www.mathcentre.ac.uk/.../arith-ext-6.mp4

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.