# Nhân số âm

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Hello. I'm Professor Von Schmohawk
and welcome to Why U.
In our lectures, so far we have discussed
multiplying positive numbers.
But what happens when we multiply
negative numbers?
With the invention of negative numbers
the rules of multiplication had to be expanded
to allow the operands to be either
positive or negative.
The rules of multiplication were picked so
as to keep everything consistent.
For instance, since one is the
multiplicative identity
if a number is multiplied by one
we should expect that the number’s
value and sign will not change.
Therefore, multiplying a positive number
times a positive number
must produce a positive result
and multiplying a positive number times a
negative number must produce a negative result.
Because of the commutative property of multiplication
we should be able to swap the operands and
get the same result.
So if either operand is negative
we must still get a negative result.
But what if both operands are negative?
If two negative numbers are multiplied,
should the product be positive or negative?
Let’s try it both ways and see what happens.
Let’s multiply six minus four,
times negative one.
We know what the answer should be.
Six minus four is two.
And we already have shown that the product
of a negative and positive number
must be negative.
So the answer must be negative two.
But instead, let’s say we use the
distributive property
and multiply negative one times
each number in the parentheses separately.
We then have negative one times six
plus negative one times negative four.
We know that negative one times six
is negative six.
But we don’t know what sign the product
should be when we multiply two negative numbers.
Is negative one times negative four
negative four or positive four?
Let’s try both possibilities and see which
one gives us the correct answer.
If we assume that multiplying two negative
numbers results in a negative product
then we end up adding negative four to
negative six
which equals negative ten.
But the answer should be negative two
so this is not correct.
The other possibility is that multiplying
two negative numbers gives a positive result.
In that case, negative one times
negative four would be positive four.
We then add positive four to negative six
which gives us negative two
the correct answer.
So, we get the correct answer
if we make the rule that
the product of two negative numbers is positive.
Now we know what sign the result should be
when we multiply two numbers of any sign.
Multiplying two numbers with the same sign
always gives a positive result.
And multiplying two numbers with opposite
signs always gives a negative result.
Understanding this, can help us simplify
multiplication problems
involving multiple numbers of different signs.
Let's say that we have a bunch of positive
and negative numbers which are multiplied.
The commutative property of multiplication
says that we can arrange these numbers
in any way we like
so let's group pairs of negative numbers together.
Each pair of negative numbers creates
the same result as if the pair was positive
so we can change their signs to positive without
changing the result of the multiplication.
Since there was an even number of
negative numbers
after each pair of negatives was
changed to positives
there were no negative numbers left over.
Therefore, the result of the multiplication
is positive.
Now, let's see what happens if we have
an odd number of negative numbers.
Once again, we group the negative numbers
into pairs and change their signs
but one unpaired negative number is left.
So this time, the result of the multiplication
is negative.
Here is another interesting trick
which can come in handy.
Multiplying any positive number,
which we will call A, by negative one
switches its sign to negative.
Likewise, multiplying any negative number by
negative one will switch its sign to positive.
Now, let’s say that we have a sum of several
numbers of various signs.
If we enclose the sum in parentheses
and multiply by negative one
the distributive property says
that this is the same
as multiplying each number individually
by negative one
which switches the sign of each number.
So multiplying a sum of numbers in parentheses
by negative one
switches the sign of each number.
Instead of multiplying by negative one
we could just put a negative sign in front
of the parentheses
which means exactly the same thing.
So a negative sign in front of a parentheses
has the same effect as switching the sign
of each number summed in the parentheses.
Here’s one more trick using the
distributive property.
Let's say we start with A minus B.
If we enclose this in parentheses
with a negative sign in front
it switches the sign of each number.
Using the commutative property, we can then
swap the positions of the two numbers
and the result is that we now have
B minus A.
So placing a negative sign in front of
two numbers which are subtracted
swaps the two numbers.
We have seen that with the invention of
negative numbers
the rules of multiplication had to be expanded
to allow numbers of any sign to be multiplied.
We have shown that these rules were picked
in a way that kept things consistent.
Since positive one is the multiplicative identity
it made sense that multiplying a number
by a positive number
should not change that number’s sign.
Since we don’t want to break the
commutative property
we must also make the rule that
if either one of the operands is negative
the result must be negative.
And if we don’t want to break the
distributive property
we must make the rule that if both operands
are negative, the result must be positive.
So we now have a set of rules
which allow us to multiply integers of any sign.
In the next few lectures we will explore the
properties of division
and see how this operation forced the world
to create a new type of number.