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So you, the ancient philosopher-mathematician
you have come to the conclusion that in order for the multiplication of positive to negative number to be
in agreement with everything you have built so far,
with all other properties of the multiplication, which has so far know,
that you have a negative number for positive
positive or negative number in order to get a negative number,
and that negative on negative
It gives a positive number.
So far, everything is logical, but still do not fully understand,
you want to have a little more extensive intuition than simply accept that it is in agreement
for distribution property or something else, so you try another
mental experiment by saying: "what is the simple multiplication?"
Say 2. 3
one way to imagine
just multiplication is to repeat
collection. You can see it as two threes,
Let write 3 +3
Note that there are only two trios, has two of these.
Or you can look at it as three couples
this is the same as the 2 + 2 + 2, there are three of them.
Whichever way you imagine it,
you get the same answer. This will be equal to
six well!
That you know before you try to deal with negative numbers.
Let us now try to make one of these negative and see
what will happen.
Making 2. (-3)
I want to do negative in a different color.
2. (-3)
One way to see it is with this same analogy,
this is twice -3, you will be
negative ... I'll try to distinguish them by color.
- 3 then again -3
or you can say -3 minus 3
or, it is interesting instead
Here we have two. (+3)
and added 2 three times.
But because we have two. (-3)
We can imagine it as subtracting 2 three times.
Instead ... Up here,
I could write 2 + 2 + 2 + 3 because it is
But because we believe in -3,
can be imagined as to subtract 2 three times,
then will be subtracting 2.
Subtract another 2 ... subtract 2.
Finally once again subtract 2.
Note that you do it again
three times. This means that it is -3; what you do is essentially the
To remove 2 three times. In any way to get it IDEAS
You will receive -6.
-6 Is the answer.
No longer worry about this part here -
positive or negative in a positive negative.
will give you negative. Now let's move to really counterintuitive part:
have negatively negative and cons suddenly somehow cancel each other out,
to give you a positive. Why is that? May start
of this example here. Let's say you have
- 2 to say that we have -2.
Let me do it in another color.
Let's say you have a -2 already used this color.
- 2 ...
- 2. (-3)
Napravim-- will now actually going to do this first.
Still multiply something -3,
So we subtract the number 3 times
Now it is not a positive number that will subtract
is -2, let it clear.
It says it will pull out something three times.
So subtract something 3 times
subtract anything, subtract anything, subtract anything - three times.
This tells us that part here.
And we will do it exactly three times.
Here we had a + 2, subtract 3 times
Now we will have a -2, -2 will have.
And we know from the removal of negative numbers, this intuition
we've already built that the removal of the negative is the same as ...
... same as to add positive
then it will be the same as a 2 + 2 + 2
which will once again give you +6.
It can use the same logic here.
Now instead of adding -3 twice, I can save it as
-3 ... In this example.
-3
-3. We collected them.
We collected them, we added a plus sign to become clearer.
Here we added it twice, added -3
twice, and here, because we now have a -2, -3 we will subtract twice.
Will pull out something and then
We will subtract something again, and this thing will be
our -3 will be our -3, then
minus, minus, put troika here.
And again, subtract -3 like to get out
someone's debt, which means to give them money.
This also collect 3 and 3, which is 6.
Now you, the ancient philosopher feel pretty confident. These things are not only
in agreement with all the mathematics that you know:
distribution property associativity,
multiplying the thing ... All these things already know them, and now
This makes sense as an idea for you, it is consistent with
your ideas, your initial or present one of the possible ideas for multiplication -
that it is repeating the multiplication of collection. Subtitles End: mo.dbxdb.com