Sal Khan talks with OPB's Rob Manning

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Kind: captions
Language: en

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Rob Manning: For the few
people who aren't familiar

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with your work how do
you describe what you do

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and what you do online, or
what you do in education?

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Sal Khan: Khan Academy is
most known for this collection

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of videos that I somewhat famously started

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making for my cousins, and
then they caught on, and

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people who were not my
cousins started watching

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them, but by the time it
kind of got serious traction,

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and I had quit my former
job in finance to do

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this full-time, and set
it up as a not-for-profit.

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Our support has really
kind of allowed us to focus

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on not only the videos,
not only translating the

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videos, but trying to
build an entire interactive

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platform so students
can get instruction, do

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practice, if they have a
teacher or parent who can

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be their coach. Those folks
will get information on

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where the students are at,
what are their strengths,

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what are their weaknesses.
And so, the Khan Academy

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on one level it's kind of
free tutoring for anybody.

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It's a not-for-profit, everything is free.

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Our mission is a free
world class education for

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anyone anywhere, but on
a deeper level we've kind

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of fallen into this situation
and it kind of speaks

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to where we are in history,
maybe, where it seems

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like we're a catalyst for
helping people think about

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what a classroom could
be now. I've never been a

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big believer that it should
be virtual only, but it feels

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like now the virtual can handle
certain parts of the equation.

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It frees up the classroom to do
maybe more interesting things.

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Rob: And what dimensions
exactly do you think virtual

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education or online does
arguably better than

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direct instruction from
a classroom teacher?

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Sal: It's kind of a
careful answer just because

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I think the ideal is you
have both. The real ideal

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is if you had a one-on-one
with a human being,

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but that's not practical,
you know, only kind of

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lords and dukes had that
hundreds of years ago.

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What we're seeing is students
when they are getting

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together with people in
the physical environment

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we think the best use of
that is not the lecture,

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the one pace fits all
lecture, the best use of

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that time is for the
human beings to actually

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interact. They're all
in the room together.

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And so, where online
strength is, is on delivering

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what used to be that lecture
in a way that it's not

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one way pace fits all any
more, so that a student

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can pause and repeat at their own time,

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at their own pace. If
they're embarrassed by

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maybe they're in 8th grade
and they forgot something

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from 4th grade they don't
have to have the guts

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to raise their hand in
class. They can just

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look at those concepts
and no one is going to

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judge them. If they're
ready to move ahead they

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can move ahead. Software
is also really good at

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kind of generating exercises,
grading the exercises,

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doing a lot of the
logistics that frankly take

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teachers away from the
actual teaching part of

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their job, and so what we're
saying is in a traditional

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classroom a lot of a
teacher's time is spent

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giving this lecture to
a bunch of students.

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Often times the classroom
is fairly passsive at

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those moments. A lot of
the teacher's time is

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kind of the administrative
work of writing exams,

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grading exams, and maybe
10-15% of their time is

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spent with the actual
one-on-one interaction,

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or small group interaction with students.

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We're saying with technology
we can take a lot of

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the information delivery,
we can take a lot of

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the logistics, a lot of
the problems, set creation,

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and grading off of a
teacher's plate. A 100 percent

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of their time could be
the one-on-one connections

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with students in the
small group interactions.

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Rob: I wonder is part of
the issue that it's the

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sort of one too many
kind of dissemination of

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information which tends
to be the prevailing

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model of instruction that
is somewhat imperfect

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anyway. It's sort of,
like, whether you're doing

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it the lecture hall in
front of 500 people,

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or you've got 500 students online.

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Sal: Yeah, and the irony here and

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you're talking to someone who has sat

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down and made 3,000
videos, or 3,200 now, but

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the most important part
of an education is not the

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lecture/videos. The most
important part of the education

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is doing problem solving,
interacting with people,

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and asking questions and answering other

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peoples questions, and
in traditional classrooms

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that tends to get squeezed
out. You get a little bit

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out of it, but it gets
squeezed out by the passive

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lecture, so I agree with
you. I think a lecture is

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an imperfect thing. I
think it gets a little bit

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better when you make it
on demand, when you allow

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it to cater to a student's
pace, and then technology

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can also understand what's working,

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what's not working. You can
start to collect data on

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what's effective, what's
not. Traditional lecture

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gets very little of that
data. Textbooks get very

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little of that data. Now
with online you can have it.

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But I think the main focus
is, and it's a bit of the

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irony here is that the
technology enables the

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classroom to become more
interactive, more human,

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and for me, you know, there
are always these debates.

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Is technology good? Is it a
cure all? Is it a silver bullet?

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Or, is it bad? and, you
know, [unintelligible]

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For me the question isn't should or should

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you not use technology. The
question is well, what is

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the ideal learning
experience? And you try to

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answer that question. To
me it's, you know, it's

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experience focused on
the student, cater to the

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students needs, works
at the students pace,

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and then when they're with other

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human beings they're
interacting with them, and so

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you say, okay, that's the
ideal. Now can we do it?

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and 30, 50, 100 years ago
you couldn't economically

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do it, you would have to
be quite well-off to have

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that type of attention
to personalize education,

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but now I think the
answer is we can do it,

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we can personalize it in a practical way.

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Rob: The instructors
have not been brought up

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obviously, in an online
environment, the vast

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majority of them, but
they have been brought up

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with the concept of
differentiated instruction,

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and I wonder there are
some who would argue

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well, I can adapt my lecture
to students questions,

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to blank stares in the
desks that I'm looking at.

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Are there instances where
perhaps in a smaller

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environment where maybe a
lecture can be equal to,

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or better than, you know, a
video lecture style of delivery?

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Sal: Yeah absolutely. There
are certain circumstances.

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I think if you have a
very small seminar type

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situation that's highly
interactive than that wins,

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and actually I would say
that's not a lecture any more,

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that's actually what we would want to see

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class time used for. I think there is also

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moments where shared
experience is very powerful,

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and I would say that falls
more in the category of

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inspiring, your TED talks
and inspirational speech.

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I think there is value
of having many people in

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the room and having that
shared moment, but I think

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for things that are
intellectually deep, things that

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are hard for people to
get their head around,

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some person might get it
faster than another person.

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I mean me and you we can watch a speech,

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a high level speech and
we'll all get it, but

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if we're doing calculus,
we're doing chemistry,

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you might get it faster
than me. I might get it

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faster than you. It's
nearly impossible. Frankly,

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if you get even more than
10 or 15 students to cater

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to where every student is,
and the only information

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that a teacher currently
has is trying to read

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the blank stares, trying
to read how engaged the

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students are, and you
know, there have been

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studies that the average
lecturer is teaching

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at the 23rd percentile,
because they're making

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sure that no one is falling
behind, and I point out

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you might say, well at least 23% of the

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students are getting
it, but the reality is

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77% are bored, and then 22% it's

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going to fast for, so you're only catering

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to 1%, and so I won't make a blanket

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statement that it's always
better, but I will say

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that most of the time
when you have human beings

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together it's better for
them to interact with

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each other, to ask questions
of each other, than

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for them to just kind
of sit passively and be

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given a lecture. And a lot of times when

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people say, well in a
live lecture you can ask

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questions, and I say, yeah absolutely.

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Today in a traditional
lecture hall that's maybe

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5% of class time, and that's usually

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only the bravest people in the class, and

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sometimes even the question
might not be appropriate

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for everyone. We'd like
to see where it's 100%

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of the class time, and it's not just

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asking the teacher questions,
it's not just asking

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the TA questions, but it's
also asking your peers

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questions where the
students themselves get

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that deeper experience of explaining
things to their colleagues.

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Rob: You see there still
being a 50/50 split,

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and maybe that's not the
right percentage, but

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at least a part that's online,
a part that's in class,

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have you heard people
or what's your response

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to people saying, well,
you know, maybe we can

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do more of the discussion
part online as well,

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whether it's through
chat rooms and skyping,

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or whatever, what's the
problem, or why should

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that remain in the classroom,
or why not, you know,

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ship that into the online
environment as well?

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Sal: I think we should
all experiment on what

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type of social experiences,
constructive social

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experiences you can
have online, and I think

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in some domains, you
know, doing skype could be

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very valuable for foreign
language. I mean the

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goal standard has always
been immersion. If you

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can't afford immersion
you skype with somebody

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who speaks that language
and maybe you could get

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interesting results, you
know, we see on our site

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we have 60 million students
use the site every month.

