Mathematical closure

WEBVTT
Kind: captions
Language: en

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- So I have a very simple
equation written here

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a plus b is equal to c and
what I want to think about

00:00:06.366 --> 00:00:09.443
if we know that a is an
integer, so let's say that this

00:00:09.443 --> 00:00:14.443
is the set of all integers
right over here. Integers.

00:00:15.224 --> 00:00:18.997
We know that a is an integer
so let's say that's a there.

00:00:18.997 --> 00:00:21.841
We also know that b is an
integer, could be the same integer

00:00:21.841 --> 00:00:24.002
but I'll draw it so it
looks like it's another one,

00:00:24.002 --> 00:00:26.322
it doesn't have to be but they
could be the same integer.

00:00:26.322 --> 00:00:27.599
B is an integer.

00:00:27.599 --> 00:00:29.341
If we're adding these two integers,

00:00:29.341 --> 00:00:31.989
are we definitely going
to get another integer?

00:00:31.989 --> 00:00:36.120
Do we know that c is
going to be an integer?

00:00:36.120 --> 00:00:38.722
And I encourage you to pause
the video and think about it.

00:00:38.722 --> 00:00:39.940
And when you've thought
about it, you might have

00:00:39.940 --> 00:00:40.858
come up with some cases.

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Well, what if a and b are both one,

00:00:41.925 --> 00:00:43.981
then for sure, c is going to be two.

00:00:43.981 --> 00:00:46.046
And c is going to be an integer.

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Well is that always going to be the case?

00:00:48.206 --> 00:00:50.040
Well, it is. There are ways to prove it.

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I'm not going to go into
it now but when you have

00:00:52.257 --> 00:00:53.720
a situation like this, when you're able

00:00:53.720 --> 00:00:55.927
to take two members of
a set, in this case,

00:00:55.927 --> 00:00:58.665
two members of a set and
perform an operation on it,

00:00:58.665 --> 00:01:02.380
and always get another member
of the set and it could be

00:01:02.380 --> 00:01:06.038
a or b again or it could
be another distinct member

00:01:06.038 --> 00:01:08.684
of the set, I'll draw it
as another distinct member

00:01:08.684 --> 00:01:11.377
of the set, whenever you have this,

00:01:11.377 --> 00:01:14.362
so you have some operation,
in this case, it is addition.

00:01:14.362 --> 00:01:16.944
So when you're performing
the addition operation

00:01:16.944 --> 00:01:20.542
on two members of the set,
you get, you stay in the set.

00:01:20.542 --> 00:01:23.550
C is also going to be a member of that set

00:01:23.550 --> 00:01:25.964
so one way to think about it, we're going,

00:01:25.964 --> 00:01:27.623
we're going like this and we are

00:01:27.623 --> 00:01:29.969
performing this addition operation.

00:01:29.969 --> 00:01:32.279
Whenever you see a circumstance like this,

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we call, we say that this set,

00:01:35.518 --> 00:01:39.083
the set of integers is
closed under addition.

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Let me write that down.

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So this means that the
set of integers, integers

00:01:47.405 --> 00:01:52.405
is closed under, under addition.

00:01:54.200 --> 00:01:56.265
So once again, you're
going to hear this idea

00:01:56.265 --> 00:01:58.482
of closure, it's the general term

00:01:58.482 --> 00:02:00.340
that we're talking about is closure

00:02:00.340 --> 00:02:02.476
when you take some, especially in higher

00:02:02.476 --> 00:02:03.764
mathematics course but
you might even hear it

00:02:03.764 --> 00:02:05.598
some of your high school courses

00:02:05.598 --> 00:02:08.478
and it might seem like
this really advanced thing

00:02:08.478 --> 00:02:10.603
but all it's saying is you have a set,

00:02:10.603 --> 00:02:12.356
you have some operation
that you can perform

00:02:12.356 --> 00:02:14.154
on members of the set and if every time

00:02:14.154 --> 00:02:16.023
you perform that operation
on members of the set

00:02:16.023 --> 00:02:18.903
you get, you stay in
the set, the result is

00:02:18.903 --> 00:02:22.317
a member of the set, then you
say that the set is closed

00:02:22.317 --> 00:02:26.078
under that operation, the
set is closed, in this case,

00:02:26.078 --> 00:02:27.934
the operation is addition, you say

00:02:27.934 --> 00:02:31.197
the set is closed under addition
and we can ask ourselves

00:02:31.197 --> 00:02:32.636
that same question about other things.

00:02:32.636 --> 00:02:37.636
Is the set of, is the set
of integers closed under,

00:02:41.517 --> 00:02:44.788
closed under, actually,
let's take some examples.

00:02:44.788 --> 00:02:49.788
Is it closed under
multiplication, multi-plication.

00:02:50.272 --> 00:02:53.418
And I encourage you to pause
the video and think about it.

00:02:53.418 --> 00:02:56.738
And then think about, is
it closed under division?

00:02:56.738 --> 00:03:01.738
Is it closed under division?
Is it closed under subtraction?

00:03:02.114 --> 00:03:05.300
Is it closed under subtraction?

00:03:05.300 --> 00:03:05.908
I encourage you to think about

00:03:05.908 --> 00:03:07.568
all these different operations.

00:03:07.568 --> 00:03:09.119
Let's take it one by one. Multiplication.

00:03:09.119 --> 00:03:11.836
If you take two integers
and you were to multiply it,

00:03:11.836 --> 00:03:14.552
you actually always
will get another integer

00:03:14.552 --> 00:03:19.115
so the set of integers is
closed under multiplication.

00:03:19.115 --> 00:03:20.554
Now, what about division?

00:03:20.554 --> 00:03:22.319
Well, it's very easy to
prove a case, in which,

00:03:22.319 --> 00:03:26.370
I'm dividing two integers and
not getting another integer.

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For example, I could say, let's
just do a very simple one.

00:03:30.677 --> 00:03:34.382
One divided by two. One
and two are both integers

00:03:34.382 --> 00:03:35.855
but what's that going to give me?

00:03:35.855 --> 00:03:39.918
That's going to give me 1/2
which is not, another integer.

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This is a rational number but
not an integer so if we were

00:03:42.136 --> 00:03:44.318
to extend our sets right over here.

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All integers are also rational but

00:03:46.036 --> 00:03:48.885
all rationals are not integers necessarily

00:03:48.885 --> 00:03:52.344
so this is rational,
rational numbers here.

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This case of one divided
by two so that's one

00:03:56.234 --> 00:03:59.066
and this is two when
we performed division,

00:03:59.066 --> 00:04:02.363
we went out of the set, so
when we performed division,

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we out of that set to
something that's not an integer

00:04:05.288 --> 00:04:08.051
but it is a rational number
so the set of integers

00:04:08.051 --> 00:04:12.428
is not closed under
division. Subtraction? Sure!

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And once again, I'm
not going to rigorously

00:04:13.925 --> 00:04:16.108
prove it here but take any two integers,

00:04:16.108 --> 00:04:19.209
you indeed are going to get another,

00:04:19.209 --> 00:04:22.137
you are going to get another integer.

00:04:22.137 --> 00:04:24.320
So the whole idea here is to just give you

00:04:24.320 --> 00:04:27.930
a brief introduction to
the notion of closure.

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The idea that if you have
a set and if you perform

00:04:31.100 --> 00:04:33.119
an operation on members of that set

00:04:33.119 --> 00:04:36.033
and you always get a
member of your original set

00:04:36.033 --> 00:04:40.375
then that set is closed
under that operation.

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What matters is that the set
that you're talking about

00:04:42.171 --> 00:04:43.949
in this case, we talked
a lot about integers

00:04:43.949 --> 00:04:45.492
and we talked about
addition, multiplication,

00:04:45.492 --> 00:04:46.966
division, subtraction but you could think

00:04:46.966 --> 00:04:50.042
about the set of polynomials
and whether they are

00:04:50.042 --> 00:04:52.899
closed under addition,
which they actually are

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or the set of rational numbers.

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Are they closed under exponentiation?

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There's all sorts of
things we can think about

00:05:00.979 --> 00:05:02.268
but hopefully, this gives you a good sense

00:05:02.268 --> 00:05:05.019
of what we mean when we say closure

00:05:05.019 --> 00:05:08.003
or for a set being closed
under some operation.


Arithmetic properties with negative numbers

WEBVTT
Kind: captions
Language: en

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- [Voiceover] What I
hope to do in this video

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is a bunch of examples to show us that

00:00:03.738 --> 00:00:05.503
a lot of the properties
we've been dealing with

00:00:05.503 --> 00:00:07.243
in arithmetic, the distributive property,

00:00:07.243 --> 00:00:09.427
associative property,
commutative property,

00:00:09.427 --> 00:00:11.772
that these apply just as
well to negative numbers.

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But with that said, it is
good to actually see it used

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using negative numbers,

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or see these properties applied

00:00:17.461 --> 00:00:20.851
just to make sure that we
understand what's going on.

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So these exercises, these
are all from Kahn Academy.

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Oops.

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So this first one says,

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which of the following
expressions are equivalent

00:00:27.561 --> 00:00:30.696
to negative two times the
quantity five minus three?

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Now you could, of course, just figure out

00:00:32.041 --> 00:00:34.968
that five minus three is two and then

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multiply that times negative two

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and you would get negative four.

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And you could see which of
these is equal to negative four.

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And that would be fair, but
the whole point of this video

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is to understand that
look, maybe I could apply

00:00:45.323 --> 00:00:47.344
the distributive property here.

00:00:47.344 --> 00:00:48.853
So let's do that.

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So what I could do is I could
distribute this negative two.

00:00:51.407 --> 00:00:53.658
I could multiply it times five

00:00:53.658 --> 00:00:56.005
and then I could multiply it by,

00:00:56.005 --> 00:00:58.442
I could either do it as it's going to be

00:00:58.442 --> 00:01:01.368
negative two times five plus

00:01:01.368 --> 00:01:03.248
negative two times negative three.

00:01:03.248 --> 00:01:05.083
Or you could view it as
negative two times five

00:01:05.083 --> 00:01:07.845
minus negative two times positive three.

00:01:07.845 --> 00:01:09.727
Now let me write those two things down.

00:01:09.727 --> 00:01:12.577
So you could do this as negative two

00:01:13.274 --> 00:01:14.544
times five,

00:01:15.708 --> 00:01:18.239
negative two times five

00:01:18.239 --> 00:01:19.160
plus

00:01:19.976 --> 00:01:24.976
negative plus negative
two times negative three.

00:01:27.028 --> 00:01:28.329
You could view it that way.

00:01:28.329 --> 00:01:31.475
Or you could view it as negative two,

00:01:32.196 --> 00:01:34.326
negative two times five

00:01:35.000 --> 00:01:36.418
minus

00:01:36.418 --> 00:01:37.718
minus

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and if I'm putting a minus here

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then I'm gonna view this
as a positive three,

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that we're subtracting a positive three.

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So minus

00:01:44.520 --> 00:01:46.396
negative two times positive three.

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Notice I either wrote the negative here

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and wrote a positive here,

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or I wrote the negative here
and made this a positive three.

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But these are going to be equivalent.

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Either way I've distributed
this negative two.

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Notice I have a negative two,

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negative two.

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And what are these going to be equal to?

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Well, negative two times
five is negative 10

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and then,

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and then negative two times negative three

00:02:09.106 --> 00:02:10.661
is positive six,

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or over here negative two
times three is negative six,

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but then we subtract it so
it's just gonna get positive,

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you're gonna get positive six either way.

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This right over here is positive six

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and this over here,
subtracting a negative six

00:02:23.368 --> 00:02:25.644
would give you positive six.

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So you get negative ten plus six.

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And that's this choice right over here.

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Which of course does
evaluate to negative four,

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which this expression does evaluate to.

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This one up here evaluates to negative 16.

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And of course I won't select
this because I found an answer.

00:02:40.341 --> 00:02:41.774
Let's do several more.

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Which of the following
expressions are equivalent

00:02:44.956 --> 00:02:49.344
to negative S times T times S?

00:02:49.344 --> 00:02:51.619
Select all that apply.

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And here we can't just substitute,

00:02:53.500 --> 00:02:54.964
we can't just evaluate it and see,

00:02:54.964 --> 00:02:57.007
oh what do these evaluate to.

00:02:57.007 --> 00:02:58.887
We need to do a little bit of manipulation

00:02:58.887 --> 00:03:00.768
of these variables.

00:03:00.768 --> 00:03:02.718
Well there's a couple of
ways to think about it.

00:03:02.718 --> 00:03:04.879
One, we could change the order in which

00:03:04.879 --> 00:03:06.643
we multiply these things.

00:03:06.643 --> 00:03:08.710
So we could view this as

00:03:08.710 --> 00:03:11.984
negative S, that's that,

00:03:11.984 --> 00:03:14.142
let me write it a little bit neater.

00:03:14.142 --> 00:03:15.783
We could view it as

00:03:16.877 --> 00:03:21.166
negative S times S,

00:03:22.143 --> 00:03:25.502
times S times T.

00:03:26.201 --> 00:03:29.010
Times, oops, let me do
that in a different color,

00:03:29.010 --> 00:03:31.719
times S times T.

00:03:32.464 --> 00:03:33.857
Times T.

00:03:33.857 --> 00:03:36.319
And do any of the choices look like that?

00:03:36.319 --> 00:03:37.728
Well almost.

