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Why does minus one times minus
one equal plus one or more

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generally why when we take a
negative number and multiply it

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by another negative number, do
we get a positive answer?

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This is a question that has
plagued every student of

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arithmetic. It disturbs us.

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It disturbs us because it seems
to lie outside our familiar

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experience. It doesn't seem to

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fit. Let me explain by reviewing
the rules of arithmetic for

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multiplying together 2 numbers.

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And we should start by
multiplying 2 positive numbers.

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Multiply positive 5.

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By plus 3.

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And we know the answer.

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Is 15.

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This. We're comfortable with it
matches our experience. When,

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for example, were counting
counting money, so we can think

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of 3 * 5 as representing three
piles, three separate piles.

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In each pile, there being 51
pound coins. So in total when we

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have them all together, we have

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15. So 3 * 5 is 15
were quite happy with.

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Next Let's see what happens when
we take a negative number.

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Negative one for example, and
I'll put brackets around for

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convenience. When we multiply
negative one by one, the answer

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is minus one.

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If we then multiply negative one
by two, the answer is minus 2.

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And we can go on multiply
minus one by three, and the

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answer is minus 3.

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And you can see where developing
what really is a Times table for

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minus one, but where convertible
with this, because again it

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matches our experience. We can
think of it again in terms of

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our bank account when dealing

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with money. We can think of 1 *

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1. As taking £1 out of our
account on one occasion only and

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so our account is in deficit by

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one pound. Two times minus one
we can think of as taking £1 out

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of our account on two separate
occasions on what happens is our

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account is in deficit by 2
pounds, and so on.

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I multiplying a positive number
by a negative number, giving

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rise to a negative answer is OK,

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it fits. What then when we
multiply 2 negative numbers

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together? Minus one times minus
one, but so the answer is

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plus one. Why is

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this so? Where on earth does it

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come from? It didn't seem
to correspond to anything

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in our familiar experience.

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So what can you do?

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Well, you could phone a friend.
That's if you've got a friend

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who is a math teacher.

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Or you could ask the math

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teacher. And I recall doing
precisely that many, many years

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ago. I asked him why does minus
one times minus one equals plus

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one. And what he said was just
accept it. For now. You'll

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understand it later on.

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Very unsatisfactory, I
thought I ask a question but

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I don't get an answer.

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But

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When you think of it, this
happens very often in life. A

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question is posed, but the
answer is out of reach. For

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example, when a small child asks
her parents what is a black hole

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or where on earth where is

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Infinity? The answer isn't
necessarily clear. In order

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to appreciate the answer,
more information, more

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knowledge is required.

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So let's return to minus one
times minus one equal plus one.

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What extra information is
required in order to

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understand this?

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It turns out that we need 2
extra bits of information, 2

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rules of arithmetic.

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And these rules are one the
rule of precedence.

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What is presidents? Well,
presidents tells us.

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Which operation to do first?

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Well, next in any given

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calculation. So if we look at an
example with positive numbers 3.

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Times bracket 4 + 2. You can see

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that. We've got multiple
multiplication to do, and we

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have an addition to do. Which do
we do first? President says you

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do what's in the brackets first.
4 + 2 is 6.

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3 * 6.

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Is 18. No problem.

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So that's one.

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Piece of information that we're
going to make yourself a second

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that we're going to make use of.

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Is the fact that multiplication
is distributive over addition?

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Now what does that mean?
Multiplication is distributive

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over addition. Well, that's best
appreciated again by an example.

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And we can use the same example
that we've got here.

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3 * 4 + 2.

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It involves multiplication and

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addition. If multiplication is
distributive over addition, it

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means that this calculation is
equivalent to multiplying 3 by

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4. And then adding three by
two 3 * 2.

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And we can check
that this is so.

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We've already worked out three.

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Times bracket 4 + 2 using the
rule of presidents and the

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answers 18. On the right hand
side with 3 * 4, which is 12.

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Also we've got 3 * 2 which is
612 + 6 is 18, right outside

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equals left outside.

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So this fact that
multiplication is

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distributive over addition
works for the numbers 3,

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four and two.

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As a second example, I'm going
to show it works when we have a

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negative number. Say minus one.

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Times 2 + 1.

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Got two and one
both positive numbers.

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And we multiply that bracket by

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minus one. If multiplication is
distributive over addition, it

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means that minus 1 * 2.

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It means that the left hand side
is minus 1 * 2.

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Plus minus 1 * 1.

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There are precedents, the left
hand side we do what's in

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the brackets first 2 plus one
is 3 times minus one.

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Is minus 3.

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On the right hand side, minus 1
* 2 is minus 2.

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Minus 1 * 1 is minus one.

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And you can see there on
the right hand side we have

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minus two and minus one
added together, which is

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minus three. So left hand
side equals right hand

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side. I multiplication is
distributive over addition

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for the numbers minus 1,
two and one.

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Now the key.

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For understanding why minus one
times minus one equals plus one.

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Is that we insist that
multiplication is distributive

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over addition for all numbers.

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Whether negative or positive and
what we need to consider is

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a particular calculation minus

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one times. Same as we had up
there, but instead of two we put

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minus one. Then if
multiplication is distributive

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over addition, this is equal to
minus one times minus one.

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Plus
Minus one times plus one.

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Now for the left hand side using
precedents we do what's in the

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brackets first minus one plus
one is 0.

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Anything times zero, are you
minus one is itself Sarah, so

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I left outside is 0 on the
right hand side. The first

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term is minus one times minus
one, which is what we're

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trying to determine.

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And the end term. The last term
minus 1 * 1 is minus one.

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So you can see that if we now
take one to the left hand side.

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We have shown that minus one
times minus one is equal to plus

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one. Why this result follows as
a direct consequence of these

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two rules of arithmetic. The
rule of precedence and the rule

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that multiplication has to be
distributed over addition.

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So you see.

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But my old school teacher, Mr
Dennison, was quite right when

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he said. Accept it for
now. Lab you will only

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understand it later on.