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We want them to interact
with each each other.

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We want them to answer their
questions. With that said,

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and that's incredibly
valuable for someone who

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maybe does not have
someone to interact with.

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They don't have access to a kind of a rich

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environment, but I think
the technology would have

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to go a very, very long
way to come close to

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what you could get with
a very rich classroom

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environment that's led
by an inspiring teacher.

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But when I say led, not
meaning lectured by an

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inspiring teacher, but
really mentored and coached

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by an inspiring teacher.

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Rob: Given that these are a different set

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of skills from teachers,
from what teachers have

00:09:37.676 --> 00:09:40.752
been really brought up
in, do you have a vision

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for how to kind of get to
the sort of 21st century

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flipped classroom instructor that

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we would need to do this?

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Sal: Yeah, and it's an open question.

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I don't know 100%, I mean, it's actually

00:09:52.846 --> 00:09:55.309
exciting to be here at a school that is so

00:09:55.309 --> 00:09:58.995
focused on education. I
suspect a lot of the answer

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might be for the future
teachers to experience

00:10:01.814 --> 00:10:04.000
it themselves, for them
to understand it, and

00:10:04.000 --> 00:10:06.077
for them to viscerally
feel the difference between

00:10:06.077 --> 00:10:08.588
being lectured at, and
being able to engage with

00:10:08.588 --> 00:10:10.344
the content, and have
conversations with their

00:10:10.344 --> 00:10:13.179
peers, and discuss it
at a deeper level, but

00:10:13.179 --> 00:10:15.059
I think we're just at
the very early stages.

00:10:15.059 --> 00:10:19.051
In my opinion it seems
like a lot of the adoption

00:10:19.051 --> 00:10:20.783
is happening faster than
I would have guessed.

00:10:20.783 --> 00:10:23.175
We're being used in 20,000
classrooms right now.

00:10:23.175 --> 00:10:25.427
This is really just over
the last couple of years,

00:10:25.427 --> 00:10:28.260
and we have two people who
engage directly with classrooms.

00:10:28.260 --> 00:10:30.268
We're a very small
organization, so it seems like

00:10:30.268 --> 00:10:32.173
there's a hunger out there.
A lot of the teachers

00:10:32.173 --> 00:10:34.130
we talked to this is kind
of the direction they've

00:10:34.130 --> 00:10:35.698
always wanted to be in, they've wanted to.

00:10:35.698 --> 00:10:37.176
They got into the profession
because they wanted

00:10:37.176 --> 00:10:39.759
to connect with students.
They wanted to be

00:10:39.759 --> 00:10:42.012
that teacher that takes
that student from a

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certain attitude to where
they're taking ownership

00:10:44.074 --> 00:10:46.261
of their life, and they
found it difficult when they

00:10:46.261 --> 00:10:48.842
have 30 faces that they
have to kind of lecture at.

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Rob: All right, and one
thing that I've heard

00:10:53.562 --> 00:10:57.330
as sort of a concern from
teachers is that there

00:10:57.330 --> 00:11:04.058
might be a ... Maybe I'll put it this way.

00:11:04.058 --> 00:11:06.679
The concerns have not
come just from teachers,

00:11:06.679 --> 00:11:10.260
but that online learning
often gets sold as

00:11:10.260 --> 00:11:13.925
a cost saving, you know,
is something that we ...

00:11:13.925 --> 00:11:16.649
Public education is
very expensive. What can

00:11:16.649 --> 00:11:20.552
we do to save money? Do
you think it is a viable

00:11:20.552 --> 00:11:22.845
way to do education less expensively?

00:11:22.845 --> 00:11:28.398
Sal: I kind of ... I don't
know, is the simple answer.

00:11:28.398 --> 00:11:30.507
I mean, I know for my own
kids I don't want them

00:11:30.507 --> 00:11:33.865
online only, and I, you
know the way I view it is if

00:11:33.865 --> 00:11:36.291
education was perfect
we should think about

00:11:36.291 --> 00:11:38.348
how we can then, you
know, if we're making a

00:11:38.348 --> 00:11:40.098
great car we can then think
about how to streamline

00:11:40.098 --> 00:11:42.926
the factory or whatever
we're doing, but, you know,

00:11:42.926 --> 00:11:45.046
the reality and there's a million alarmist

00:11:45.046 --> 00:11:46.870
articles that come out
about all the reasons

00:11:46.870 --> 00:11:49.390
why we should worry about
education, and so in my mind

00:11:49.390 --> 00:11:52.317
I would not try to
de-resource the problem.

00:11:52.317 --> 00:11:54.650
I would try to maybe hold resources costs,

00:11:54.650 --> 00:11:56.618
or maybe even potentially
increase them, but

00:11:56.618 --> 00:11:59.347
try to get, but try to
have better outcomes,

00:11:59.347 --> 00:12:02.055
and so, I view technology
as much more of given

00:12:02.055 --> 00:12:04.341
whatever resources you
have and it's hard to

00:12:04.341 --> 00:12:06.468
argue that you're ever
putting too many resources

00:12:06.468 --> 00:12:08.507
in education. Given
whatever resources you have

00:12:08.507 --> 00:12:10.344
how do we actually optimize outcomes?

00:12:10.344 --> 00:12:13.923
Rob: You've had a good
time doing the videos.

00:12:13.923 --> 00:12:17.208
I mean, I don't think
you'd do 3,000 of them

00:12:17.208 --> 00:12:18.756
if you didn't like doing them.

00:12:18.756 --> 00:12:22.094
Sal: Yes, no. I like
doing them a lot, yes.

00:12:22.094 --> 00:12:24.149
Rob: Do you have any particular favorites

00:12:24.149 --> 00:12:27.478
that you were sort of
like, you know, or have

00:12:27.478 --> 00:12:29.502
you heard from people
what are particularly

00:12:29.502 --> 00:12:30.819
fond of a certain video?

00:12:30.819 --> 00:12:32.723
Sal: Oh, you know,
there's a bunch. I mean,

00:12:32.723 --> 00:12:34.322
well I think one of the ... people ask me.

00:12:34.322 --> 00:12:36.222
What's the trick? How do you ... ? and

00:12:36.222 --> 00:12:38.845
I say the video that
I'm doing I could try to

00:12:38.845 --> 00:12:40.800
convince myself that's
my favorite video, so

00:12:40.800 --> 00:12:42.755
that you literally have,
like, you know, you

00:12:42.755 --> 00:12:44.180
hear the energy, you hear the excitement,

00:12:44.180 --> 00:12:46.038
and you can't understate
how important those

00:12:46.038 --> 00:12:48.173
cues are for the student.
If the lecturer themselves

00:12:48.173 --> 00:12:52.095
does not care about it the
student immediately ...

00:12:52.095 --> 00:12:53.934
We're very sensitive to
that. If you're saying

00:12:53.934 --> 00:12:56.344
the next step in this
equation, I mean, whose

00:12:56.344 --> 00:12:58.315
going to care about
that, and so I think it's

00:12:58.315 --> 00:13:00.592
very important for the
lecturer themselves to

00:13:00.592 --> 00:13:03.013
feel excited that they get to do this, but

00:13:03.013 --> 00:13:05.179
there's all sorts of
topics. I'm a real big fan

00:13:05.179 --> 00:13:07.930
of the cosmology videos.
It was a reason for me

00:13:07.930 --> 00:13:10.173
to learn a lot of cosmology
and really understand

00:13:10.173 --> 00:13:12.180
it, and it's something
that always fascinated me.

00:13:12.180 --> 00:13:15.754
But I get into the finance
videos. My whole life

00:13:15.754 --> 00:13:18.027
I was an analyst at an
investment firm, and

00:13:18.027 --> 00:13:20.041
obviously there's a
certain part of me that

00:13:20.041 --> 00:13:22.304
gets very passionate
about that. I love doing

00:13:22.304 --> 00:13:23.875
the history videos. I
love doing ... I mean

00:13:23.875 --> 00:13:25.673
I like doing it all.