00:03:37.728 --> 00:03:39.377
Instead of saying negative S times S,

00:03:39.377 --> 00:03:41.374
this says S times negative S.

00:03:41.374 --> 00:03:43.348
And because multiplication, once again,

00:03:43.348 --> 00:03:45.113
I'm not a big fan of using the word

00:03:45.113 --> 00:03:46.135
because it sounds complicated,

00:03:46.135 --> 00:03:47.806
but it's commutative.

00:03:47.806 --> 00:03:50.638
A times B is the same thing as B times A.

00:03:50.638 --> 00:03:52.798
So I can rewrite this as,

00:03:52.798 --> 00:03:53.791
I can rewrite this,

00:03:53.791 --> 00:03:55.598
I can swap these two and write this as

00:03:55.598 --> 00:03:57.472
S times,

00:03:58.194 --> 00:04:02.367
S times negative S times T,

00:04:03.087 --> 00:04:04.781
times T.

00:04:05.757 --> 00:04:07.268
Times T.

00:04:07.268 --> 00:04:09.937
All I did is I swapped these two.

00:04:09.937 --> 00:04:12.724
This negative S and this S.

00:04:12.724 --> 00:04:14.164
I just swapped them and I got exactly

00:04:14.164 --> 00:04:15.690
what I have right over here.

00:04:16.620 --> 00:04:19.244
Now let's just make sure
that this one does not apply

00:04:19.244 --> 00:04:21.612
and maybe the easiest way
is to try to simplify this.

00:04:21.612 --> 00:04:22.889
And the best way I could think about that

00:04:22.889 --> 00:04:25.071
is by distributing this S.

00:04:25.071 --> 00:04:26.969
So if I distribute this S,

00:04:27.645 --> 00:04:28.899
what I'm going to get,

00:04:28.899 --> 00:04:30.385
this is going to be equal to

00:04:30.385 --> 00:04:34.216
S times T, which is ST.

00:04:34.216 --> 00:04:35.948
Or I could even, I can write it like this.

00:04:35.948 --> 00:04:38.503
I could write it S times T, like that.

00:04:38.503 --> 00:04:41.428
And then I have minus S times S,

00:04:41.428 --> 00:04:44.609
so minus S times S.

00:04:44.609 --> 00:04:46.832
I could write it that was
or I could write minus

00:04:47.511 --> 00:04:49.202
S squared if I want to.

00:04:49.202 --> 00:04:50.897
That's the same thing as S times S.

00:04:50.897 --> 00:04:52.383
But this is very different.

00:04:52.383 --> 00:04:53.706
This is very different.

00:04:53.706 --> 00:04:56.168
Here I'm just taking the
product of three variables here,

00:04:56.168 --> 00:04:57.747
I have two different terms,

00:04:57.747 --> 00:04:59.836
taking the product of two variables here

00:04:59.836 --> 00:05:01.182
and then the product I guess you could say

00:05:01.182 --> 00:05:03.203
I'm taking S squared or
I'm taking S times S.

00:05:03.203 --> 00:05:04.341
So this is not,

00:05:04.341 --> 00:05:06.568
this is not the same thing.

00:05:07.753 --> 00:05:09.495
Which of the following
expressions are equal

00:05:09.495 --> 00:05:12.025
to negative X times,
and then in parentheses

00:05:12.025 --> 00:05:13.535
negative Y times X?

00:05:13.535 --> 00:05:15.602
And I forgot to mention
it, but like always,

00:05:15.602 --> 00:05:17.621
pause the video, try to
work them out by yourself

00:05:17.621 --> 00:05:19.155
before I do them.

00:05:19.155 --> 00:05:21.282
Alright, select all that apply.

00:05:22.120 --> 00:05:24.234
So let's just try to
manipulate this a little bit.

00:05:24.234 --> 00:05:27.113
So once again, multiplication,
it's associative.

00:05:27.113 --> 00:05:30.374
I could, so it's negative X times,

00:05:30.979 --> 00:05:32.693
times negative Y

00:05:32.693 --> 00:05:34.433
times X.

00:05:34.433 --> 00:05:37.011
So, the way it's written here,

00:05:37.011 --> 00:05:39.073
I could do these first, that's essentially

00:05:39.073 --> 00:05:40.558
what's written over here.

00:05:40.558 --> 00:05:41.417
Or it's associative.

00:05:41.417 --> 00:05:43.856
Instead, I could do these first.

00:05:43.856 --> 00:05:46.178
And the reason why I find this interesting

00:05:46.178 --> 00:05:47.617
is a negative times a negative

00:05:47.617 --> 00:05:49.127
is going to be a positive.

00:05:49.127 --> 00:05:51.069
So this is going to be the same thing,

00:05:51.721 --> 00:05:54.042
this thing over here is
going to be the same thing

00:05:54.042 --> 00:05:56.017
as positive X times positive Y.

00:05:56.017 --> 00:05:58.060
Negative times a negative is a positive.

00:05:58.060 --> 00:06:00.986
So you're gonna get positive X times Y

00:06:00.986 --> 00:06:03.092
and then you're multiplying by an X again.

00:06:03.999 --> 00:06:05.880
Multiplying by an X again.

00:06:05.880 --> 00:06:07.993
Now the other thing we
know about multiplication

00:06:07.993 --> 00:06:09.037
is it's commutative.

00:06:09.037 --> 00:06:11.546
We can change the order
in which we multiply.

00:06:11.546 --> 00:06:13.751
Because I don't see this quite,

00:06:13.751 --> 00:06:16.097
I don't see this going on over here yet.

00:06:16.097 --> 00:06:17.350
So let's see, if I change the order,

00:06:17.350 --> 00:06:19.440
if I put the Xs, if I
multiply the Xs first,

00:06:19.440 --> 00:06:21.159
I could write this as,

00:06:21.159 --> 00:06:26.159
I could write this as X times X,

00:06:27.032 --> 00:06:29.417
X times X times Y,

00:06:30.422 --> 00:06:31.950
times Y.

00:06:31.950 --> 00:06:33.413
All I did is I swapped these two.

00:06:33.413 --> 00:06:36.687
Once again, I can swap the
order when I'm multiplying.

00:06:36.687 --> 00:06:39.357
And I don't quite see
this yet, but X times X,

00:06:39.357 --> 00:06:41.399
that's the same thing as X squared.

00:06:41.399 --> 00:06:43.257
This thing right over
here, that's X squared.

00:06:43.257 --> 00:06:45.081
So this is going to be X squared,

00:06:45.919 --> 00:06:49.658
X squared times Y, which is exactly

00:06:49.658 --> 00:06:51.586
what we have over there.

00:06:51.586 --> 00:06:53.744
Now what does this one evaluate to?

00:06:53.744 --> 00:06:55.603
Now this one actually
evaluates to a number.

00:06:55.603 --> 00:06:57.390
Because regardless of what X you pick,

00:06:57.390 --> 00:06:59.782
X minus X, that's going to be zero.

00:06:59.782 --> 00:07:01.756
Zero times anything is going to be zero.

00:07:01.756 --> 00:07:05.169
So this thing is going
to be equal to zero.

00:07:05.169 --> 00:07:07.769
So this is different than
what's going on over here.

00:07:07.769 --> 00:07:09.735
So I definitely would not pick that.

00:07:10.712 --> 00:07:12.430
Let's do one more.

00:07:12.430 --> 00:07:13.870
Which of the following expressions

00:07:13.870 --> 00:07:17.748
are equivalent to A times
negative 10 plus 11?

00:07:17.748 --> 00:07:19.443
Well once again, you know the instinct

00:07:19.443 --> 00:07:20.256
when you see something like this

00:07:20.256 --> 00:07:22.135
well I'd love to distribute that A,

00:07:22.135 --> 00:07:23.506
it's just sitting there.

00:07:23.506 --> 00:07:24.389
So let's do that.

00:07:24.389 --> 00:07:28.962
A times negative 10 would
be negative 10 times A,

00:07:28.962 --> 00:07:30.610
or negative 10A.

00:07:30.610 --> 00:07:33.228
And then A times 11,

00:07:34.047 --> 00:07:35.435
so it's gonna be plus,

00:07:35.435 --> 00:07:37.942
A times 11 is the same
thing as 11 times A,

00:07:37.942 --> 00:07:40.753
which we could write as 11A.

00:07:40.753 --> 00:07:42.262
Now which of these choices are that?

00:07:42.262 --> 00:07:44.948
Negative 10A plus 11A.

00:07:45.623 --> 00:07:48.247
So this is negative 10A plus 11A,

00:07:48.247 --> 00:07:49.756
so this one looks right.

00:07:49.756 --> 00:07:51.074
Now what about this one?

00:07:51.912 --> 00:07:53.257
Well here they just swapped the order.

00:07:53.257 --> 00:07:56.067
If you put the 11A first,
you could write it this way.

00:07:56.067 --> 00:07:58.197
If you write the 11A first,

00:07:59.476 --> 00:08:00.699
we could write

00:08:01.398 --> 00:08:02.558
11A

00:08:03.326 --> 00:08:04.464
and then we have,

00:08:04.464 --> 00:08:05.531
instead of saying it negative 10A,

00:08:05.531 --> 00:08:07.644
we just say minus 10A.

00:08:07.644 --> 00:08:08.874
Once again I just took,

00:08:08.874 --> 00:08:10.803
all I did is I took

00:08:10.803 --> 00:08:13.310
this thing and I put it

00:08:13.310 --> 00:08:14.354
out front.

00:08:14.354 --> 00:08:17.118
So these two things are
actually equivalent.

00:08:17.118 --> 00:08:19.763
So I would select that one as well.


Thinking about the sign of expressions

WEBVTT
Kind: captions
Language: en

00:00:00.259 --> 00:00:02.443
- [Voiceover] Let's now
do some examples that

00:00:02.443 --> 00:00:04.387
test our understanding of what happens

00:00:04.387 --> 00:00:06.163
when we multiply or divide a bunch of

00:00:06.163 --> 00:00:08.515
positive or negative numbers together.

00:00:08.515 --> 00:00:11.045
And like always, pause the
video and see if you can

00:00:11.045 --> 00:00:13.973
work through it on your
own before I do it.

00:00:13.973 --> 00:00:18.866
So first question: What is the
sign of s to the 67th power

00:00:18.866 --> 00:00:22.913
divided by t to the
ninth power, divided by

00:00:22.913 --> 00:00:27.511
the expression 3s times s over 4t,

00:00:27.511 --> 00:00:32.278
when s is less than zero,
and t is greater than zero?

00:00:33.094 --> 00:00:34.866
So the really interesting
thing here is that

00:00:34.866 --> 00:00:38.181
s is less than zero, s is negative.

00:00:38.181 --> 00:00:41.650
So if s is a negative number,

00:00:41.650 --> 00:00:45.250
we first have s to the 67th power.

00:00:45.250 --> 00:00:47.529
And we just have to remind ourselves,

00:00:47.529 --> 00:00:50.202
a negative number being multiplied,

00:00:50.202 --> 00:00:52.169
if you have an odd number
of negative numbers

00:00:52.169 --> 00:00:53.698
being multiplied together,

00:00:53.698 --> 00:00:55.553
you're going to have a negative number.

00:00:55.553 --> 00:00:59.233
And s to the 67th power, that's 67 ses,

00:00:59.233 --> 00:01:02.353
so it's literally s times s times s,

00:01:02.353 --> 00:01:05.073
and you literally are going
to have 67 of these ses,

00:01:05.073 --> 00:01:06.603
and you're multiplying them all together.

00:01:06.603 --> 00:01:09.059
So you have an odd number of ses,

00:01:09.059 --> 00:01:10.555
an odd number of negative numbers

00:01:10.555 --> 00:01:12.259
that you're multiplying together.

00:01:12.259 --> 00:01:13.403
So if you have an odd number of

00:01:13.403 --> 00:01:15.426
negative numbers multiplied together?

00:01:15.426 --> 00:01:18.235
Well that means that the
product is going to be negative.

00:01:18.235 --> 00:01:21.514
So this thing right over
here is going to be negative.

00:01:21.514 --> 00:01:23.466
Now what's t to the ninth?

00:01:23.466 --> 00:01:25.905
Well, t is a positive
number, they tell us that.

00:01:25.905 --> 00:01:28.893
They tell us that t is greater than zero.

00:01:28.893 --> 00:01:31.410
Well, a positive number,
really to any power,

00:01:31.410 --> 00:01:33.586
however many you have
that multiply together

00:01:33.586 --> 00:01:35.658
with an even or odd, if
it's just all positive,

00:01:35.658 --> 00:01:37.038
well this is going to be positive.

00:01:37.038 --> 00:01:39.594
So this thing right over
here is going to be positive.

00:01:39.594 --> 00:01:42.770
Now what's a negative
divided by a positive?

00:01:42.770 --> 00:01:45.162
Well a negative divided by a positive

00:01:45.162 --> 00:01:46.986
is going to be negative.

00:01:46.986 --> 00:01:49.354
We've already seen that show before.

00:01:49.354 --> 00:01:52.913
So this whole thing,
I'll circle it like that,

00:01:52.913 --> 00:01:55.593
is going to be a negative.

00:01:55.593 --> 00:01:57.212
And so that this thing
doesn't get too messy,

00:01:57.212 --> 00:01:58.849
actually let me just rewrite this.

