Lets say you are an Ancient Philosopher who was building up mathematics who was building mathematics from the ground up
And you already have a reasonable of what a negative number could or should represent and you know how to add and subtract negative numbers
But now you are faced with a conundrum
What happens when you multiply negative numbers?
Either when you multiply a positive number times a negative number
Or when you multiply two negative numbers
So, for example
You aren't quite sure what should happen if you were to multiply (and im just picking two numbers where one is positive and one is negative)
What would happen if you were to multiply 5 times negative 3
You're not quite sure about this just yet
You're also not quite sure what would happen if you multiply two negative numbers.
So lets say negative two times negative 6
This is also unclear to you
What you do know, because you are a mathematician, is however you define this or whatever this should be
It should hopefully be consistant with all of the other properties of mathematics that you already know
And preferably all of the other properties of multiplication
That would make you feel comfortable that you are getting this right.
and later we can think about other ways to get the intuition
for what these might be allowed you to actually make sense
but to make this make consistent with the rest of mathematics
that you know, you go into a little bit of a thought experiment
you say, well, what should five times
three plus negative three equal
well you already have a philosophy of adding negative numbers
or adding positive numbers to negative numbers, you know negative three is
the opposite of three, but you add three to negative three
you're going to get zero, so this is going to be equal to
five times zero
based on how you already thought about adding a negative number
to a positive, and anything times zero is going to be
zero, so this expression right over here should be zero
but you see, I want to multiply positive and negative
numbers to be consistent with this distributive property so
I should should be able to distribute this five
and for math to be consistent, and math should be consistent, I should
get the exact same answer, so let's distribute this five so
we get five times three
is going to write out as five times three
let me write this multiplication sign, not this dot
five times three, so i distributed there
plus five times negative three
i'll do that in yellow, five times negative three
and this whole thing we just said should be equal to zero
it should be equal to zero, well five times three
those are two positive numbers, we should know what should should be, that is going to be fifteen
now we get this thing, fifteen plus
times whatever five times negative three is
needs to be equal to zero in order to be consistent with all the other mathematics
that we know, well what plus fifteen is going to
be equal to zero, well the opposite of fifteen in order for this
to be true, in order for this to be consistent with all the other mathematics we know
this right over here needs to be equal to
negative fifteen until you say five times negative three
in order to be consistent with all other mathematics we know, needs to
be equal to negative fifteen. That's also consistent
with the intuition of adding negative three repeatedly
five times, now look above above us
slightly higher so you can see ideas of multiplying
two negatives, but we can do the exact same product experiment.
We want whatever this answer to be consistent with the rest of mathematics
that we know so we can
do the same product experiment. What would negative two times
six plus negative six to be equal to.
Well, six plus negative six is going to be zero.
Negative two times zero, anything times zero, needs to be equal to
zero, but then once again, we can distribute
negative two times six so we get
negative two times six, then plus negative two times
negative six plus negative two
times negative six, then once again all of this is going to be
equal to zero, now based on the five experiment we just
did, we said "well this needs to be equal to negative twelve"
or we can view this as going to the six twice
left direction on the number line which gets us to negative twelve
or you could say repeatedly adding negative twos
times six would also get you to negative twelve and now
we also saw over here we want to multiply a positive and a negative
we got the negative so
this could be, you know, going to be equal to negative twelve
so we have negative twelve plus
whatever this business is
going to have to be equal to zero (repeated)
in order to be consistent with all the other mathematics that we know
and so what plus negative twelve is going to equal to be zero
Well, positive twelve plus negative twelve is going to equal to zero
so this needs to be equal to positive twelve in
order to be consistent with all the other mathematics we know
so there we get the idea that this is going positive
twelve. I'll leave you there and I'll see
if I can make a few other videos that can also give you a conceptual understanding
of why these are true