00:13:25.673 --> 00:13:28.141
Rob: You get some pushback
from educators who

00:13:28.141 --> 00:13:30.373
are, like, well he didn't
go to education school,

00:13:30.373 --> 00:13:33.682
he didn't, you know,
do you feel like maybe

00:13:33.682 --> 00:13:37.817
teaching is not as hard as
people make it out to be?

00:13:37.817 --> 00:13:41.513
Sal: Well no, actually
I think it's probably

00:13:41.513 --> 00:13:43.340
harder than people make it out to be.

00:13:43.340 --> 00:13:45.931
And, you know, depending
on what you call a

00:13:45.931 --> 00:13:48.304
teacher it's, you know, you can determine

00:13:48.304 --> 00:13:50.595
whether I am one, or whether
I'm a content creator.

00:13:50.595 --> 00:13:54.762
I don't have to, or I
mean depending on how

00:13:54.762 --> 00:13:57.375
you do it, you know, I
don't have 30 live students

00:13:57.375 --> 00:13:59.549
that I'm trying to cater
to that I'm ... Some of

00:13:59.549 --> 00:14:00.869
them might have some problem at home.

00:14:00.869 --> 00:14:02.929
Some of them have some
issue. Some are struggling.

00:14:02.929 --> 00:14:08.813
That is a nuanced and
deep and important role

00:14:08.813 --> 00:14:12.255
that I am not doing when I
just make a nice resource,

00:14:12.255 --> 00:14:14.624
when I make this nice
video. Hopefully, it's an

00:14:14.624 --> 00:14:16.388
engaging video and gets
people excited about

00:14:16.388 --> 00:14:18.598
the subject matter and, you
know, I want to stress this.

00:14:18.598 --> 00:14:21.426
I recently met Wendy Kopp, at dinner, the

00:14:21.426 --> 00:14:24.300
founder of Teach for America, and she was

00:14:24.300 --> 00:14:25.962
telling me that the
single biggest determinant

00:14:25.962 --> 00:14:28.373
of the teachers that they
have, and they studied

00:14:28.373 --> 00:14:30.209
this very carefully. What are the teachers

00:14:30.209 --> 00:14:32.343
that are showing outsized
gains? And most people

00:14:32.343 --> 00:14:34.473
think it's going to be
content delivery, and

00:14:34.473 --> 00:14:36.096
there is some correlation. If someone can

00:14:36.096 --> 00:14:38.007
deliver a good lecture
then their students tend

00:14:38.007 --> 00:14:40.285
to do well, but the thing that really

00:14:40.285 --> 00:14:42.394
moves the dial is the
teachers ability to change

00:14:42.394 --> 00:14:44.677
the students mindsets.
To make the students

00:14:44.677 --> 00:14:46.384
realize that they really
have to take ownership

00:14:46.384 --> 00:14:48.173
of their learning, that
they really have to

00:14:48.173 --> 00:14:52.130
start thinking about
their futures, and when

00:14:52.130 --> 00:14:53.846
she said that I was, like,
well I need to quote you,

00:14:53.846 --> 00:14:55.461
and I'm quoting her
right now because that's

00:14:55.461 --> 00:14:56.997
very consistent with what we're saying.

00:14:56.997 --> 00:14:59.594
Instead of 5 or 10% of a teacher's time

00:14:59.594 --> 00:15:02.875
being that coach, that
inspiration, and 90%

00:15:02.875 --> 00:15:05.256
being the content delivery we want 100%

00:15:05.256 --> 00:15:07.068
of a teacher's time being that coach, that

00:15:07.068 --> 00:15:09.214
inspiration, that person
in the student's life

00:15:09.214 --> 00:15:11.818
that can change their
mindset, that can change

00:15:11.818 --> 00:15:14.209
their outlook, that can
be that mentor, and that's

00:15:14.209 --> 00:15:16.675
something I'll never be able
to do as a video creator,

00:15:16.675 --> 00:15:18.829
and you need a live human being for.

00:15:18.829 --> 00:15:20.782
Rob: Do you think
there's much of a stretch

00:15:20.782 --> 00:15:24.386
between standing in front
of a group of 25 students,

00:15:24.386 --> 00:15:27.133
and doing an exercise at the board versus

00:15:27.133 --> 00:15:30.184
doing an exercise in
a YouTube video that's

00:15:30.184 --> 00:15:31.465
going to be seen at home?

00:15:31.465 --> 00:15:34.703
Sal: You know we've been,
and we're constantly

00:15:34.703 --> 00:15:36.644
looking for folks to make
more and more videos,

00:15:36.644 --> 00:15:38.262
and obviously there's
some incredible teachers

00:15:38.262 --> 00:15:39.640
out there, many, many incredible teachers,

00:15:39.640 --> 00:15:44.435
but we found that there's
a certain nervousness

00:15:44.435 --> 00:15:46.150
once you record yourself
and you put it out there

00:15:46.150 --> 00:15:49.889
for the world to see
and comment on, and so,

00:15:49.889 --> 00:15:53.932
yes, there is a difference
when you, I mean I

00:15:53.932 --> 00:15:55.897
actually don't think
about it. When I make a

00:15:55.897 --> 00:15:57.512
video I kind of just imagine
my cousins listening,

00:15:57.512 --> 00:16:00.135
and I think it just makes
it more intimate and warm,

00:16:00.135 --> 00:16:03.634
hopefully, but we have seen
folks who are, I think,

00:16:03.634 --> 00:16:07.096
amazing teachers in
physical form, but when they

00:16:07.096 --> 00:16:09.931
know they're being recorded they
tend to get a little nervous.

00:16:09.931 --> 00:16:13.679
Rob: And I think teachers
are accustomed to making

00:16:13.679 --> 00:16:15.850
relationships, or forming
relationships with

00:16:15.850 --> 00:16:17.643
students that they have
right in front of them,

00:16:17.643 --> 00:16:22.179
and there's this
uncertainty around well how

00:16:22.179 --> 00:16:25.250
am I relating, how are
students relating to me

00:16:25.250 --> 00:16:29.885
when they see me on their
computer and is that,

00:16:29.885 --> 00:16:31.762
obviously you started
with your cousins, so

00:16:31.762 --> 00:16:33.345
I don't know if you ever
sort of went through

00:16:33.345 --> 00:16:34.636
that period of uncertainty?

00:16:34.636 --> 00:16:36.484
Sal: Yeah. No, I go through
that period of uncertainty

00:16:36.484 --> 00:16:38.650
every time I upload a
video, and this is what's

00:16:38.650 --> 00:16:41.952
exciting these days is I
can upload a video now,

00:16:41.952 --> 00:16:44.317
and within an hour I'll
get 50 comments from

00:16:44.317 --> 00:16:46.897
kids of all ages. I
shouldn't say just kids,

00:16:46.897 --> 00:16:50.405
all over the world, and it matters to me.

00:16:50.405 --> 00:16:52.312
I mean, it really helps me. If you say

00:16:52.312 --> 00:16:54.150
"that really helped, that
was exactly what I needed,"

00:16:54.150 --> 00:16:55.927
I've gotten through
that concept now, and so

00:16:55.927 --> 00:16:58.111
in that way I get much
more direct feedback.

00:16:58.111 --> 00:16:59.842
You know, I've done ... I've never been a

00:16:59.842 --> 00:17:02.894
professional teacher, but
I've done some teaching

00:17:02.894 --> 00:17:05.844
in college and, you know,
I used to give a lecture

00:17:05.844 --> 00:17:08.238
or teach a class and the day would go by,

00:17:08.238 --> 00:17:11.416
no feedback, I'd get
insecure about it. (laughs)

00:17:11.416 --> 00:17:13.402
And that would happen
and maybe at the end of

00:17:13.402 --> 00:17:14.948
the term you might get
one or two students,

00:17:14.948 --> 00:17:16.811
if you're lucky, who
might say, "oh that was

00:17:16.811 --> 00:17:18.787
really helpful, you
know, thank you," but you

00:17:18.787 --> 00:17:21.356
don't get that direct
feedback as often, and

00:17:21.356 --> 00:17:23.895
you know, online you get that feedback

00:17:23.895 --> 00:17:25.818
for better or for worse.
Sometimes the feedback

00:17:25.818 --> 00:17:27.971
can be harsh, but once again,
that can only make you better.