00:01:58.849 --> 00:02:01.673
So all of this stuff that
we've written over here,

00:02:01.673 --> 00:02:03.384
I'll just write this
as this is going to be

00:02:03.384 --> 00:02:05.559
some negative number,

00:02:05.559 --> 00:02:07.956
and then we're going to divide it by

00:02:07.956 --> 00:02:12.533
3s times s over 4t.

00:02:13.856 --> 00:02:15.829
Now this expression right over here.

00:02:15.829 --> 00:02:18.573
What is s divided by 4t going to be?

00:02:18.573 --> 00:02:20.467
Is that going to be positive or negative?

00:02:20.467 --> 00:02:22.932
Well we know that s is negative.

00:02:22.932 --> 00:02:26.195
T is positive, so four times
t is going to be positive.

00:02:26.195 --> 00:02:29.076
So this thing right over
here is going to be positive.

00:02:29.076 --> 00:02:31.549
Negative divided by positive,

00:02:31.549 --> 00:02:33.429
well that's going to be a negative.

00:02:33.429 --> 00:02:36.293
So actually let me just write it this way.

00:02:36.293 --> 00:02:38.764
So all of this stuff, I
could just replace with

00:02:38.764 --> 00:02:41.722
s divided by 4t is going to be a negative.

00:02:41.722 --> 00:02:44.850
So times a negative, times
some other negative number.

00:02:44.850 --> 00:02:46.707
Now what's 3s?

00:02:46.707 --> 00:02:49.066
Well s is negative, so 3s is negative.

00:02:49.066 --> 00:02:50.698
So that is negative, so you have

00:02:50.698 --> 00:02:52.154
a negative times a negative.

00:02:52.154 --> 00:02:54.364
Which is a positive.

00:02:54.364 --> 00:02:56.431
So all of this business right over here

00:02:56.431 --> 00:02:58.318
is going to be positive.

00:02:58.318 --> 00:02:59.754
Another way you could think about it is

00:02:59.754 --> 00:03:02.708
you're going to have a
negative times a negative,

00:03:02.708 --> 00:03:05.849
which is positive, and then
divide that by a positive,

00:03:05.849 --> 00:03:07.662
it's just going to be positive.

00:03:09.932 --> 00:03:12.205
So all of that,

00:03:12.205 --> 00:03:14.903
so this whole thing, all of
this business right over here,

00:03:14.903 --> 00:03:16.904
is going to be a positive.

00:03:16.904 --> 00:03:19.301
This is all a negative,
this is a positive.

00:03:19.301 --> 00:03:21.869
You're gonna have a negative
divided by a positive.

00:03:21.869 --> 00:03:25.101
Well that's just going to be negative.

00:03:25.101 --> 00:03:28.373
So the sign of this crazy
expression right over here?

00:03:28.373 --> 00:03:31.154
It's going to be negative
if s is less than zero

00:03:31.154 --> 00:03:33.500
and t is greater than zero.

00:03:33.500 --> 00:03:35.357
Let's keep going, this is fun.

00:03:35.357 --> 00:03:36.764
All right.

00:03:36.764 --> 00:03:41.173
So what is the sign of p times q over p,

00:03:41.173 --> 00:03:44.759
times four and 2/7, when
p is greater than zero,

00:03:44.759 --> 00:03:47.452
and q is equal to zero.

00:03:47.452 --> 00:03:50.339
And the way I just
stressed q is equal to zero

00:03:50.339 --> 00:03:51.836
should be a big hint.

00:03:51.836 --> 00:03:53.012
And let me give you another hint.

00:03:53.012 --> 00:03:55.116
Pause the video, if you really think about

00:03:55.116 --> 00:03:57.580
what I just said, that q is equal to zero,

00:03:57.580 --> 00:04:00.963
you should be able to do
this in less than a second.

00:04:02.459 --> 00:04:03.810
So why did I say that you could do this

00:04:03.810 --> 00:04:05.434
in less than a second?

00:04:05.434 --> 00:04:07.034
Well if you just have a bunch of numbers,

00:04:07.034 --> 00:04:09.425
and then you multiply them times
zero at some point in that,

00:04:09.425 --> 00:04:12.546
the whole thing is just
going to be equal to zero.

00:04:12.546 --> 00:04:17.041
I could literally have a
times b times c times d,

00:04:17.041 --> 00:04:19.435
I could multiply a bunch of numbers.

00:04:19.435 --> 00:04:21.889
And if I just knew that one
of these numbers is zero,

00:04:21.889 --> 00:04:24.195
then the whole product is
going to be equal to zero.

00:04:24.195 --> 00:04:26.634
You know if c is equal
to zero, I could multiply

00:04:26.634 --> 00:04:29.338
a times b times d times
e and get some number,

00:04:29.338 --> 00:04:30.929
and then multiply that times c,

00:04:30.929 --> 00:04:32.657
it's gonna be anything times zero is zero.

00:04:32.657 --> 00:04:36.889
Now notice, q is equal to
zero, and q is right over here.

00:04:36.889 --> 00:04:40.937
So I could take p divided
by p, that's gonna be one.

00:04:40.937 --> 00:04:43.598
One times q, that's just gonna be zero,

00:04:43.598 --> 00:04:44.601
as one times zero is zero,

00:04:44.601 --> 00:04:46.361
then you're gonna have
zero times four and 2/7,

00:04:46.361 --> 00:04:48.665
it's just all gonna be zero.

00:04:48.665 --> 00:04:51.196
And the key is, there's
a zero right over here.

00:04:51.196 --> 00:04:52.504
At some point you're multiplying

00:04:52.504 --> 00:04:53.985
this entire product times zero,

00:04:53.985 --> 00:04:55.712
so the whole product is going,

00:04:55.712 --> 00:04:57.257
or you're multiplying
these numbers times zero,

00:04:57.257 --> 00:04:59.656
so the whole product is going to be zero.

00:04:59.656 --> 00:05:02.696
So this side isn't positive
or negative, it's zero.

00:05:03.915 --> 00:05:05.730
All right, there's no sign there.

00:05:05.730 --> 00:05:07.261
It's neither positive nor negative.

00:05:07.261 --> 00:05:08.662
All right, let's do another one.

00:05:08.662 --> 00:05:10.339
What is the sign of negative 3/4

00:05:10.339 --> 00:05:13.114
times negative a to the fourth over three,

00:05:13.114 --> 00:05:14.819
when a is less than zero?

00:05:14.819 --> 00:05:17.315
So they're telling us that a is negative.

00:05:17.315 --> 00:05:19.706
So a is negative.

00:05:19.706 --> 00:05:22.962
So let me use a more dramatic color.

00:05:22.962 --> 00:05:25.684
So right over here, I have a

00:05:25.684 --> 00:05:27.899
negative number to the fourth power.

00:05:27.899 --> 00:05:30.913
So it's a times a times a times a.

00:05:30.913 --> 00:05:33.789
And we've already said if
you have an even number

00:05:33.789 --> 00:05:36.730
of negatives being
multiplied by each other,

00:05:36.730 --> 00:05:39.121
then that's going to be a positive.

00:05:39.121 --> 00:05:40.313
You could even see it over here,

00:05:40.313 --> 00:05:41.900
I mean they're all
negative, but a negative

00:05:41.900 --> 00:05:44.249
times a negative is
going to be a positive.

00:05:44.249 --> 00:05:46.233
A negative times the
negative, I take the third

00:05:46.233 --> 00:05:48.488
and the fourth one here,
that's gonna be a positive.

00:05:48.488 --> 00:05:49.921
And then a positive times a positive,

00:05:49.921 --> 00:05:53.122
the whole thing is going to be positive,

00:05:53.122 --> 00:05:54.991
even though each of these are negative.

00:05:54.991 --> 00:05:57.041
Negative times negative, or
another way you could say it,

00:05:57.041 --> 00:05:59.472
a times a is going to be positive,

00:05:59.472 --> 00:06:01.097
then times a is going to be negative,

00:06:01.097 --> 00:06:02.640
but then you multiply times a again,

00:06:02.640 --> 00:06:04.498
it's going to be positive again.

00:06:04.498 --> 00:06:07.779
So a to the fourth power
is going to be positive.

00:06:07.779 --> 00:06:10.460
But then you have a
negative in front of it.

00:06:10.460 --> 00:06:12.084
So you're taking the negative of a

00:06:12.084 --> 00:06:14.371
positive divided by a positive,

00:06:14.371 --> 00:06:16.841
Well, so this, all of this stuff

00:06:16.841 --> 00:06:18.603
right over here is all gonna be positive,

00:06:18.603 --> 00:06:20.363
but then you have this negative in front.

00:06:20.363 --> 00:06:23.678
So this entire thing
inside the parentheses

00:06:23.678 --> 00:06:26.394
is going to be a negative.

00:06:26.394 --> 00:06:29.627
And then you have a
negative, negative 3/4,

00:06:29.627 --> 00:06:32.938
times a negative, well a
negative times a negative,

00:06:32.938 --> 00:06:36.005
the whole thing is going to be positive.

00:06:36.005 --> 00:06:37.267
So what's the sign of this?

00:06:37.267 --> 00:06:39.770
It's going to be positive.

00:06:39.770 --> 00:06:42.210
Let's do another one,
I'm having too much fun.

00:06:42.210 --> 00:06:45.394
All right, what is the sign
of x to the 59th power,

00:06:45.394 --> 00:06:48.964
divided by 2.3x times 4/5,

00:06:48.964 --> 00:06:51.387
when x is less than zero?

00:06:51.387 --> 00:06:53.786
So x is once again a negative number.

00:06:53.786 --> 00:06:57.132
So x to the 59th power,
that's an odd number

00:06:57.132 --> 00:06:59.681
of negatives being multiplied together.

00:06:59.681 --> 00:07:02.697
So this thing right over
here is going to be negative.

00:07:02.697 --> 00:07:05.115
That's going to be
negative right over there.

00:07:05.115 --> 00:07:08.135
And this x right over here
is going to be negative.

00:07:08.135 --> 00:07:11.985
And so 2.3 times that x,
well that's going to be

00:07:11.985 --> 00:07:15.678
negative as well, so this
thing is going to be negative.

00:07:15.678 --> 00:07:17.739
That's going to be negative.

00:07:17.739 --> 00:07:20.811
A negative divided by a
negative, is a positive.

00:07:20.811 --> 00:07:24.102
So all of this business
is going to be a positive.

00:07:24.102 --> 00:07:26.372
A positive times a positive,

00:07:26.372 --> 00:07:28.143
well that's just gonna be positive.

00:07:28.143 --> 00:07:32.052
So all of this whole thing
is going to be positive.

00:07:32.052 --> 00:07:35.174
I can't stop, this is, all right.

00:07:35.174 --> 00:07:40.089
What is the sign of negative
x, times y divided by 7/8?

00:07:40.089 --> 00:07:43.547
And here they're showing us
x and y on a number line.

00:07:43.547 --> 00:07:46.761
And the key takeaways here
is that x is positive,

00:07:46.761 --> 00:07:50.283
x is to the right of zero, y is negative.

00:07:52.122 --> 00:07:53.999
So let's think about this.

00:07:53.999 --> 00:07:56.210
X is positive, but then we're taking

00:07:56.210 --> 00:07:58.843
the negative of x right over here.

00:07:58.843 --> 00:08:02.515
So x is positive but negative
of x is going to be negative.

00:08:02.515 --> 00:08:04.669
You bring a negative number
in front of a positive number,

00:08:04.669 --> 00:08:06.771
that's going to be a negative number.

00:08:06.771 --> 00:08:09.091
Now what's y divided by 7/8?

00:08:09.091 --> 00:08:11.283
Y is a negative number.

00:08:13.666 --> 00:08:16.450
I'm dividing it by a positive number.

00:08:16.450 --> 00:08:18.580
So a negative divided by a positive

00:08:18.580 --> 00:08:20.322
is going to be a negative.

00:08:20.322 --> 00:08:23.226
So this whole expression,
this whole expression

00:08:23.226 --> 00:08:25.754
right over here, is going to be negative.

00:08:25.754 --> 00:08:28.537
And then we have a
negative, all of this stuff,

00:08:28.537 --> 00:08:31.084
times a negative, well
that's going to give us

00:08:31.084 --> 00:08:33.081
a positive.

00:08:33.081 --> 00:08:36.990
So the whole thing here
is going to be positive.


Multiplying and dividing even and odd numbers of negatives

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.533
- [Voiceover] When we first learned

00:00:01.533 --> 00:00:04.078
to multiply and divide
positive and negative numbers,

00:00:04.078 --> 00:00:05.132
we saw, well look, if you have

00:00:05.132 --> 00:00:08.360
a positive number times a positive number.

00:00:08.360 --> 00:00:10.682
I'm just writing "pos" short for positive.

00:00:10.682 --> 00:00:11.626
Well that, of course, is going

00:00:11.626 --> 00:00:13.143
to be equal to a positive number

00:00:13.143 --> 00:00:14.180
that you saw that even before

00:00:14.180 --> 00:00:15.720
you learned about negative numbers.

00:00:15.720 --> 00:00:17.375
There's also true if
you had the same sign,

00:00:17.375 --> 00:00:21.153
a negative times a negative,
that that was also going

00:00:21.153 --> 00:00:23.173
to be equal to a positive number.

00:00:23.173 --> 00:00:25.310
And that the way that you
get a negative number,

00:00:25.310 --> 00:00:28.932
is if you multiply or divide
something of different signs.

