead of two we put
minus one. Then if
multiplication is distributive
over addition, this is equal to
minus one times minus one.
Plus
Minus one times plus one.
Now for the left hand side using
precedents we do what's in the
brackets first minus one plus
one is 0.
Anything times zero, are you
minus one is itself Sarah, so
I left outside is 0 on the
right hand side. The first
term is minus one times minus
one, which is what we're
trying to determine.
And the end term. The last term
minus 1 * 1 is minus one.
So you can see that if we now
take one to the left hand side.
We have shown that minus one
times minus one is equal to plus
one. Why this result follows as
a direct consequence of these
two rules of arithmetic. The
rule of precedence and the rule
that multiplication has to be
distributed over addition.
So you see.
But my old school teacher, Mr
Dennison, was quite right when
he said. Accept it for
now. Lab you will only
understand it later on.