00:17:27.971 --> 00:17:31.442
Rob: Would you expect every instructor to

00:17:31.442 --> 00:17:34.564
produce their own content,
or do you think that

00:17:34.564 --> 00:17:36.817
there could be just a standardized

00:17:36.817 --> 00:17:38.950
repository that teachers would draw from?

00:17:38.950 --> 00:17:41.563
Sal: I think both. I mean,
I'd like to see a world

00:17:41.563 --> 00:17:44.400
where every instructor
if they're not happy

00:17:44.400 --> 00:17:46.482
with what they're seeing
out in the world that they

00:17:46.482 --> 00:17:48.894
are creating their own, and say hey, I can

00:17:48.894 --> 00:17:50.694
do better than that,
that persons not covering

00:17:50.694 --> 00:17:52.926
this nuance or this angle,
let me do something.

00:17:52.926 --> 00:17:56.121
With that said, I can
imagine over time the

00:17:56.121 --> 00:17:58.705
content delivery, you know, just as every

00:17:58.705 --> 00:18:00.655
teacher is not writing
their own textbook that

00:18:00.655 --> 00:18:02.806
you kind of start to settle on it becomes

00:18:02.806 --> 00:18:05.060
reasonably resource intensive. Now we can

00:18:05.060 --> 00:18:07.228
actually use data and
analytics to figure out

00:18:07.228 --> 00:18:09.397
which of the videos,
and once again, I think

00:18:09.397 --> 00:18:11.702
the videos in some ways
they're an analogist

00:18:11.702 --> 00:18:13.345
to the text in the
textbook, but we can start

00:18:13.345 --> 00:18:14.983
to measure how effective
is it, how engaging is it,

00:18:14.983 --> 00:18:19.681
and in 5 or 10 years
we won't need thousands

00:18:19.681 --> 00:18:22.617
of explanations of factoring a polynomial.

00:18:22.617 --> 00:18:24.954
We could have a handful
that have proven themselves

00:18:24.954 --> 00:18:28.401
to really engage students
and resonate with them.

00:18:28.401 --> 00:18:33.127
Rob: I have one last question,
if I could, which is the

00:18:33.127 --> 00:18:36.823
level of work that this
type of education requires

00:18:36.823 --> 00:18:38.784
as opposed to what
teachers are accustomed to

00:18:38.784 --> 00:18:41.176
doing now. What I've
heard from one instructor

00:18:41.176 --> 00:18:43.933
who does sort of a
hybrid kind of, you know,

00:18:43.933 --> 00:18:46.314
does a little bit of
both, he's like, well the

00:18:46.314 --> 00:18:48.611
classes that have an
online component you really

00:18:48.611 --> 00:18:50.945
have to develop all those
online pieces in advance,

00:18:50.945 --> 00:18:54.292
and then you still have
all of the work of the,

00:18:54.292 --> 00:18:56.234
you know, regular
interaction with students.

00:18:56.234 --> 00:18:59.520
In short, it sounds like it is more work.

00:18:59.520 --> 00:19:02.103
It may be worth it, but
do you think that it is

00:19:02.103 --> 00:19:03.650
more work and perhaps worth it,

00:19:03.650 --> 00:19:05.103
or what's your view on that?

00:19:05.103 --> 00:19:07.270
Sal: What we're seeing
is, yeah, I think, today,

00:19:07.270 --> 00:19:09.656
in most schools especially
I would say public

00:19:09.656 --> 00:19:13.484
schools where a teacher
wants to do something

00:19:13.484 --> 00:19:16.397
in this direction it is
more work for them, today,

00:19:16.397 --> 00:19:18.450
because to some degree they
have to serve two masters.

00:19:18.450 --> 00:19:19.987
They have to serve the
traditional master of

00:19:19.987 --> 00:19:21.731
this is the state
curriculum. You have to do

00:19:21.731 --> 00:19:23.399
this in this week, but
at the same time, they

00:19:23.399 --> 00:19:25.613
want to have the flexibility and allow the

00:19:25.613 --> 00:19:27.488
students to work at
their own pace and cater

00:19:27.488 --> 00:19:29.786
to the student, so it is
more work for them, today.

00:19:29.786 --> 00:19:32.182
It's less work then if they wanted to do a

00:19:32.182 --> 00:19:34.695
student centered curriculum
10 years ago without

00:19:34.695 --> 00:19:37.573
any technology. With
that said, the direction

00:19:37.573 --> 00:19:39.736
that everything seems
to be going in is that

00:19:39.736 --> 00:19:41.636
the tools are getting better, you know,

00:19:41.636 --> 00:19:43.358
Khan Academy itself is
nowhere near being a

00:19:43.358 --> 00:19:46.359
perfect tool. We have a
list of 200 things that

00:19:46.359 --> 00:19:48.102
we need to fix about
it, make it easier for

00:19:48.102 --> 00:19:49.768
teachers to use, make
it easier for them to

00:19:49.768 --> 00:19:51.842
plan their curriculum.
We have to add a lot

00:19:51.842 --> 00:19:53.817
more content so that
people don't have to go

00:19:53.817 --> 00:19:55.454
and fill in the gaps in other places.

00:19:55.454 --> 00:19:57.403
There's other people
making more and more tools.

00:19:57.403 --> 00:20:01.445
So I suspect as the tools
get better, as kind of

00:20:01.445 --> 00:20:03.957
the curriculum becomes
clearer, I think this

00:20:03.957 --> 00:20:05.848
adoption of the common core is actually a

00:20:05.848 --> 00:20:08.429
huge opportunity because
states have adopted it.

00:20:08.429 --> 00:20:11.925
It hasn't kind of fully gone
into implementation yet.

00:20:11.925 --> 00:20:14.704
I think all of these things
are kind of converged.

00:20:14.704 --> 00:20:16.315
The technology is going
to become more usable,

00:20:16.315 --> 00:20:18.395
and more accessible.
Right now teachers have to

00:20:18.395 --> 00:20:19.903
struggle with trying
to get the laptop cart

00:20:19.903 --> 00:20:21.339
for their class at the right period.

00:20:21.339 --> 00:20:23.176
Hopefully, a lot of
those things are going to

00:20:23.176 --> 00:20:24.896
solve themselves over
the next 4 - 5 years.

00:20:24.896 --> 00:20:28.509
So, yes, I would say
it's harder today, but

00:20:28.509 --> 00:20:31.375
the teachers think it's
worth it, but I would

00:20:31.375 --> 00:20:33.734
suspect 5 - 10 years from now it might be

00:20:33.734 --> 00:20:35.526
easier than a traditional model.

00:20:35.526 --> 00:20:38.483
Rob: Well thanks. I should
probably wrap up now.

00:20:38.483 --> 00:20:39.982
I could ask you lots more questions.

00:20:39.982 --> 00:20:42.275
Sal: Oh no, no, no, great, great,
yeah, those were great questions.

00:20:42.275 --> 00:20:44.000
Rob: I appreciate it.


Matrix addition and subtraction

WEBVTT
Kind: captions
Language: en

00:00:00.762 --> 00:00:03.839
Let's think about how we can define "Matrix Addition."

00:00:03.839 --> 00:00:06.609
And mathematicians could have chosen any

00:00:06.609 --> 00:00:09.406
of an arbitrary number of ways to define addition.

00:00:09.406 --> 00:00:11.691
But they've picked a way to define addition

00:00:11.691 --> 00:00:13.269
that seems – one – to make sense,

00:00:13.269 --> 00:00:16.280
and it also has nice properties that allow us to do

00:00:16.280 --> 00:00:19.049
interesting things with matrices later on.

00:00:19.049 --> 00:00:20.719
So if you were one of these mathematicians

00:00:20.719 --> 00:00:23.409
who were first defining how matrices should be added,

00:00:23.409 --> 00:00:24.425
how would you define

00:00:24.425 --> 00:00:27.543
adding this first matrix over here to the second one?

00:00:29.389 --> 00:00:31.418
Well, the most common-sense thing

00:00:31.418 --> 00:00:32.724
that might have jumped out at you –

00:00:32.740 --> 00:00:34.023
especially because these two matrices

00:00:34.023 --> 00:00:35.152
have the same dimensions –

00:00:35.152 --> 00:00:36.589
(This is a 2-by-3 matrix.

00:00:36.589 --> 00:00:38.170
It has 2 rows and 3 columns.

00:00:38.170 --> 00:00:39.840
This is also a 2-by-3 matrix.