00:00:28.932 --> 00:00:32.670
So if you have a positive times a negative

00:00:32.670 --> 00:00:34.965
that would give you a negative number.

00:00:34.965 --> 00:00:38.982
Or if you had a negative,
negative times a positive,

00:00:38.982 --> 00:00:40.955
that also would give
you a negative number.

00:00:40.955 --> 00:00:42.752
And here I wrote it for multiplication,

00:00:42.752 --> 00:00:44.435
but it also applies to division.

00:00:44.435 --> 00:00:47.946
A negative divided by
a negative is positive.

00:00:47.946 --> 00:00:51.586
A negative divided by a
positive would be negative.

00:00:51.586 --> 00:00:52.700
Now all of these, we thought about

00:00:52.700 --> 00:00:55.255
if we're only multiplying
or dividing two things.

00:00:55.255 --> 00:00:56.578
But I was just thinking about what happens

00:00:56.578 --> 00:00:58.051
if we multiply or divide three things,

00:00:58.051 --> 00:00:59.491
or four things, or five things,

00:00:59.491 --> 00:01:03.742
or n things together,
what we might expect?

00:01:03.742 --> 00:01:08.107
So let's say that we
were to multiply a times,

00:01:08.107 --> 00:01:10.155
and now I'm going to
write the dot for times.

00:01:10.155 --> 00:01:15.155
A times b times c,

00:01:15.219 --> 00:01:18.581
a times b times c.

00:01:18.581 --> 00:01:21.716
Now if I told you that these
were all positive numbers,

00:01:21.716 --> 00:01:23.364
then you say, "OK a
times b times c is going

00:01:23.364 --> 00:01:24.220
"to be positive."

00:01:24.220 --> 00:01:26.974
"A times b would be positive,
and that times c is positive."

00:01:27.497 --> 00:01:29.796
Now what would happen
if I were to tell you

00:01:29.796 --> 00:01:31.862
that they were all negative numbers.

00:01:31.862 --> 00:01:34.858
What if a, b, and c were all negative?

00:01:34.858 --> 00:01:36.354
Well if there were all negative,
let me write it that way.

00:01:36.354 --> 00:01:38.080
Let me actually write,

00:01:38.080 --> 00:01:42.222
let me write, a, b, and c,

00:01:42.222 --> 00:01:46.026
a, b, and c, they're all
going to be negative.

00:01:46.026 --> 00:01:47.698
So if that's the case,
what is this product

00:01:47.698 --> 00:01:49.462
going to be equal to?

00:01:49.462 --> 00:01:53.340
Well you're going to have a
negative here, times a negative.

00:01:53.340 --> 00:01:55.940
So a negative times a negative, a times b,

00:01:55.940 --> 00:01:58.402
if you do that first, and we can

00:01:58.402 --> 00:01:59.725
when we multiply these numbers.

00:01:59.725 --> 00:02:01.490
That's going to give you a positive.

00:02:01.490 --> 00:02:04.183
So a times b is going
to give you a positive.

00:02:04.183 --> 00:02:06.598
But then you're going to
multiply that times c.

00:02:06.598 --> 00:02:09.616
You're going to multiply that
times c, which is a negative.

00:02:09.616 --> 00:02:12.379
So you're going to have a
positive times a negative,

00:02:12.379 --> 00:02:15.491
which is going to be a negative.

00:02:15.491 --> 00:02:20.277
So this one, if a, b, and
c are all less than zero,

00:02:20.277 --> 00:02:22.782
then the product, a, b, c is going

00:02:22.782 --> 00:02:25.939
to be less than zero as well.

00:02:25.939 --> 00:02:28.888
This whole thing is going to be negative.

00:02:28.888 --> 00:02:30.444
Now, if I did something else.

00:02:30.444 --> 00:02:32.061
If I said, "There's other ways

00:02:32.061 --> 00:02:33.811
"that I can make the product negative."

00:02:33.811 --> 00:02:37.282
If a is, let's say that a,

00:02:37.282 --> 00:02:38.328
actually let me just write it this way.

00:02:38.328 --> 00:02:42.968
Let's say that a is
positive, b is negative.

00:02:42.968 --> 00:02:47.473
B is negative, and c is positive.

00:02:47.473 --> 00:02:49.458
And c is positive.

00:02:49.458 --> 00:02:54.458
Well here, positive times a
negative, if you do this first.

00:02:54.590 --> 00:02:57.515
Positive times a negative is
going to give you a negative.

00:02:57.515 --> 00:02:59.025
And then a negative times a positive,

00:02:59.025 --> 00:03:01.486
different signs, is going
to give you a negative.

00:03:01.486 --> 00:03:03.088
So this whole thing,

00:03:03.088 --> 00:03:06.176
this whole thing is
going to be a negative.

00:03:06.176 --> 00:03:07.384
But let's keep on doing, we said,

00:03:07.384 --> 00:03:08.893
look if all of them are negative,

00:03:08.893 --> 00:03:10.100
then this thing would be negative,

00:03:10.100 --> 00:03:11.331
but that's because I
had three numbers here.

00:03:11.331 --> 00:03:12.863
What if I had four numbers here?

00:03:12.863 --> 00:03:17.182
What if I had times,

00:03:17.182 --> 00:03:19.736
What if I had times d here?

00:03:19.736 --> 00:03:22.453
And if I told you all of
these numbers were negative?

00:03:22.453 --> 00:03:24.357
Let's think about it, if negative,

00:03:24.357 --> 00:03:27.979
negative, negative, negative.

00:03:27.979 --> 00:03:29.929
And I can do the
multiplication in any order,

00:03:29.929 --> 00:03:31.624
but I'll just go left to right.

00:03:31.624 --> 00:03:34.318
A times b, negative times a negative.

00:03:34.318 --> 00:03:36.268
That would yield a positive.

00:03:36.268 --> 00:03:39.472
Now if you multiply that product times c,

00:03:39.472 --> 00:03:43.605
positive times a negative,
positive times a negative,

00:03:43.605 --> 00:03:47.159
positive times a negative that
would give you a negative.

00:03:47.159 --> 00:03:49.969
And then you multiply this
negative times this negative,

00:03:49.969 --> 00:03:52.476
so this whole product, a, b,
c, is going to be negative.

00:03:52.476 --> 00:03:54.543
But then we multiply it times a negative.

00:03:54.543 --> 00:03:56.818
Well a negative times a negative
is going to be a positive.

00:03:56.818 --> 00:03:59.697
So this whole thing is
going to be a positive.

00:03:59.697 --> 00:04:03.018
And so you're probably
seeing a pattern here.

00:04:03.018 --> 00:04:05.734
If you're multiplying a bunch of numbers,

00:04:05.734 --> 00:04:07.685
if you're multiplying a bunch of numbers,

00:04:07.685 --> 00:04:10.531
and if you have an odd
number of negatives,

00:04:10.531 --> 00:04:13.930
odd number of negatives

00:04:13.930 --> 00:04:16.298
being multiplied, and or divided.

00:04:16.298 --> 00:04:17.841
I just did multiplication here,

00:04:17.841 --> 00:04:19.072
but this would have also been true

00:04:19.072 --> 00:04:20.837
if these were all division symbols.

00:04:20.837 --> 00:04:22.810
If these were all division symbols,

00:04:22.810 --> 00:04:25.945
we would've been able to
say the exact same thing.

00:04:25.945 --> 00:04:29.056
If you have an odd number
of negative numbers

00:04:29.056 --> 00:04:31.169
in your product or in your quotient,

00:04:31.169 --> 00:04:33.778
well then you're going to have a negative,

00:04:33.778 --> 00:04:35.467
you're going to have a negative

00:04:35.467 --> 00:04:37.986
value for the entire expression.

00:04:37.986 --> 00:04:41.090
If you have an even, if you have an even

00:04:41.090 --> 00:04:45.348
number of negative,

00:04:45.348 --> 00:04:47.286
well then the whole thing is going to be,

00:04:47.286 --> 00:04:49.712
the whole thing is going to be positive.

00:04:49.712 --> 00:04:51.685
And so you can view
these as generalizations

00:04:51.685 --> 00:04:53.148
of what we just saw here.

00:04:53.148 --> 00:04:55.014
Positive times a positive.

00:04:55.014 --> 00:04:55.842
You have zero negative.

00:04:55.842 --> 00:04:59.052
Zero is actually an even number,
so this would be positive.

00:04:59.069 --> 00:05:00.857
Negative times a negative.

00:05:00.857 --> 00:05:02.436
Well that's an even number of negatives.

00:05:02.436 --> 00:05:04.956
You have two negatives,
so that's this case again.

00:05:04.956 --> 00:05:08.171
That's this case again, so
you're going to be positive.

00:05:08.171 --> 00:05:10.957
Either of these cases,
you have one negative.

00:05:10.957 --> 00:05:13.179
You have one negative in
either of these cases,

00:05:13.179 --> 00:05:15.774
so it's going to be this
case, odd number of negatives.

00:05:15.774 --> 00:05:17.410
So that's going to be a negative.

00:05:17.410 --> 00:05:19.287
And so, we can use this
knowledge to start dealing

00:05:19.287 --> 00:05:20.912
with negative numbers and exponents.

00:05:20.912 --> 00:05:25.858
So if I were to say,
I would have a to the,

00:05:25.858 --> 00:05:28.726
let me throw out a wild number here,

00:05:28.726 --> 00:05:31.442
A to the 101st power,

00:05:31.442 --> 00:05:36.272
And we know that a is less than zero.

00:05:36.272 --> 00:05:37.990
What is this going to be?

00:05:37.990 --> 00:05:42.030
Well this is taking 101 A's
and multiplying them together.

00:05:42.030 --> 00:05:45.095
You have an odd number, an odd number

00:05:45.095 --> 00:05:47.696
of negatives being multiplied together?

00:05:47.696 --> 00:05:50.089
Well this whole then is going,

00:05:50.089 --> 00:05:52.514
this whole thing is going
to be less than zero.

00:05:52.514 --> 00:05:54.749
If we knew what a was, we
could calculate it somehow,

00:05:54.749 --> 00:05:57.401
but we know that this thing
is going to be negative.

00:05:57.401 --> 00:06:00.605
Let's do something, so, and
we could do it other ways.

00:06:00.605 --> 00:06:02.607
We could say something like this.

00:06:02.607 --> 00:06:03.740
Actually, let me try another one.

00:06:03.740 --> 00:06:08.443
What if I tell you that
a is less than zero,

00:06:08.443 --> 00:06:11.402
and b is also less than zero,

00:06:11.402 --> 00:06:13.193
and if I had a

00:06:13.193 --> 00:06:16.708
to the 101 power,

00:06:16.708 --> 00:06:19.690
divided by b to the seventh power,

00:06:19.690 --> 00:06:20.809
what would this expression be?

00:06:20.809 --> 00:06:22.666
Would it be positive or negative?

00:06:22.666 --> 00:06:24.942
Or zero, maybe?

00:06:24.942 --> 00:06:27.705
Well you see here, a to the 101st power.

00:06:27.705 --> 00:06:30.422
You have an odd number of,

00:06:30.422 --> 00:06:34.908
you have an odd number of
negative numbers being multiplied.

00:06:34.908 --> 00:06:39.682
So this whole thing is
going to be negative.

00:06:39.682 --> 00:06:41.029
And the same thing applies here.

00:06:41.029 --> 00:06:42.863
You have a negative
number to an odd power.

00:06:42.863 --> 00:06:43.978
You have an odd number of this

00:06:43.978 --> 00:06:46.230
negative number being multiplied together.

00:06:46.230 --> 00:06:48.088
So that's going to be negative,

00:06:48.088 --> 00:06:50.897
but then you have a negative
divided by a negative.

00:06:50.897 --> 00:06:54.775
A negative divided by a negative
is going to be a positive.

00:06:54.775 --> 00:06:58.699
So this thing, right here,
is going to be positive.

00:06:58.699 --> 00:07:01.137
So this is just the beginning,
and in the next few videos

00:07:01.137 --> 00:07:02.483
we'll actually do a bunch of examples

00:07:02.483 --> 00:07:05.143
that really test our
understanding of this.


Exponents with negative bases

WEBVTT
Kind: captions
Language: en

00:00:00.405 --> 00:00:01.456
Let's see if we can apply

00:00:01.456 --> 00:00:04.532
what we know about negative numbers,

00:00:04.532 --> 00:00:06.031
and what we know about exponents

00:00:06.031 --> 00:00:08.879
to apply exponents to negative numbers.

00:00:08.879 --> 00:00:10.197
So let's first think about –

00:00:10.197 --> 00:00:13.723
Let's say we have -3.

00:00:13.723 --> 00:00:15.123
Let's first think about what it means

00:00:15.123 --> 00:00:18.057
to raise it to the 1st power.

00:00:18.057 --> 00:00:22.085
Well that literally means just taking a -3.

00:00:22.085 --> 00:00:23.994
And there's nothing left to multiply it with.

00:00:23.994 --> 00:00:27.642
So this is just going to be equal to -3.

00:00:27.642 --> 00:00:30.069
Now what happens if you were take a -3,

00:00:30.069 --> 00:00:34.144
and we were to raise it to the 2nd power?

00:00:34.144 --> 00:00:37.660
Well that's equivalent to taking 2 -3's,

00:00:37.660 --> 00:00:44.661
so a -3 and a -3, and then multiplying them together.

