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.