00:00:39.840 --> 00:00:41.275
It also has 2 rows and 3 columns.)

00:00:41.275 --> 00:00:44.010
– is to just add the corresponding entries.

00:00:44.010 --> 00:00:46.270
And if that was your intuition,

00:00:46.270 --> 00:00:47.668
then you had the same intuition

00:00:47.668 --> 00:00:50.205
as the mathematical mainstream.

00:00:50.205 --> 00:00:51.835
That the addition of matrices

00:00:51.835 --> 00:00:53.102
should literally just be

00:00:53.102 --> 00:00:55.009
adding the corresponding entries.

00:00:55.009 --> 00:00:58.384
So in this situation, we would add 1 + 5

00:00:58.384 --> 00:01:00.224
to get the corresponding entry in the sum –

00:01:00.224 --> 00:01:01.205
which is 6.

00:01:01.205 --> 00:01:04.941
You can add negative seven plus zero

00:01:04.941 --> 00:01:06.836
to get negative seven.

00:01:06.836 --> 00:01:12.940
You can add five plus three to get eight.

00:01:12.940 --> 00:01:15.088
You can add

00:01:15.088 --> 00:01:16.500
-and I'm running out of colours here-

00:01:16.500 --> 00:01:19.921
you could add zero plus eleven

00:01:19.921 --> 00:01:21.532
to get eleven.

00:01:21.871 --> 00:01:25.210
You can add three to negative one

00:01:25.210 --> 00:01:26.486
to get two.

00:01:27.010 --> 00:01:30.404
And you could add -and you could add

00:01:30.404 --> 00:01:32.755
negative ten plus seven

00:01:32.755 --> 00:01:34.926
to get negative three.

00:01:34.926 --> 00:01:37.271
And if you see this definition of matrix addition

00:01:37.271 --> 00:01:39.004
you see that it actually does not matter

00:01:39.004 --> 00:01:41.877
in what order that we actually add these matrices.

00:01:41.877 --> 00:01:43.538
I could've done this the other way around,

00:01:43.538 --> 00:01:45.404
if I did this the other way around

00:01:45.404 --> 00:01:46.588
-so let me copy and paste this-

00:01:46.588 --> 00:01:47.936
so if I were to add this matrix

00:01:47.936 --> 00:01:51.344
-so let me copy, let me paste it-

00:01:51.344 --> 00:01:53.418
if I were to add that matrix to

00:01:53.418 --> 00:01:54.606
-let me copy and paste the other one-

00:01:54.606 --> 00:02:02.085
this matrix, copy and paste,

00:02:02.085 --> 00:02:04.512
you'll see that the order

00:02:04.512 --> 00:02:07.418
in which I'm adding the matrices does not matter

00:02:07.418 --> 00:02:09.004
So this is just like adding numbers.

00:02:09.004 --> 00:02:10.272
A plus B is just the same thing

00:02:10.272 --> 00:02:11.272
as B plus A.

00:02:11.272 --> 00:02:13.200
What we'll see is this won't be true

00:02:13.200 --> 00:02:15.203
for every matrix operation that we study

00:02:15.203 --> 00:02:16.802
and in particular this will not be true

00:02:16.802 --> 00:02:18.355
for matrix multiplication.

00:02:18.355 --> 00:02:20.010
But if you add these two things,

00:02:20.010 --> 00:02:22.092
using the definition we just came up with,

00:02:22.092 --> 00:02:23.417
adding corresponding terms,

00:02:23.417 --> 00:02:25.466
you'll get the exact same result.

00:02:25.466 --> 00:02:27.509
Up here we added one plus five and we got six

00:02:27.509 --> 00:02:30.346
Her we'll add five plus one and we'll get six.

00:02:30.346 --> 00:02:31.405
We get the same result

00:02:31.405 --> 00:02:34.168
because one plus five is the same thing as five plus one.

00:02:34.168 --> 00:02:35.602
Here we have zero plus negative seven

00:02:35.602 --> 00:02:37.260
you get negative seven.

00:02:37.260 --> 00:02:39.090
So you're going to get the exact same thing

00:02:39.090 --> 00:02:41.535
as we got up here.

00:02:41.535 --> 00:02:43.938
So when you're adding matrices, if you were to call

00:02:43.938 --> 00:02:46.936
-if you were to call this matrix right over here matrix A

00:02:46.936 --> 00:02:49.133
which we normally denote with a capital, bolder letter,

00:02:49.133 --> 00:02:52.670
and you call this matrix right over here Matrix B

00:02:52.670 --> 00:02:55.546
Then when we take the sum of A plus B

00:02:55.546 --> 00:02:57.206
which is this thing right over here,

00:02:57.206 --> 00:02:58.675
and we see it's the exact same thing

00:02:58.675 --> 00:03:05.123
as B, as Matrix B plus Matrix A.

00:03:05.123 --> 00:03:08.542
Now let me ask you an interesting question.

00:03:08.542 --> 00:03:10.539
What if I wanted to subtract matrices?

00:03:10.539 --> 00:03:12.923
So let's once again think about matrices

00:03:12.923 --> 00:03:14.868
that have the same dimensions.

00:03:14.868 --> 00:03:19.425
So let's say I'm gonna do then two two-by-two matrices.

00:03:19.425 --> 00:03:22.810
So let's say it's zero, one, three, two,

00:03:22.810 --> 00:03:25.605
and from that I want to subtract

00:03:25.605 --> 00:03:31.270
negative one, three, zero, and five.

00:03:31.270 --> 00:03:32.927
So you might say well maybe

00:03:32.927 --> 00:03:34.745
we just subtract corresponding entries.

00:03:34.745 --> 00:03:36.422
And that indeed is how you can define

00:03:36.422 --> 00:03:38.172
matrix subtraction.

00:03:38.172 --> 00:03:39.942
In fact you don't even have to define matrix subtraction,

00:03:39.942 --> 00:03:41.670
you can let this fall out of

00:03:41.670 --> 00:03:44.179
what we did with scalar multiplication

00:03:44.179 --> 00:03:45.603
and matrix addition.

00:03:45.603 --> 00:03:47.756
We can view as the exact same thing

00:03:47.756 --> 00:03:48.758
-this as the exact same thing-

00:03:48.758 --> 00:03:50.670
as taking zero, one, three, two

00:03:50.670 --> 00:03:54.761
and to that we add negative one, negative one times

00:03:54.761 --> 00:03:58.603
negative one, three, zero, five.

00:03:58.603 --> 00:04:01.088
And if you work out the math you're going to get

00:04:01.088 --> 00:04:02.511
the exact same result as just

00:04:02.511 --> 00:04:04.809
subtracting the corresponding terms.

00:04:04.809 --> 00:04:06.337
So this is going to be

00:04:06.737 --> 00:04:08.761
-what is this going to be?

00:04:08.761 --> 00:04:11.663
Zero minus negative one is positive one,

00:04:11.832 --> 00:04:14.473
one minus three is negative two,

00:04:14.473 --> 00:04:16.799
three minus zero is three,

00:04:16.799 --> 00:04:19.844
two minus five is negative three.

00:04:20.137 --> 00:04:21.133
And you'll see that you get

00:04:21.133 --> 00:04:22.347
the exact same thing here.

00:04:22.347 --> 00:04:24.468
When you multiply negative one times negative one

00:04:24.468 --> 00:04:27.273
you get positive one, positive one plus zero is one.

00:04:27.273 --> 00:04:32.073
Negative one times three plus one is negative two.

00:04:32.195 --> 00:04:33.468
Fair enough.

00:04:33.669 --> 00:04:34.874
There might be a question

00:04:34.874 --> 00:04:36.273
that is lingering in your brain right now.

00:04:36.273 --> 00:04:37.422
"Okay Sal, I understand

00:04:37.422 --> 00:04:39.175
when I'm adding or subtracting matrices

00:04:39.175 --> 00:04:40.673
with the same dimensions

00:04:40.673 --> 00:04:43.092
I just add or subtract the corresponding terms.

00:04:43.092 --> 00:04:45.070
But what happens when I have

00:04:45.070 --> 00:04:47.205
matrices with different dimensions?"