00:00:44.661 --> 00:00:46.457
What's that going to be?

00:00:46.457 --> 00:00:48.930
Well a negative times a negative is a positive.

00:00:48.930 --> 00:00:53.670
So that is going to be positive 9.

00:00:53.670 --> 00:00:55.918
Let me write this. It's going to be positive 9.

00:00:55.918 --> 00:00:56.798
Well, let's keep going.

00:00:56.798 --> 00:00:59.054
Let's see if there is some type of pattern here.

00:00:59.054 --> 00:01:04.798
Let's take -3 and raise it to the 3rd power.

00:01:04.798 --> 00:01:06.921
What is this going to be equal to?

00:01:06.921 --> 00:01:12.659
Well, we're going to take 3 -3's, [WRITING] –

00:01:12.659 --> 00:01:14.629
and we're going to multiply them together.

00:01:14.629 --> 00:01:16.582
So we're going to multiply them together.

00:01:16.582 --> 00:01:20.195
-3 × -3, we already figured out is positive 9.

00:01:20.195 --> 00:01:26.806
But positive 9 × -3, well that's that's -27.

00:01:26.806 --> 00:01:28.720
And so you might notice a pattern here.

00:01:28.720 --> 00:01:33.057
Whenever we raised raised a negative base

00:01:33.057 --> 00:01:41.188
to an exponent, if we raise it to an odd exponent,

00:01:41.188 --> 00:01:47.910
we are going to get a negative value.

00:01:47.910 --> 00:01:49.402
And that's because when you multiply

00:01:49.402 --> 00:01:50.853
negative numbers an even number of times,

00:01:50.853 --> 00:01:52.122
a negative number times

00:01:52.122 --> 00:01:53.006
a negative number is a positive.

00:01:53.006 --> 00:01:54.521
But then you have one more negative number

00:01:54.521 --> 00:01:56.696
to multiply the result by – which makes it negative.

00:01:56.696 --> 00:01:59.594
And if you take a negative base,

00:01:59.594 --> 00:02:04.471
and you raise it to an even power,

00:02:04.471 --> 00:02:07.327
that's because if you multiply a negative

00:02:07.327 --> 00:02:09.726
times a negative, you're going to get a positive.

00:02:09.726 --> 00:02:12.055
And so when you do it an even number of times,

00:02:12.055 --> 00:02:14.133
doing it a multiple-of-two number of times.

00:02:14.133 --> 00:02:16.736
So the negatives and the negatives all cancel out,

00:02:16.736 --> 00:02:17.529
I guess you could say.

00:02:17.529 --> 00:02:19.735
Or when you take the product of the two negatives,

00:02:19.735 --> 00:02:21.267
you keep getting positives.

00:02:21.267 --> 00:02:23.867
So this right over here is

00:02:23.867 --> 00:02:27.660
going to give you a positive value.

00:02:27.660 --> 00:02:29.497
So there's really nothing new about

00:02:29.497 --> 00:02:32.489
taking powers of negative numbers.

00:02:32.489 --> 00:02:33.659
It's really the same idea.

00:02:33.659 --> 00:02:35.409
And you just really have to remember that

00:02:35.409 --> 00:02:38.136
a negative times a negative is a positive.

00:02:38.136 --> 00:02:41.663
And a negative times a positive is a negative,

00:02:41.663 --> 00:02:43.297
which we already learned from

00:02:43.297 --> 00:02:45.545
multiplying negative numbers.

00:02:45.545 --> 00:02:48.072
Now there's one other thing that I want to clarify –

00:02:48.072 --> 00:02:50.549
because sometimes there might be ambiguity

00:02:50.549 --> 00:02:54.227
if someone writes this.

00:02:54.227 --> 00:02:55.895
Let's say someone writes that.

00:02:55.895 --> 00:02:58.328
And I encourage you to actually pause the video

00:02:58.328 --> 00:03:01.472
and think about with this right over here

00:03:01.472 --> 00:03:03.332
would evaluate to.

00:03:03.332 --> 00:03:05.963
And, if you given a go at that, think about whether

00:03:05.963 --> 00:03:11.793
this should mean something different then that.

00:03:11.793 --> 00:03:14.664
Well this one can be a little bit and big ambiguous

00:03:14.664 --> 00:03:17.031
and if people are strict about order of operations,

00:03:17.031 --> 00:03:19.469
you should really be thinking about the exponent

00:03:19.485 --> 00:03:21.786
before you multiply by this -1.

00:03:21.786 --> 00:03:25.895
You could this is implicitly saying -1 × 2^3.

00:03:25.895 --> 00:03:29.003
So many times, this will usually be interpreted

00:03:29.003 --> 00:03:32.060
as negative 2 to the third power,

00:03:32.060 --> 00:03:34.501
which is equal to -8,

00:03:34.501 --> 00:03:36.382
while this is going to be interpreted

00:03:36.382 --> 00:03:38.388
as -2 to the third power.

00:03:38.388 --> 00:03:40.525
Now that also is equal to -8.

00:03:40.525 --> 00:03:43.461
You might say well what's what's the big deal here?

00:03:43.461 --> 00:03:46.929
Well what if this was what if these were even exponents.

00:03:46.929 --> 00:03:51.486
So what if someone had give myself some more space here.

00:03:51.486 --> 00:04:00.614
What if someone had these to express its -4 or a -4 squared or -4 squared.

00:04:00.614 --> 00:04:04.558
This one clearly evaluates to 16 – positive 16.

00:04:04.558 --> 00:04:07.189
It's a negative 4 times a *4.

00:04:07.189 --> 00:04:09.073
This one could be interpreted as is.

00:04:09.073 --> 00:04:11.003
Especially if you look at order of operations,

00:04:11.003 --> 00:04:12.564
and you do your exponent first,

00:04:12.564 --> 00:04:15.766
this would be interpreted as -4 times 4,

00:04:15.766 --> 00:04:18.077
which would be -16.

00:04:18.077 --> 00:04:20.964
So it's really important to think about this properly.

00:04:20.964 --> 00:04:23.438
And if you want to write the number negative

00:04:23.438 --> 00:04:25.474
if you want the base to be negative 4,

00:04:25.474 --> 00:04:29.000
put parentheses around it and then write the exponent.


Expressions with rational numbers

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:02.480
We have four different
expressions here,

00:00:02.480 --> 00:00:03.990
and what I want
you to do is think

00:00:03.990 --> 00:00:11.385
about which of these expressions
are equal to negative 2/3.

00:00:11.385 --> 00:00:13.930
And I encourage you now
to pause this video,

00:00:13.930 --> 00:00:16.570
and try this on your own.

00:00:16.570 --> 00:00:18.860
So let's go to this first
expression right over here.

00:00:18.860 --> 00:00:23.610
I have 1/9, and I'm
going to add to that 5/9.

00:00:23.610 --> 00:00:26.380
So how many ninths
am I going to have?

00:00:26.380 --> 00:00:32.162
Well, I had 1/9, now I'm adding
5/9, so I'm going to have 6/9.

00:00:32.162 --> 00:00:34.120
If I have one of something
and I have five more

00:00:34.120 --> 00:00:35.828
of that same something--
so in this case,

00:00:35.828 --> 00:00:39.480
that something is a ninth--
1/9 plus 5/9 is 6/9.

00:00:39.480 --> 00:00:42.170
Now, can we simplify
this in any way?

00:00:42.170 --> 00:00:45.320
Well, both six and nine are
divisible by 3, so let's

00:00:45.320 --> 00:00:50.200
divide them both by 3 to try to
get this fraction in a simpler

00:00:50.200 --> 00:00:51.000
form.

00:00:51.000 --> 00:00:53.300
6 divided by 3 is 2.

00:00:53.300 --> 00:00:56.120
9 divided by 3 is 3.

00:00:56.120 --> 00:00:58.620
So this is 2/3, while what
we're trying to get to

00:00:58.620 --> 00:00:59.940
is negative 2/3.

00:00:59.940 --> 00:01:01.510
So these are not equal.

00:01:01.510 --> 00:01:05.630
This expression does
not equal negative 2/3,

00:01:05.630 --> 00:01:07.960
so I'll write "no" for that one.

00:01:07.960 --> 00:01:11.120
Now let's go to this green
expression right over here.

00:01:11.120 --> 00:01:14.020
Give myself a little bit
more real estate to work in.

00:01:14.020 --> 00:01:17.790
Now, we have negative
1/6 plus negative 1/2.

00:01:17.790 --> 00:01:19.665
Now, we can view this
as being the same thing

00:01:19.665 --> 00:01:21.998
as-- just to clarify, right
now the negative is in front

00:01:21.998 --> 00:01:23.550
of the entire 1/6,
the negative's

00:01:23.550 --> 00:01:25.290
in front of the entire 1/2.

00:01:25.290 --> 00:01:29.690
But this is the same
thing as negative 1/6,

00:01:29.690 --> 00:01:35.080
plus negative 1/2.

00:01:35.080 --> 00:01:38.160
Negative 1/2 is the same thing
as negative 1 divided by 2

00:01:38.160 --> 00:01:39.462
is one way to think about it.

00:01:39.462 --> 00:01:40.920
And the whole reason
why I did this

00:01:40.920 --> 00:01:44.790
is so we can simplify what the
negatives are right now only

00:01:44.790 --> 00:01:46.830
in our numerator.

00:01:46.830 --> 00:01:49.260
So whenever we
add two fractions,

00:01:49.260 --> 00:01:51.540
we want to have the
same denominator.

00:01:51.540 --> 00:01:53.970
And we see that 6 is
already a multiple of 2,

00:01:53.970 --> 00:01:56.510
so we could leave this first
fraction the way it is.

00:01:56.510 --> 00:01:59.716
We can rewrite it
as negative 1/6.

00:01:59.716 --> 00:02:01.590
And then the second
fraction, we can write it

00:02:01.590 --> 00:02:03.390
as something over 6.

00:02:03.390 --> 00:02:07.390
Well, to go from 2 to 6,
we have to multiply by 3.

00:02:07.390 --> 00:02:09.660
So let's also multiply
the numerator by 3,

00:02:09.660 --> 00:02:13.700
negative 1 times
3 is negative 3.

00:02:13.700 --> 00:02:18.120
So if I have negative 1/6 sixth
and I add to that negative 3/6,

00:02:18.120 --> 00:02:22.740
this is going to be negative 1
plus negative 3 sixths, which

00:02:22.740 --> 00:02:27.390
is equal to negative 4/6.

00:02:27.390 --> 00:02:29.790
Now, let's see if
we can simplify it.

00:02:29.790 --> 00:02:32.310
Both negative 4-- I guess
we can say both four

00:02:32.310 --> 00:02:35.840
and six are divisible by 2, so
let's divide them both by 2.

00:02:38.430 --> 00:02:41.140
And in the numerator, we're
left with negative 4 divided

00:02:41.140 --> 00:02:43.080
by 2 is negative 2.

00:02:43.080 --> 00:02:45.650
6 divided by 2 is 3.

00:02:45.650 --> 00:02:47.230
Negative 2 divided by 3.

00:02:47.230 --> 00:02:50.590
Well, that's the same thing
as negative 2/3, which

00:02:50.590 --> 00:02:54.140
is exactly what our goal
value we're trying to get to.

00:02:54.140 --> 00:02:59.360
So, yes, this thing in green
is equal to negative 2/3.

00:02:59.360 --> 00:03:00.810
Now let's go over here.

00:03:00.810 --> 00:03:04.170
So we have negative
1.3 times negative 2.

00:03:04.170 --> 00:03:06.360
Well, if you multiply a
negative times a negative,

00:03:06.360 --> 00:03:08.120
we're going to get a
positive, and we're

00:03:08.120 --> 00:03:11.306
going to get a
positive 1/3 times 2.

00:03:11.306 --> 00:03:12.680
So one way to
think about this is

00:03:12.680 --> 00:03:16.170
going to be the same thing
as one third times 2,

00:03:16.170 --> 00:03:17.782
which is the same thing.

00:03:17.782 --> 00:03:19.740
And there's a couple of
ways to think about it.

00:03:19.740 --> 00:03:22.700
If you have 1/3 and now you're
going to multiply it by 2,

00:03:22.700 --> 00:03:24.390
we now have 2/3.

00:03:24.390 --> 00:03:26.340
You now have 2/3.

00:03:26.340 --> 00:03:28.370
Another way to
think about this is

00:03:28.370 --> 00:03:32.970
that this is the same thing
as 2/3 times 2 over 1.

00:03:32.970 --> 00:03:36.680
And you know that when we
multiply two fractions,

00:03:36.680 --> 00:03:39.330
so this time we've expressed
the 2 as a fraction,

00:03:39.330 --> 00:03:41.120
we can multiply the numerators.

00:03:41.120 --> 00:03:45.150
So it's 1 times 2 over the
product of their denominators,

00:03:45.150 --> 00:03:50.080
3 times 1, which is 2 over 3.

00:03:50.080 --> 00:03:52.980
So either way you look at it,
this goes to positive 2/3.

00:03:52.980 --> 00:03:56.150
A negative times a
negative is a positive.

00:03:56.150 --> 00:03:59.690
So it gets us to positive
2/3 not negative 2/3,

00:03:59.690 --> 00:04:06.115
so like this first one, no, it
does not equal negative 2/3.

00:04:06.115 --> 00:04:07.240
Now let's look at this one.