00:04:47.205 --> 00:04:49.093
So, for example, what about

00:04:49.093 --> 00:04:52.022
the scenario where I want to add the matrix

00:04:52.022 --> 00:04:58.837
one, zero, three, five, zero, one to the matrix

00:04:58.837 --> 00:05:01.009
-so this a three-by-two matrix-

00:05:01.009 --> 00:05:05.424
and I wanna add it to, let's say, a two-by-two matrix.

00:05:05.424 --> 00:05:08.735
Five, seven, negative one, zero.

00:05:08.735 --> 00:05:12.344
What would we define this as?

00:05:12.344 --> 00:05:13.542
Well it turns out that

00:05:13.603 --> 00:05:16.133
the mathematical mainstream does not define this.

00:05:16.133 --> 00:05:17.857
This is undefined.

00:05:18.088 --> 00:05:20.604
This is undefined.

00:05:20.604 --> 00:05:22.601
So we do not define matrix addition,

00:05:22.601 --> 00:05:24.091
or matrix subtraction,

00:05:24.091 --> 00:05:27.342
when the matrices have different dimensions.

00:05:27.342 --> 00:05:29.341
There didn't seem to be any reasonable way to do this,

00:05:29.341 --> 00:05:31.090
that would actually be useful

00:05:31.090 --> 00:05:33.469
and logically consistent in some nice way.


Introduction to the matrix

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:01.990
What I want to do
in this video is

00:00:01.990 --> 00:00:05.970
explore the notion
of a matrix outside

00:00:05.970 --> 00:00:09.290
of the context of a
surprisingly good movie that

00:00:09.290 --> 00:00:10.289
involves Keanu Reeves.

00:00:10.289 --> 00:00:11.830
And it's actually
the first of three.

00:00:11.830 --> 00:00:15.250
I guess we could call the three
movies combined The Matrices.

00:00:15.250 --> 00:00:18.100
And there is a relationship
between the movie, which

00:00:18.100 --> 00:00:19.990
is about a virtual
reality constructed

00:00:19.990 --> 00:00:24.080
by super-smart
computers, and the notion

00:00:24.080 --> 00:00:27.090
of what a matrix is when
you study it in mathematics,

00:00:27.090 --> 00:00:28.800
or when you study it
in computer science.

00:00:28.800 --> 00:00:31.020
And the connection
really is that matrices

00:00:31.020 --> 00:00:33.390
are used a lot when you
are simulating things

00:00:33.390 --> 00:00:35.770
or when you're constructing
things in computer science,

00:00:35.770 --> 00:00:38.000
especially in, frankly,
computer graphics.

00:00:38.000 --> 00:00:40.740
So the super-intelligent
robots that

00:00:40.740 --> 00:00:43.610
made the matrix in
the movie Matrix

00:00:43.610 --> 00:00:47.230
were probably using
matrices in order to do it,

00:00:47.230 --> 00:00:49.500
if they actually did exist.

00:00:49.500 --> 00:00:51.460
Now, what is a matrix then?

00:00:51.460 --> 00:00:53.640
Well, that's a
fairly simple answer.

00:00:53.640 --> 00:00:56.330
It's just a rectangular
array of numbers.

00:00:56.330 --> 00:00:58.170
So for example, this
right over here.

00:00:58.170 --> 00:01:05.470
If I have 1, 0, negative 7,
pi, 5, and-- I don't know-- 11,

00:01:05.470 --> 00:01:07.670
this is a matrix.

00:01:07.670 --> 00:01:11.570
This is a matrix where 1, 0,
negative 7, pi-- each of those

00:01:11.570 --> 00:01:13.310
are an entry in the matrix.

00:01:13.310 --> 00:01:17.450
This matrix right over
here has two rows.

00:01:17.450 --> 00:01:20.680
And it has three columns.

00:01:20.680 --> 00:01:23.290
And because it has two
rows and three columns,

00:01:23.290 --> 00:01:27.660
people will often say that
this is a 2 by 3 matrix.

00:01:27.660 --> 00:01:30.470
Whenever they say it's
something by something matrix,

00:01:30.470 --> 00:01:33.370
they're telling you that it has
two rows-- so you see the two

00:01:33.370 --> 00:01:34.790
rows right over there.

00:01:34.790 --> 00:01:38.820
And they are telling you
that it has three columns.

00:01:38.820 --> 00:01:41.010
You see the three
columns right over there.

00:01:41.010 --> 00:01:43.580
I could give you other
examples of a matrix.

00:01:43.580 --> 00:01:46.270
So I could have a 1 by 1 matrix.

00:01:46.270 --> 00:01:48.360
So I could have the matrix 1.

00:01:48.360 --> 00:01:50.930
This right over here
is a 1 by 1 matrix.

00:01:50.930 --> 00:01:52.860
It has one row, one column.

00:01:52.860 --> 00:01:58.450
I could have a matrix
like this-- 3, 7, and 17.

00:01:58.450 --> 00:01:59.870
What is this?

00:01:59.870 --> 00:02:01.539
Well, this has one row.

00:02:01.539 --> 00:02:03.080
This is the one row
that we see here.

00:02:03.080 --> 00:02:04.820
And it has three columns.

00:02:04.820 --> 00:02:06.509
This is a 1 by 3 matrix.

00:02:06.509 --> 00:02:09.050
I could have a matrix-- and I
think you see where all of this

00:02:09.050 --> 00:02:09.919
is going.

00:02:09.919 --> 00:02:12.950
Figuring out the dimensions of
a matrix are not too difficult.

00:02:12.950 --> 00:02:17.230
I could have a matrix that looks
like this, where it's 3, 5, 0,

00:02:17.230 --> 00:02:19.850
0, negative 1, negative 7.

00:02:22.620 --> 00:02:25.450
This right over
here has three rows.

00:02:25.450 --> 00:02:30.910
So it's three rows,
and it has two columns.

00:02:30.910 --> 00:02:35.340
So we would call this a 3 by 2.

00:02:35.340 --> 00:02:37.150
Let me do that in
that same color.

00:02:37.150 --> 00:02:44.490
We would call it a 3
by 2 matrix, three rows

00:02:44.490 --> 00:02:46.115
and two columns.

00:02:46.115 --> 00:02:46.740
So fair enough.

00:02:46.740 --> 00:02:51.441
You know that a matrix is just
a rectangular array of numbers.

00:02:51.441 --> 00:02:52.940
You can say what
its dimensions are.

00:02:52.940 --> 00:02:54.910
You know that each
of these numbers that

00:02:54.910 --> 00:02:57.569
take one of these positions--
we just call those entries.

00:02:57.569 --> 00:02:58.860
But what are matrices good for?

00:02:58.860 --> 00:03:00.780
I still might not be
clear what the connection

00:03:00.780 --> 00:03:05.040
is between this and
this right over here.

00:03:05.040 --> 00:03:07.180
And at the most
fundamental level,

00:03:07.180 --> 00:03:08.860
this is just a
compact representation

00:03:08.860 --> 00:03:09.780
of a bunch of numbers.

00:03:09.780 --> 00:03:11.730
It's a way of
representing information.

00:03:11.730 --> 00:03:13.950
They become very valuable
in computer graphics

00:03:13.950 --> 00:03:17.750
because these numbers could
represent the color intensity

00:03:17.750 --> 00:03:18.572
at a certain point.

00:03:18.572 --> 00:03:20.530
They could represent
whether an object is there

00:03:20.530 --> 00:03:21.590
at a certain point.

00:03:21.590 --> 00:03:23.662
And as we develop an
algebra around matrices,

00:03:23.662 --> 00:03:25.495
and when we talk about
developing an algebra

00:03:25.495 --> 00:03:27.777
around matrices, we're going
to talk about operations

00:03:27.777 --> 00:03:30.360
that we're going to perform on
matrices that we would normally

00:03:30.360 --> 00:03:31.470
perform with numbers.

00:03:31.470 --> 00:03:33.790
So we're going to
essentially define

00:03:33.790 --> 00:03:39.040
how to multiply matrices,
how to add matrices.

00:03:39.040 --> 00:03:42.010
We'll learn about taking
an inverse of a matrix.