00:04:07.240 --> 00:04:10.970
Negative 2/3
divided by one half.

00:04:10.970 --> 00:04:13.245
So when you divide
by a fraction,

00:04:13.245 --> 00:04:16.589
so when you take
negative 1/3, dividing--

00:04:16.589 --> 00:04:18.540
let me write it this way.

00:04:18.540 --> 00:04:21.250
So negative 1/3
divided by 1/2, this

00:04:21.250 --> 00:04:24.500
is the same thing
as negative 1/3--

00:04:24.500 --> 00:04:27.860
and let me color code it just
so you see what I'm doing.

00:04:27.860 --> 00:04:30.600
So let me make that green color.

00:04:30.600 --> 00:04:33.080
Let me make this a blue color.

00:04:33.080 --> 00:04:36.350
So negative 1/3
divided by 1/2 half

00:04:36.350 --> 00:04:38.280
is the same thing
as negative 1/3

00:04:38.280 --> 00:04:47.840
times the reciprocal of
1/2, so times 2 over 1.

00:04:50.510 --> 00:04:53.340
And what is this
going to be equal to?

00:04:53.340 --> 00:04:56.130
Well, we could assume
instead of just doing this

00:04:56.130 --> 00:04:59.470
as negative 1/3, we could do
this as negative 1 divided

00:04:59.470 --> 00:05:02.620
by 3, that might help us keep
track of the signs a little bit

00:05:02.620 --> 00:05:04.409
more.

00:05:04.409 --> 00:05:06.200
And let me actually
write it that way, just

00:05:06.200 --> 00:05:07.533
to make it a little bit clearer.

00:05:09.980 --> 00:05:14.260
Let me write this as
negative 1 divided by 3.

00:05:14.260 --> 00:05:21.770
So our numerator is now going
to be negative 1 times 2.

00:05:21.770 --> 00:05:26.250
When you multiply two
fractions, you just

00:05:26.250 --> 00:05:28.750
multiply the two numerators
to get the new numerator,

00:05:28.750 --> 00:05:31.370
and it's over 3 times 1.

00:05:33.722 --> 00:05:35.930
And you normally wouldn't
have to do all these steps,

00:05:35.930 --> 00:05:37.596
but I'm just doing
them to make sure you

00:05:37.596 --> 00:05:38.844
understand what's going on.

00:05:38.844 --> 00:05:41.010
And so this is going to be
equal to negative 1 times

00:05:41.010 --> 00:05:43.790
2 is negative 1.

00:05:43.790 --> 00:05:47.580
And 3 times 1 is positive 3.

00:05:47.580 --> 00:05:49.060
Negative 2 over 3?

00:05:49.060 --> 00:05:53.890
Well, that's the same
thing as negative 2/3.

00:05:53.890 --> 00:05:55.160
So this one works out.

00:05:55.160 --> 00:05:58.200
It is equal to negative 2/3.


Making sense of hairy fractions

WEBVTT
Kind: captions
Language: en

00:00:00.290 --> 00:00:01.113
- What I hope to do in this video

00:00:01.113 --> 00:00:03.691
is emphasize the relation, the connection,

00:00:03.691 --> 00:00:05.435
between fractions and division

00:00:05.435 --> 00:00:07.199
and then using that knowledge

00:00:07.199 --> 00:00:10.926
to help us simplify some
hairy looking fractions.

00:00:10.926 --> 00:00:12.958
So, let's say, let's just
do a little bit of a review.

00:00:12.958 --> 00:00:17.958
So, if I say two divided
by, two divided by three,

00:00:18.762 --> 00:00:21.893
two divided by three,
I could write this as,

00:00:21.893 --> 00:00:25.678
I could write this as two over three.

00:00:25.678 --> 00:00:29.975
Two divided by, two divided by three.

00:00:29.975 --> 00:00:31.856
So, this expression here,
sometimes we might say,

00:00:31.856 --> 00:00:34.234
"Hey, this is 2/3. Think
of it as a fraction."

00:00:34.234 --> 00:00:38.821
But you can also view this
as two divided by three.

00:00:38.821 --> 00:00:40.537
So, if we went the other way around.

00:00:40.537 --> 00:00:43.885
If someone were to write four over,

00:00:43.885 --> 00:00:45.740
actually, let me just do
it in a different color,

00:00:45.740 --> 00:00:50.353
if someone were to say four, 4/9,

00:00:50.353 --> 00:00:55.332
four, four over nine,
we could interpret this,

00:00:55.332 --> 00:00:59.500
this could be interpreted
as four divided by nine.

00:00:59.500 --> 00:01:01.616
Four divided by nine.

00:01:01.616 --> 00:01:04.400
In fact, if we wanted
to, if wanted to convert

00:01:04.400 --> 00:01:06.594
this into a decimal that's
exactly what we would do.

00:01:06.594 --> 00:01:08.779
We would calculate what
four divided by nine is.

00:01:08.779 --> 00:01:10.857
We would divide nine into four.

00:01:10.857 --> 00:01:14.036
Four divided by nine.

00:01:14.036 --> 00:01:15.115
So, this is all review.

00:01:15.115 --> 00:01:16.671
So, let's just use this knowledge

00:01:16.671 --> 00:01:20.655
and I'll think more complex fractions.

00:01:20.655 --> 00:01:22.673
So, if someone were to say,

00:01:22.673 --> 00:01:25.910
actually, let me just take
this example right over here,

00:01:25.910 --> 00:01:28.932
if someone were to say two over,

00:01:28.932 --> 00:01:30.941
instead of saying two over three.

00:01:30.941 --> 00:01:33.516
If they said two over, I don't know,

00:01:33.516 --> 00:01:37.338
let me say two over 2/3.

00:01:37.338 --> 00:01:41.018
Two over 2/3. Well, what
would this simplify to?

00:01:41.018 --> 00:01:42.422
Well, we could go the other way around.

00:01:42.422 --> 00:01:45.799
This is the same thing
as two divided by 2/3.

00:01:45.799 --> 00:01:47.285
So, let's write it like that.

00:01:47.285 --> 00:01:50.781
This is the same thing as two

00:01:50.781 --> 00:01:55.781
divided by, two divided by
2/3, two divided by 2/3,

00:01:58.105 --> 00:02:00.538
which is the same thing as,

00:02:00.538 --> 00:02:02.281
dividing by a fraction's the same thing

00:02:02.281 --> 00:02:04.914
as multiplying by its reciprocal.

00:02:04.914 --> 00:02:06.391
So, this is going to be the same thing

00:02:06.391 --> 00:02:11.195
as two times, two times
the reciprocal of 2/3,

00:02:12.195 --> 00:02:14.808
which is, we swap the
denominator and the numerator,

00:02:14.808 --> 00:02:17.722
two times three over two.

00:02:17.722 --> 00:02:19.821
So, two times three halves,

00:02:19.821 --> 00:02:21.518
if I have three halves twice, well,

00:02:21.518 --> 00:02:22.870
that's just going to be equal to,

00:02:22.870 --> 00:02:25.232
that's going to be equal to six halves,

00:02:25.232 --> 00:02:27.029
and six halves, you can view that as,

00:02:27.029 --> 00:02:28.891
well, two halves is a whole,

00:02:28.891 --> 00:02:30.458
so this is gonna be three wholes.

00:02:30.458 --> 00:02:31.816
Or you can say six divided by two.

00:02:31.816 --> 00:02:33.834
Well, that's just gonna be equal to three.

00:02:33.834 --> 00:02:34.949
You can view it either way.

00:02:34.949 --> 00:02:36.911
And then that is equal to three.

00:02:36.911 --> 00:02:38.259
Let's do a few more of these.

00:02:38.259 --> 00:02:41.229
And let's keep making them a
little bit more complicated.

00:02:41.229 --> 00:02:42.694
Just let me get some good practice.

00:02:42.694 --> 00:02:44.027
And like always, pause the video.

00:02:44.027 --> 00:02:46.395
You should get excited when
you see one of these things.

00:02:46.395 --> 00:02:50.068
And pause the video and see
if you can do it on your own.

00:02:50.068 --> 00:02:52.292
All right, let's do
something really interesting.

00:02:52.292 --> 00:02:57.179
Let's say negative 16 over nine

00:02:58.262 --> 00:03:03.241
over, over, I'll do this in a tan color,

00:03:03.695 --> 00:03:08.695
over, let's say, three over seven.

00:03:09.012 --> 00:03:12.272
What is this? Can you simplify
this complex fraction?

00:03:12.272 --> 00:03:14.933
Can you simplify this
expression over here?

00:03:14.933 --> 00:03:16.860
Well, once again, we can view

00:03:16.860 --> 00:03:20.854
this as negative 16/9 divided by 3/7.

00:03:20.854 --> 00:03:23.733
So, this could be rewritten as,

00:03:23.733 --> 00:03:27.109
and I can write it either
as negative 16 over 9

00:03:27.109 --> 00:03:31.474
or I could rewrite it as negative 16/9.

00:03:31.474 --> 00:03:33.099
So, I can put the negative
in front of the whole,

00:03:33.099 --> 00:03:35.352
in front of the whole fraction like that,

00:03:35.352 --> 00:03:38.357
or I could say that it's
negative 16 over nine,

00:03:38.357 --> 00:03:40.822
or I can even write this as
negative 16 over negative nine.

00:03:40.822 --> 00:03:41.901
Those would all be equivalent.

00:03:41.901 --> 00:03:44.653
But right now I'm writing at negative 16/9

00:03:44.653 --> 00:03:47.300
and I am going to now divide,

00:03:47.300 --> 00:03:49.991
we can interpret this complex fraction

00:03:49.991 --> 00:03:52.176
as dividing it by 3/7.

00:03:52.176 --> 00:03:55.192
Dividing by 3/7.

00:03:55.192 --> 00:03:57.574
And so, this is going to be equal to,

00:03:57.574 --> 00:04:01.113
this is going to be
equal to negative 16/9,

00:04:01.113 --> 00:04:03.598
negative 16 over nine.

00:04:03.598 --> 00:04:06.837
Actually, let me just rewrite
it as negative 16 over nine.

00:04:06.837 --> 00:04:09.973
Negative 16 over nine just
to show that we can do it.

00:04:09.973 --> 00:04:12.051
Times the reciprocal of this.

00:04:12.051 --> 00:04:15.859
So, times 7/3.

00:04:15.859 --> 00:04:18.109
I just swap the numerator
and the denominator.

00:04:18.109 --> 00:04:20.816
Times, do that same brown color,

00:04:20.816 --> 00:04:25.443
times, times 7/3, and now
what am I going to get?

00:04:25.443 --> 00:04:29.112
In the numerator, I have
negative 16 times seven.

00:04:29.112 --> 00:04:30.182
Let me think about it.

00:04:30.182 --> 00:04:31.935
10 times seven is 70.

00:04:31.935 --> 00:04:34.254
Six times seven is 42.

00:04:34.254 --> 00:04:37.101
70 plus 42 is 112.

00:04:37.101 --> 00:04:39.980
So, this, so this part right over here,

00:04:39.980 --> 00:04:43.078
this part right over here
would be negative 112.

00:04:43.078 --> 00:04:48.078
Negative 112 over nine times three.

00:04:49.128 --> 00:04:54.128
Nine times three is 20,
nine times three is 27.

00:04:55.627 --> 00:04:56.301
And there you have it.

00:04:56.301 --> 00:04:58.601
You can view this as negative 112 over 27

00:04:58.601 --> 00:05:00.169
or you can put the negative up front

00:05:00.169 --> 00:05:01.446
and you can say, "Hey, this is hinting

00:05:01.446 --> 00:05:05.589
"as negative 112, 112/27."

00:05:06.980 --> 00:05:10.339
112/27 and just to make
everything come full circle

00:05:10.339 --> 00:05:14.251
you can view this as
negative 112 divided by 27

00:05:14.251 --> 00:05:18.342
or negative 112/27. Either way.


Negative signs in numerators and denominators

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:02.667
- We already know a good
bit about negative numbers.

00:00:02.667 --> 00:00:04.000
And we know a good bit of fractions.

00:00:04.000 --> 00:00:05.567
So you can imagine we're
going to start seeing

00:00:05.567 --> 00:00:09.100
negatives and fractions together a lot.

00:00:09.100 --> 00:00:10.067
What I want to do in this video

00:00:10.067 --> 00:00:11.566
is just make sure we have
a decent understanding

00:00:11.566 --> 00:00:13.433
how to manipulate negative signs

00:00:13.433 --> 00:00:15.267
when we see them in fractions.

00:00:15.267 --> 00:00:19.866
For example, if I have
the fraction negative 1/2.

00:00:20.166 --> 00:00:22.868
Here I have the negative out
in front of the entire 1/2.

00:00:22.868 --> 00:00:26.500
This is the same thing
as negative one over two.

00:00:26.500 --> 00:00:27.800
And it's going to be the same thing

00:00:27.800 --> 00:00:31.200
as one over negative two.

00:00:31.566 --> 00:00:33.899
Now I could also think
about something like

00:00:33.899 --> 00:00:37.566
negative one over negative two.

00:00:37.566 --> 00:00:39.515
Now it's important to realize
one way to think about

00:00:39.515 --> 00:00:40.933
this as a fraction is you could view this

00:00:40.933 --> 00:00:44.233
as negative one divided by negative two.

00:00:44.233 --> 00:00:45.801
And we already know, if
you divide a negative

00:00:45.801 --> 00:00:47.734
by a negative it would be a positive.