00:03:42.010 --> 00:03:44.630
And by coming up with an
algebra of how we manipulate

00:03:44.630 --> 00:03:46.319
these things, it'll
become very useful

00:03:46.319 --> 00:03:48.860
in the future when you're trying
to write a computer graphics

00:03:48.860 --> 00:03:51.090
program or you're trying to
do an economic simulation

00:03:51.090 --> 00:03:53.530
or a probability
simulation, to say, oh,

00:03:53.530 --> 00:03:55.531
I have this matrix
that represents where

00:03:55.531 --> 00:03:56.905
different particles
are in space.

00:03:56.905 --> 00:03:58.510
Or I have this matrix
that represents

00:03:58.510 --> 00:04:01.840
the state of some
type of a game.

00:04:01.840 --> 00:04:04.670
And I know the
algebra of matrices.

00:04:04.670 --> 00:04:06.770
And I know ways of doing
it very efficiently

00:04:06.770 --> 00:04:08.590
so that I can multiply
a bunch of them.

00:04:08.590 --> 00:04:10.040
Or I could come
run a simulation,

00:04:10.040 --> 00:04:12.420
and I can actually come
up with useful results.

00:04:12.420 --> 00:04:13.740
So that's all matrices are.

00:04:13.740 --> 00:04:17.010
But as you'll see
through this, we

00:04:17.010 --> 00:04:18.790
can define operations on them.

00:04:18.790 --> 00:04:21.019
And then later on, when
you take a linear algebra

00:04:21.019 --> 00:04:23.940
course in college, you'll
learn a lot more of the depth

00:04:23.940 --> 00:04:25.490
of how they can be
applied and what

00:04:25.490 --> 00:04:27.527
you can use them to represent.


Scalar multiplication

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:02.402
Now that we know
what a matrix is,

00:00:02.402 --> 00:00:03.860
let's see if we
can start to define

00:00:03.860 --> 00:00:07.380
some operations on matrices.

00:00:07.380 --> 00:00:10.810
So let's say I have the 2 by 3
matrix, so two rows and three

00:00:10.810 --> 00:00:19.840
columns, and the entries are
7, 5, negative 10, 3, 8, and 0.

00:00:19.840 --> 00:00:23.530
And I want to define what
happens when I multiply 3 times

00:00:23.530 --> 00:00:24.795
this whole thing.

00:00:24.795 --> 00:00:26.920
So first of all, let's get
a little terminology out

00:00:26.920 --> 00:00:27.880
of the way.

00:00:27.880 --> 00:00:29.850
The number three, in
just the everyday world,

00:00:29.850 --> 00:00:32.290
if you weren't dealing
with matrices or vectors,

00:00:32.290 --> 00:00:33.360
and if you don't know
what vectors are,

00:00:33.360 --> 00:00:34.710
don't worry about
them just now, you

00:00:34.710 --> 00:00:35.680
would just call that a number.

00:00:35.680 --> 00:00:36.840
You would call
this a real number.

00:00:36.840 --> 00:00:38.960
It's just a regular
number sitting out there.

00:00:38.960 --> 00:00:40.376
But now in the
world where we have

00:00:40.376 --> 00:00:42.620
these new structured
things, these matrices,

00:00:42.620 --> 00:00:45.560
these arrays of
numbers, we will refer

00:00:45.560 --> 00:00:47.720
to these just plain
old real numbers that

00:00:47.720 --> 00:00:49.770
aren't part of some
type of an array here,

00:00:49.770 --> 00:00:51.270
we call these scalars.

00:00:55.730 --> 00:00:57.550
So essentially what
we're defining here,

00:00:57.550 --> 00:00:59.880
we don't know-- I haven't
said what this is actually

00:00:59.880 --> 00:01:02.140
going to turn out to be,
but whatever this turns out

00:01:02.140 --> 00:01:05.155
to be will be a product
of scalar multiplication,

00:01:05.155 --> 00:01:09.270
where we're multiplying
a scalar times a matrix.

00:01:09.270 --> 00:01:10.656
And so how would
you define this?

00:01:10.656 --> 00:01:12.030
What do you think
this should be?

00:01:12.030 --> 00:01:15.650
3 times this stuff
right over here.

00:01:15.650 --> 00:01:18.170
Well, the world could have
defined scalar multiplication

00:01:18.170 --> 00:01:21.060
however it saw fit, but
one way that we find,

00:01:21.060 --> 00:01:23.820
perhaps, the most obvious
and the most useful,

00:01:23.820 --> 00:01:26.930
is to multiply this
scalar quantity times each

00:01:26.930 --> 00:01:28.390
of the entries.

00:01:28.390 --> 00:01:32.030
So this is going to
be equal to 3 times 7

00:01:32.030 --> 00:01:39.190
in the top left, 3 times
5, 3 times negative 10,

00:01:39.190 --> 00:01:46.410
3 times 3, 3 times
8, and 3 times 0,

00:01:46.410 --> 00:01:48.070
which will give us--
it didn't change

00:01:48.070 --> 00:01:49.342
the dimensions of the matrix.

00:01:49.342 --> 00:01:50.800
It didn't change,
I guess you could

00:01:50.800 --> 00:01:51.920
say, the structure
of the matrix,

00:01:51.920 --> 00:01:53.920
it just multiplied each
of the entries times 3.

00:01:53.920 --> 00:01:55.640
So the top left
entry is now going

00:01:55.640 --> 00:02:01.980
to be 21, the entry
in the middle row,

00:02:01.980 --> 00:02:10.030
top column is going to be
15, negative 30, 9, 24 and 0.

00:02:10.030 --> 00:02:12.230
So when you multiply a
matrix times a scalar,

00:02:12.230 --> 00:02:15.670
you just multiply each of
those entries times that scalar

00:02:15.670 --> 00:02:16.938
quantity.


Transpose of a matrix

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:02.780
Let's say that I
have the matrix A.

00:00:02.780 --> 00:00:06.770
And we know we denote matrices
with bolded capital letters.

00:00:06.770 --> 00:00:10.830
So I have the matrix A.
And it is a 3 by 2 matrix.

00:00:10.830 --> 00:00:16.940
And so let's say it's 1,
negative 2, 3, 0, 7, 5.

00:00:16.940 --> 00:00:19.220
It's a 3 by 2 matrix.

00:00:19.220 --> 00:00:22.400
And what my question
to myself-- or to you,

00:00:22.400 --> 00:00:26.350
if you're listening-- is
what is the transpose of A?

00:00:26.350 --> 00:00:29.400
And so we denote the
transpose like this,

00:00:29.400 --> 00:00:32.700
A with this little
superscript T right over here.

00:00:32.700 --> 00:00:43.922
And all this is talking about
is the transpose of matrix A.

00:00:43.922 --> 00:00:45.380
And so your natural
question should

00:00:45.380 --> 00:00:48.430
be, well, what does it mean to
take the transpose of a matrix?

00:00:48.430 --> 00:00:50.270
It sounds very fancy.

00:00:50.270 --> 00:00:52.460
Well, it is not that fancy.

00:00:52.460 --> 00:00:56.985
All this means is
rows become columns.

00:00:59.750 --> 00:01:02.280
And you could imagine, when
the rows become columns,

00:01:02.280 --> 00:01:07.190
the columns become the rows.

00:01:07.190 --> 00:01:09.100
What do I mean by that?

00:01:09.100 --> 00:01:11.030
Well, let's do it
right over here.

00:01:11.030 --> 00:01:14.210
So if I have three rows.

00:01:14.210 --> 00:01:15.030
So let's be clear.

00:01:15.030 --> 00:01:18.350
This is one, two, three rows.

00:01:18.350 --> 00:01:22.150
So this is three rows
by one, two columns.

00:01:22.150 --> 00:01:24.300
So this is a 3 by 2 matrix.

00:01:24.300 --> 00:01:27.170
If all the rows are
going to become columns

00:01:27.170 --> 00:01:29.230
and all the columns are
going to become rows,

00:01:29.230 --> 00:01:32.360
the transpose is going
to be a 2 by 3 matrix.

00:01:32.360 --> 00:01:34.390
It's going to be two rows.

00:01:34.390 --> 00:01:36.710
The two columns here are
going to become the two rows,

00:01:36.710 --> 00:01:38.900
by three columns.

00:01:38.900 --> 00:01:41.820
These three rows are going
to become three columns.

00:01:41.820 --> 00:01:43.420
And so it's going
to look something

00:01:43.420 --> 00:01:45.970
like-- let me do it
in that blue color.

00:01:45.970 --> 00:01:47.500
It's going to have two rows now.