00:00:47.734 --> 00:00:49.100
So this right over here is going to be

00:00:49.100 --> 00:00:51.767
the same thing as 1/2.

00:00:51.767 --> 00:00:56.033
This is going to be the
same thing as positive 1/2.

00:00:56.400 --> 00:00:58.766
Now with that out of the way,
let's think a little bit.

00:00:58.766 --> 00:01:01.068
Let's do some example
problems that might push

00:01:01.068 --> 00:01:03.533
our thinking on this a little bit more.

00:01:03.833 --> 00:01:06.866
So this first question.

00:01:07.133 --> 00:01:08.366
Which of the following expressions

00:01:08.366 --> 00:01:12.533
are equivalent to negative g over h?

00:01:12.700 --> 00:01:14.733
Negative G over H.

00:01:14.733 --> 00:01:16.433
Select all that apply.

00:01:16.700 --> 00:01:20.500
All right, so this has all
sorts of negatives here.

00:01:20.500 --> 00:01:23.267
So at first it looks a little bit unusual.

00:01:23.267 --> 00:01:25.233
But then we need to just
realize that this part.

00:01:25.233 --> 00:01:27.766
Actually, let me just square this off in

00:01:27.766 --> 00:01:29.266
blue right over here.

00:01:29.266 --> 00:01:31.800
Negative g over negative h.

00:01:31.800 --> 00:01:32.800
We've already figured that out.

00:01:32.800 --> 00:01:34.666
We actually looked at
that right over here.

00:01:34.967 --> 00:01:36.667
If you have a negative
divided by a negative,

00:01:36.667 --> 00:01:38.800
that's the same thing as a positive value

00:01:38.800 --> 00:01:40.534
divided by the positive value.

00:01:40.767 --> 00:01:43.433
So negative g over negative h,

00:01:43.433 --> 00:01:46.166
is the same thing as g over h.

00:01:46.166 --> 00:01:50.233
And then you still have
this negative out front.

00:01:50.233 --> 00:01:52.400
You still have that negative out front.

00:01:52.400 --> 00:01:54.500
So this one right over here is actually

00:01:54.500 --> 00:01:56.801
equal to negative g over h.

00:01:56.801 --> 00:01:58.300
When we think about it,
negative divided by a negative

00:01:58.300 --> 00:02:00.967
is a positive and you still
have this negative out here.

00:02:00.967 --> 00:02:02.533
So that's the same thing.

00:02:02.867 --> 00:02:05.033
And this right over
here, negative in front.

00:02:05.033 --> 00:02:07.233
And then you have g over negative h.

00:02:07.233 --> 00:02:08.534
This is going to be the same thing.

00:02:08.534 --> 00:02:10.600
You could rewrite this, you
could put the negative on top

00:02:10.600 --> 00:02:14.000
as negative g over negative h.

00:02:14.000 --> 00:02:17.100
And then this would be equal to g over h,

00:02:17.100 --> 00:02:17.968
which is different.

00:02:17.968 --> 00:02:19.100
This is positive g over h.

00:02:19.100 --> 00:02:20.366
This is negative g over h.

00:02:20.366 --> 00:02:21.933
So we wouldn't select that.

00:02:21.933 --> 00:02:24.034
And of course we wouldn't
select "None of the above."

00:02:24.034 --> 00:02:26.133
Cause we found a choice that we liked.

00:02:26.133 --> 00:02:27.400
All right.

00:02:27.966 --> 00:02:30.000
Which of the following
expressions are equivalent

00:02:30.000 --> 00:02:32.634
to five over b.

00:02:33.132 --> 00:02:35.533
Select all that apply.

00:02:35.533 --> 00:02:36.933
All right, so this one over here.

00:02:36.933 --> 00:02:40.068
Negative five over negative b.

00:02:40.068 --> 00:02:41.700
Well we could remember that this negative,

00:02:41.700 --> 00:02:44.433
we could write this is the
same thing as negative five

00:02:44.634 --> 00:02:46.033
over negative b.

00:02:46.033 --> 00:02:47.933
And I just want to make it
clear we're that negative.

00:02:47.933 --> 00:02:49.234
So this is negative

00:02:49.234 --> 00:02:50.433
Instead of writing it negative

00:02:50.433 --> 00:02:51.334
in front of the entire fraction,

00:02:51.334 --> 00:02:53.867
I could essentially
multiply the negative one

00:02:53.867 --> 00:02:55.233
times just the numerator.

00:02:55.233 --> 00:02:58.333
So you could write this as
negative five over negative b.

00:02:58.333 --> 00:02:59.767
And negative divided by a negative

00:02:59.767 --> 00:03:00.867
is going to be a positive.

00:03:00.867 --> 00:03:02.966
So this actually is going to be equal to

00:03:03.100 --> 00:03:05.734
positive five over b, which
is what we're looking for.

00:03:06.034 --> 00:03:07.900
So this is going to be right.

00:03:08.300 --> 00:03:10.133
Now this one, negative
divided by a negative,

00:03:10.133 --> 00:03:11.434
well that's just going to be positive.

00:03:11.434 --> 00:03:13.668
So that's the same thing as five over b.

00:03:13.668 --> 00:03:16.100
One way to think about it
is that well the negatives

00:03:16.100 --> 00:03:17.966
kind of cancel each other out.

00:03:17.966 --> 00:03:20.967
So five over b, that looks good too.

00:03:20.967 --> 00:03:22.267
And of course I won't
select none of the above

00:03:22.267 --> 00:03:24.400
because I found two choices that worked.

00:03:24.633 --> 00:03:26.433
All right, let's do one more.

00:03:26.767 --> 00:03:30.085
Which of the following expressions

00:03:30.085 --> 00:03:33.733
are equal to negative e over negative f?

00:03:33.733 --> 00:03:35.666
And remember we just have to
take this step by step here.

00:03:35.666 --> 00:03:37.400
Actually let's try to just
simplify this directly.

00:03:37.400 --> 00:03:39.922
So negative e

00:03:39.922 --> 00:03:42.933
over negative f.

00:03:43.367 --> 00:03:44.967
Well we just need to remind ourselves

00:03:44.967 --> 00:03:46.566
that this part right over here.

00:03:46.566 --> 00:03:48.566
Negative e over negative f.

00:03:48.566 --> 00:03:50.333
Let me write an equal sign.

00:03:50.333 --> 00:03:53.100
Negative e over, and I'm
gonna put this negative.

00:03:53.100 --> 00:03:54.366
Let me do this in a different color.

00:03:54.366 --> 00:03:55.800
Let me do this in purple.

00:03:56.133 --> 00:03:57.133
So we have this purple.

00:03:57.133 --> 00:03:59.333
So we have that purple
negative right over there.

00:03:59.333 --> 00:04:01.133
And negative e over negative f.

00:04:01.133 --> 00:04:02.833
We've already talked
about this multiple times.

00:04:02.833 --> 00:04:04.601
That's the same thing as negative's

00:04:04.601 --> 00:04:06.100
divided by a negative is a positive.

00:04:06.100 --> 00:04:08.400
That's the same thing as e over f,

00:04:08.400 --> 00:04:10.367
as positive e over f.

00:04:10.367 --> 00:04:12.401
So this whole thing will simplify to

00:04:12.401 --> 00:04:16.034
negative e over f.

00:04:16.233 --> 00:04:18.546
So let's see which of
these choices are that.

00:04:18.930 --> 00:04:21.267
Well this right here is positive e over f.

00:04:21.267 --> 00:04:23.100
So that's not the choice.

00:04:23.100 --> 00:04:24.667
This one over here.

00:04:24.667 --> 00:04:27.200
This one we could write
it several ways actually.

00:04:27.200 --> 00:04:31.733
We could write it negative
negative e over f.

00:04:31.900 --> 00:04:35.100
Which of course is equal
to positive e over f.

00:04:35.100 --> 00:04:36.266
We could also write this.

00:04:36.266 --> 00:04:38.133
We could put the negative
in the denominator.

00:04:38.133 --> 00:04:39.533
We could say that this thing.

00:04:39.533 --> 00:04:40.567
Actually let me write it over here

00:04:40.567 --> 00:04:44.300
as negative e over negative f.

00:04:44.533 --> 00:04:46.200
This is also a legitimate thing to do.

00:04:46.200 --> 00:04:47.400
You could take this negative

00:04:47.400 --> 00:04:50.333
and multiply it times the denominator.

00:04:50.333 --> 00:04:51.400
Right over here.

00:04:51.400 --> 00:04:54.966
But either way it's going to
be equal to positive e over f.

00:04:55.335 --> 00:04:58.766
These two are actually evaluate
to the same expression.

00:04:58.766 --> 00:05:00.400
So here, I would select.

00:05:00.400 --> 00:05:01.900
Finally, I would select.

00:05:01.900 --> 00:05:04.868
I've been waiting to
select "None of the above."

00:05:04.868 --> 00:05:06.733
All right, hopefully that helps.


Interpreting multiplicationa and division of negative numbers

WEBVTT
Kind: captions
Language: en

00:00:00.287 --> 00:00:02.771
- Grandma Millie is shrinking.

00:00:02.771 --> 00:00:07.771
Her height decreases by a
quarter centimeter each year.

00:00:08.007 --> 00:00:11.026
She want to predict the
total change in her height

00:00:12.687 --> 00:00:15.040
over the next three years.

00:00:15.228 --> 00:00:16.842
Which of the following equations

00:00:16.842 --> 00:00:19.162
matches the situation above?

00:00:19.349 --> 00:00:21.068
Her height is decreasing.

00:00:21.068 --> 00:00:23.808
We can view that as either
changing her height each year

00:00:23.808 --> 00:00:26.222
is a negative fourth of a centimeter.

00:00:26.222 --> 00:00:27.963
That's going to happen for three years.

00:00:27.963 --> 00:00:30.703
We multiply negative
one-fourth times three.

00:00:30.703 --> 00:00:33.049
Which is exactly what's
happening in this first choice

00:00:33.049 --> 00:00:34.139
right over here.

00:00:34.139 --> 00:00:38.202
Her height is decreasing by a
quarter centimeter each year,

00:00:38.202 --> 00:00:40.048
that's this part right over here.

00:00:41.186 --> 00:00:43.397
She's doing this for three years.

00:00:43.397 --> 00:00:44.371
After three years,

00:00:44.371 --> 00:00:47.081
her change in height would
be negative three-fourth

00:00:47.081 --> 00:00:48.218
of a centimeter.

00:00:48.218 --> 00:00:49.448
You just multiply this out,

00:00:49.448 --> 00:00:53.382
which means that she lost
three-fourths of a centimeter.

00:00:53.628 --> 00:00:55.404
The second choice would evaluate,

00:00:55.404 --> 00:00:57.983
if you multiply positive
one-fourth times negative three,

00:00:57.983 --> 00:00:59.700
you still get negative three-fourths,

00:00:59.700 --> 00:01:02.102
but this doesn't seem
to follow as naturally

00:01:02.102 --> 00:01:03.658
from this statement up here.

00:01:03.658 --> 00:01:05.318
We're decreasing by a fourth.

00:01:05.318 --> 00:01:06.805
I like putting the negative
in front of the fourth,

00:01:06.805 --> 00:01:09.601
and then we're going in
forward in time by three years.

00:01:09.601 --> 00:01:12.144
It feels weird to put a
negative on the three.

00:01:12.897 --> 00:01:14.850
We're not backwards in time here.

00:01:14.850 --> 00:01:17.682
Even though this expression
evaluates to the same

00:01:17.682 --> 00:01:18.843
things as this first one,

00:01:18.843 --> 00:01:20.619
the first one seems to better match

00:01:20.619 --> 00:01:23.440
what this statement is describing.

00:01:23.440 --> 00:01:25.147
I wouldn't pick none of the above

00:01:25.147 --> 00:01:27.225
just because I found one that I like,

00:01:27.225 --> 00:01:29.500
that I think that describes
the equation that matches

00:01:29.500 --> 00:01:31.240
the situation above.

00:01:31.485 --> 00:01:32.946
Let's do another one of these.

00:01:33.783 --> 00:01:37.009
A zombie lost three limbs
each day for four days.

00:01:37.009 --> 00:01:39.276
He's going to lose 12
limbs, three times four.

00:01:41.029 --> 00:01:41.984
It's not a human zombie.

00:01:41.984 --> 00:01:45.433
It's an alien zombie that has
at least 12 limbs to lose.

00:01:45.433 --> 00:01:47.929
The following equation
describes this situation.

00:01:48.713 --> 00:01:51.590
The zombie lost three limbs each day.

00:01:51.590 --> 00:01:52.769
Lost three limbs.

00:01:52.769 --> 00:01:54.291
A negative three could represent,

00:01:54.291 --> 00:01:56.763
hey I'm losing three limbs.

00:01:56.763 --> 00:01:58.109
The negative is going ...

00:01:58.109 --> 00:02:01.235
my number of limbs is going down by three.

00:02:01.235 --> 00:02:02.699
This is negative three.

00:02:04.114 --> 00:02:06.166
It's dong that for four days.

00:02:06.166 --> 00:02:07.441
For four days.

00:02:07.500 --> 00:02:08.697
We have negative 12,

00:02:08.697 --> 00:02:11.820
which is the total number of
limbs that our zombie loses.

00:02:13.035 --> 00:02:14.222
What does negative 12 tell us?

00:02:14.222 --> 00:02:15.187
Well, I just said it.

00:02:15.187 --> 00:02:19.482
The zombie lost a total of 12
limbs over this time period.

00:02:19.482 --> 00:02:20.630
That looks right.

00:02:20.630 --> 00:02:22.070
Now, let's see what are these other one.

00:02:22.070 --> 00:02:24.450
The zombie has negative 12 limbs.

00:02:25.219 --> 00:02:25.868
Well, no.

00:02:26.391 --> 00:02:27.339
To have negative 12 ...

00:02:27.339 --> 00:02:29.801
I don't even know a
negative limb would be.

00:02:29.801 --> 00:02:32.296
We don't know how many limbs
the zombie started with.

00:02:32.296 --> 00:02:33.261
The zombie, I don't know,

00:02:33.261 --> 00:02:35.792
it might be some type of
centipede zombie or something.

00:02:35.792 --> 00:02:37.811
It might have had a hundred limbs,

00:02:37.811 --> 00:02:39.692
and if it lost 12 of them,

00:02:39.692 --> 00:02:43.059
it would now have 88 limbs.

00:02:43.059 --> 00:02:44.695
The zombie might have had 12 limbs,

00:02:44.695 --> 00:02:46.298
and now it has a zero limbs.

00:02:46.298 --> 00:02:48.932
In fact, I don't even know
how you get to negative limbs.

00:02:48.932 --> 00:02:50.778
I don't think this makes sense.

00:02:50.778 --> 00:02:51.917
I'm not going to pick none of the above,

00:02:51.917 --> 00:02:53.832
because I agree with this first choice.

00:02:53.832 --> 00:02:55.734
Let's do one more of these.

00:02:58.492 --> 00:03:01.993
All right, a certain reality TV show

00:03:01.993 --> 00:03:04.013
lost, so we're losing,

00:03:04.013 --> 00:03:07.135
so a negative might be a nice
way to represent the lost,

00:03:07.135 --> 00:03:10.467
lost a total of 9,000 viewers over

00:03:10.467 --> 00:03:12.625
the past three months.

00:03:12.987 --> 00:03:14.021
Over three months.

00:03:14.021 --> 00:03:16.319
It lost the same number
of viewers each month.

00:03:16.749 --> 00:03:19.778
The following equation
describes the situation.

00:03:20.654 --> 00:03:23.215
It lost a total of 9,000 viewers,

00:03:23.215 --> 00:03:25.360
and it did it over three months.

00:03:25.606 --> 00:03:27.287
What they're figuring out here is,

00:03:28.347 --> 00:03:30.066
what was the change in viewership.

00:03:30.066 --> 00:03:32.840
What was the change in
viewership per month?

00:03:32.840 --> 00:03:33.780
It tells us ...

00:03:33.780 --> 00:03:36.705
the negative 3,000 tells us
that they lost on average ...

00:03:37.780 --> 00:03:39.752
it actually tells us they
lost the same number.

00:03:40.321 --> 00:03:40.832
This is actually ...

00:03:40.832 --> 00:03:42.051
it's the same number every month.

00:03:42.051 --> 00:03:45.707
They actually lost 3,000
viewers every month.

00:03:45.707 --> 00:03:48.411
Let's see which of these
choices match up to that.

00:03:48.587 --> 00:03:51.245
There are 3,000 fewer
people watching the show

00:03:51.245 --> 00:03:53.207
compared to three months ago.

00:03:53.207 --> 00:03:55.855
No, they lost 3,000 every month.

00:03:55.855 --> 00:03:56.852
That's what this is telling us.

00:03:56.852 --> 00:03:58.431
Every month for three months.

00:03:58.431 --> 00:03:59.872
There's a same number each month.

00:03:59.872 --> 00:04:01.764
They lost 3,000 every
month for three months

00:04:01.764 --> 00:04:03.307
for a total of 9,000.

00:04:03.307 --> 00:04:04.410
This isn't right.

00:04:04.979 --> 00:04:06.539
The total change in the number of people

00:04:06.539 --> 00:04:09.940
watching the show is negative 3,000.

00:04:09.940 --> 00:04:11.647
Over a month that would be true,

00:04:11.647 --> 00:04:13.481
but the total change was 9,000,

00:04:14.650 --> 00:04:17.149
or a negative 9,00 I guess I could say.

00:04:17.149 --> 00:04:18.419
They lost 9,000.

00:04:18.419 --> 00:04:20.716
I would say none of the above.

00:04:20.847 --> 00:04:22.657
Well, the negative 3,000 tells us

00:04:22.657 --> 00:04:24.003
that there was a choice that said,

00:04:24.003 --> 00:04:27.416
each month they lost 3,000 viewers.

00:04:27.416 --> 00:04:29.227
That's the choice that I would pick.

00:04:29.995 --> 00:04:33.139
I would say none of the above.


Breakthrough Junior Challenge

WEBVTT
Kind: captions
Language: en

00:00:02.858 --> 00:00:05.378
- Hi, this is Sal Khan,
founder of the Khan Academy.

00:00:05.378 --> 00:00:07.714
And I just wanted to let you
know about the Breakthrough

00:00:07.714 --> 00:00:11.371
Junior Challenge in
conjunction with Khan Academy.

00:00:11.371 --> 00:00:12.854
And for those of you who aren't familiar

00:00:12.854 --> 00:00:14.480
with the Breakthrough Prize,

00:00:14.480 --> 00:00:16.755
it's an incredibly prestigious
set of prizes awarded

00:00:16.755 --> 00:00:20.717
every year to the world's
leading mathematicians,

00:00:20.717 --> 00:00:23.481
biologists, chemists, physicists.

00:00:23.481 --> 00:00:26.082
And the idea is to award
people who are literally

00:00:26.082 --> 00:00:28.479
moving humanity forward
through the sciences

00:00:28.479 --> 00:00:30.165
and through mathematics.

00:00:30.165 --> 00:00:31.851
And what we're going this year
is we're going to extend it

00:00:31.851 --> 00:00:35.082
even further where we're going
to have a Breakthrough Prize

00:00:35.082 --> 00:00:36.321
for high school students.

00:00:36.321 --> 00:00:39.023
That's what the Breakthrough
Junior Challenge is.

00:00:39.023 --> 00:00:43.554
And the idea is, is to award
people who can explain things

00:00:43.554 --> 00:00:45.444
in a new and innovative ways.

00:00:45.444 --> 00:00:47.678
Ways that can open up all of our minds.

00:00:47.678 --> 00:00:49.445
Because the big breakthroughs,
the types of things

00:00:49.445 --> 00:00:51.925
that the Breakthrough Prizes award,

00:00:51.925 --> 00:00:53.754
these all start with new insights,

00:00:53.754 --> 00:00:55.724
new ways of looking at the universe.

00:00:55.724 --> 00:00:58.994
And these new ways of looking
at the universe often happen

00:00:58.994 --> 00:01:01.332
with the people who are
just starting to look at it.

00:01:01.332 --> 00:01:02.712
It might happen with young people,

00:01:02.712 --> 00:01:04.256
especially high school students.

00:01:04.256 --> 00:01:05.944
So we're going to give this prize,

00:01:05.944 --> 00:01:07.507
it's going to include a scholarship,

00:01:07.507 --> 00:01:09.336
it's going to include a
prize for your teacher,

00:01:09.336 --> 00:01:11.753
and a possible lab for your school.

00:01:11.753 --> 00:01:14.233
And we're going to give this
for students who can explain

00:01:14.233 --> 00:01:17.422
either and existing concept in a new way,

00:01:17.422 --> 00:01:18.986
in a way that opens our minds.

00:01:18.986 --> 00:01:22.075
Or a new idea that they came up with,

00:01:22.075 --> 00:01:24.492
or a new discovery that they came up with.

00:01:24.492 --> 00:01:26.910
So, it's a very exciting
thing to be part of.

00:01:26.910 --> 00:01:29.837
I really believe that we're
going to hopefully inspire

00:01:29.837 --> 00:01:32.661
another generation of potential
Breakthrough Prize winners.

00:01:32.661 --> 00:01:36.479
And I think even this competition

00:01:36.479 --> 00:01:38.348
is going to open our minds.

00:01:38.348 --> 00:01:41.945
It's going to make us all
see the world in a new

00:01:41.945 --> 00:01:43.245
and exciting way.

00:01:43.245 --> 00:01:47.898
And if you win, you will come
out here to Silicon Valley,

00:01:47.898 --> 00:01:51.068
and you will share the
stage with both people

00:01:51.068 --> 00:01:54.297
from Silicon Valley, leaders
in the tech community,

00:01:54.297 --> 00:01:55.333
leaders in science.

00:01:55.333 --> 00:01:58.381
You'll share a stage with the winners

00:01:58.381 --> 00:01:59.804
of the Breakthrough Prize.

00:01:59.804 --> 00:02:02.202
And there'll be a few
Hollywood celebrities out here.

00:02:02.202 --> 00:02:04.233
It's going to be televised
on national television.

00:02:04.233 --> 00:02:07.097
So, this is going to be a
really, really, really fun event.

00:02:07.097 --> 00:02:08.946
But most importantly, I
think we might be able

00:02:08.946 --> 00:02:10.591
to move society forward.

00:02:10.591 --> 00:02:11.857
Thank you.


Categorizing geometric shapes: practice examples

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:04.350
Which side is
perpendicular to side BC?

00:00:04.350 --> 00:00:07.720
So BC is this line
segment right over here.

00:00:07.720 --> 00:00:10.350
And for another segment
to be perpendicular to it,

00:00:10.350 --> 00:00:12.640
perpendicular just means
that the two segments

00:00:12.640 --> 00:00:16.540
need to intersect at a right
angle, or at a 90-degree angle.

00:00:16.540 --> 00:00:21.640
And we see that BC intersects
AB at a 90-degree angle.

00:00:21.640 --> 00:00:24.100
This symbol right over here
represents a 90-degree,

00:00:24.100 --> 00:00:25.740
or a right angle.

00:00:25.740 --> 00:00:28.670
So we just have to
find side AB or BA.

00:00:28.670 --> 00:00:29.860
And that's right over here.

00:00:29.860 --> 00:00:34.650
Side AB is perpendicular
to side BC.

00:00:34.650 --> 00:00:37.530
Let's do a few more of these.

00:00:37.530 --> 00:00:41.690
Put the triangles into
the correct categories,

00:00:41.690 --> 00:00:42.940
so this right over here.

00:00:42.940 --> 00:00:43.200
So let's see.

00:00:43.200 --> 00:00:44.575
Let's think about
our categories.

00:00:44.575 --> 00:00:47.410
Right triangles-- so that means
it has a 90-degree angle in it.

00:00:47.410 --> 00:00:50.590
Obtuse triangles-- that
means it has an angle larger

00:00:50.590 --> 00:00:52.920
than 90 degrees in it.

00:00:52.920 --> 00:00:55.810
Acute triangles-- that
means all three angles

00:00:55.810 --> 00:00:58.580
are less than 90 degrees.

00:00:58.580 --> 00:01:00.029
So this one has a
90-degree angle.

00:01:00.029 --> 00:01:01.570
It has a right angle
right over here.

00:01:01.570 --> 00:01:04.349
So this is a right triangle.

00:01:04.349 --> 00:01:07.040
This one right over
here, all of these angles

00:01:07.040 --> 00:01:10.960
are less than 90 degrees,
just eyeballing it.

00:01:10.960 --> 00:01:13.030
So this is going to
be an acute-- that's

00:01:13.030 --> 00:01:14.280
going to be an acute triangle.

00:01:14.280 --> 00:01:17.080
I'll put it under acute
triangles right over there.

00:01:17.080 --> 00:01:19.470
Then this one over here,
this angle up here,

00:01:19.470 --> 00:01:22.630
this is-- and we can
assume that these actually

00:01:22.630 --> 00:01:26.564
are drawn to scale, this is more
open than a 90-degree angle.

00:01:26.564 --> 00:01:28.230
This is an obtuse
angle right over here.

00:01:28.230 --> 00:01:30.290
It's going to be
more than 90 degrees.

00:01:30.290 --> 00:01:34.090
So this is an obtuse triangle.

00:01:34.090 --> 00:01:38.030
Now, this one over here,
all of them seem acute.

00:01:38.030 --> 00:01:40.280
None of them even seem
to be a right angle.

00:01:40.280 --> 00:01:46.274
So I would put this again
into acute-- acute triangles.

00:01:46.274 --> 00:01:47.940
This one here clearly
has a right angle.

00:01:47.940 --> 00:01:49.680
It's labeled as such.

00:01:49.680 --> 00:01:51.600
So we'll throw it
right over here.

00:01:51.600 --> 00:01:53.580
And then this one, this
angle right over here

00:01:53.580 --> 00:01:54.740
is clearly even larger.

00:01:54.740 --> 00:01:56.660
It has a larger measure
than a right angle.

00:01:56.660 --> 00:01:59.170
So this angle right over
here is more than 90 degrees.

00:01:59.170 --> 00:02:02.060
It's going to be
an obtuse angle.

00:02:02.060 --> 00:02:07.770
So we will throw it into
obtuse-- obtuse triangles.

00:02:07.770 --> 00:02:10.810
So we got two in each of these.

00:02:10.810 --> 00:02:12.700
And let's check our answer.

00:02:12.700 --> 00:02:14.580
We got it right.