00:01:47.500 --> 00:01:50.570
And it's going to
have three columns.

00:01:50.570 --> 00:01:52.230
So I said all the
rows became columns.

00:01:52.230 --> 00:01:53.490
All the columns became rows.

00:01:53.490 --> 00:01:55.590
Well, I could view
it exactly like that.

00:01:55.590 --> 00:01:58.940
I have this first row
here of 1, negative 2.

00:01:58.940 --> 00:02:04.090
I can turn that into my first
column now of 1, negative 2.

00:02:04.090 --> 00:02:07.790
I have the second row, 3 and 0.

00:02:07.790 --> 00:02:12.290
That will be now my
second column, 3 and 0.

00:02:12.290 --> 00:02:16.250
And I have this
third row, 7 and 5.

00:02:16.250 --> 00:02:21.010
Well, that's going to be my
third column now, 7 and 5.

00:02:21.010 --> 00:02:23.101
And you could have
just as easily viewed

00:02:23.101 --> 00:02:24.100
it the other way around.

00:02:24.100 --> 00:02:26.150
You could say, look,
I have two columns.

00:02:26.150 --> 00:02:28.570
My first column is 1, 3, 7.

00:02:28.570 --> 00:02:33.190
So my first row now is
going to be 1, 3, 7.

00:02:33.190 --> 00:02:36.740
And then you could say, my
second column is negative 2,

00:02:36.740 --> 00:02:38.230
0, 5.

00:02:38.230 --> 00:02:42.836
And so my second row
will be negative 2, 0, 5.

00:02:42.836 --> 00:02:44.210
Let's do another
example of that,

00:02:44.210 --> 00:02:46.050
just so we really make it clear.

00:02:46.050 --> 00:02:47.900
So let's say I
have matrix B. Once

00:02:47.900 --> 00:02:50.330
again, bolded capital letter.

00:02:50.330 --> 00:02:53.250
And let's just say it's a
really simple 2 by 2 matrix.

00:02:53.250 --> 00:03:00.460
So it's negative 1, 5,
pi, and 3 are its entries.

00:03:00.460 --> 00:03:05.990
What is B transpose
going to be equal to?

00:03:05.990 --> 00:03:08.290
Well, once again, we have
two rows and two columns.

00:03:08.290 --> 00:03:10.050
So if all the rows become
columns and all the columns

00:03:10.050 --> 00:03:12.310
become rows, I'm still going
to have a 2 by 2 matrix.

00:03:15.380 --> 00:03:17.630
But we just have to say,
look, my first row here

00:03:17.630 --> 00:03:20.050
is negative 1, 5.

00:03:20.050 --> 00:03:23.230
So that's now going to be my
first column, negative 1 and 5.

00:03:26.160 --> 00:03:30.540
I have my second row
here, which is pi and 3.

00:03:30.540 --> 00:03:35.030
So now that's going to become
my second column of pi and 3.

00:03:35.030 --> 00:03:37.110
And so I have taken
the transpose.

00:03:37.110 --> 00:03:38.400
All the rows became columns.

00:03:38.400 --> 00:03:41.320
All the columns became rows.


Expression for combined area of triangle and square

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:04.500
Let's say that I have
a 100 meter long wire.

00:00:04.500 --> 00:00:07.230
So that is my wire
right over there.

00:00:07.230 --> 00:00:09.850
And it is 100 meters.

00:00:09.850 --> 00:00:14.270
And I'm going to make a
cut someplace on this wire.

00:00:14.270 --> 00:00:17.180
And so let's say I make
the cut right over there.

00:00:17.180 --> 00:00:18.760
With the left
section of wire-- I'm

00:00:18.760 --> 00:00:21.760
going to obviously cut it in
two-- with the left section,

00:00:21.760 --> 00:00:24.185
I'm going to construct
an equilateral triangle.

00:00:27.250 --> 00:00:30.575
And with the right section, I'm
going to construct a square.

00:00:33.670 --> 00:00:36.250
And my question
for you and for me

00:00:36.250 --> 00:00:39.230
is, where do we make
this cut in order

00:00:39.230 --> 00:00:43.440
to minimize the combined
areas of this triangle

00:00:43.440 --> 00:00:46.620
and this square?

00:00:46.620 --> 00:00:47.730
Well, let's figure out.

00:00:47.730 --> 00:00:50.240
Let's define a variable that
we're trying to minimize,

00:00:50.240 --> 00:00:53.770
or that we're trying to
optimize with respect to.

00:00:53.770 --> 00:00:57.350
So let's say that the variable
x is the number of meters

00:00:57.350 --> 00:00:59.180
that we decide to
cut from the left.

00:00:59.180 --> 00:01:02.750
So if we did that, then
this length for the triangle

00:01:02.750 --> 00:01:05.880
would be x meters, and
the length for the square

00:01:05.880 --> 00:01:09.500
would be, well, if we use x
up for the left hand side,

00:01:09.500 --> 00:01:14.505
we're going to have 100 minus
x for the right hand side.

00:01:14.505 --> 00:01:17.130
And so what would the dimensions
of the triangle and the square

00:01:17.130 --> 00:01:17.790
be?

00:01:17.790 --> 00:01:23.220
Well, the triangle sides are
going to be x over 3, x over 3,

00:01:23.220 --> 00:01:26.090
and x over 3 as an
equilateral triangle.

00:01:26.090 --> 00:01:32.260
And the square is going
to be 100 minus x over 4

00:01:32.260 --> 00:01:36.690
by 100 minus x over 4.

00:01:36.690 --> 00:01:38.810
Now it's easy to figure
out an expression

00:01:38.810 --> 00:01:42.470
for the area of the
square in terms of x.

00:01:42.470 --> 00:01:45.290
But let's think about what the
area of an equilateral triangle

00:01:45.290 --> 00:01:48.800
might be as a function of
the length of its sides.

00:01:48.800 --> 00:01:51.440
So let me do a little bit
of an aside right over here.

00:01:51.440 --> 00:01:53.575
So let's say we have an
equilateral triangle.

00:01:57.500 --> 00:01:58.350
Just like that.

00:01:58.350 --> 00:02:02.950
And its sides are
length s, s, and s.

00:02:02.950 --> 00:02:07.130
Now we know that the
area of a triangle

00:02:07.130 --> 00:02:14.010
is 1/2 times the base
times the height.

00:02:14.010 --> 00:02:15.510
So in this case,
the height we could

00:02:15.510 --> 00:02:18.400
consider to be altitude, if we
were to drop an altitude just

00:02:18.400 --> 00:02:19.460
like this.

00:02:19.460 --> 00:02:23.510
This length right over
here, this is the height.

00:02:23.510 --> 00:02:26.410
And this would be
perpendicular, just like that.

00:02:26.410 --> 00:02:33.410
So our area is going to be equal
to one half times our base is

00:02:33.410 --> 00:02:34.230
s.

00:02:34.230 --> 00:02:39.320
1/2 times s times whatever our
height is, times our height.

00:02:39.320 --> 00:02:43.940
Now how can we express
h as a function of s?

00:02:43.940 --> 00:02:46.430
Well, to do that we just
have to remind ourselves

00:02:46.430 --> 00:02:52.540
that what we've drawn over
here is a right triangle.

00:02:52.540 --> 00:02:56.480
It's the left half of
this equilateral triangle.

00:02:56.480 --> 00:03:00.840
And we know what this bottom
side of this right triangle is.

00:03:00.840 --> 00:03:03.980
This altitude splits this
side exactly into two.

00:03:03.980 --> 00:03:07.190
So this right over here
has length s over 2.

00:03:07.190 --> 00:03:09.020
So to figure out what
h is, we could just

00:03:09.020 --> 00:03:10.830
use the Pythagorean theorem.

00:03:10.830 --> 00:03:15.925
We would have h squared
plus s over 2 squared plus s

00:03:15.925 --> 00:03:18.980
over 2 squared is going to
be equal to the hypotenuse

00:03:18.980 --> 00:03:22.240
squared, is going to
be equal to s squared.

00:03:22.240 --> 00:03:26.090
So you would get h squared
plus s squared over 4

00:03:26.090 --> 00:03:27.760
is equal to s squared.

00:03:27.760 --> 00:03:30.450
Subtract s squared
over 4 from both sides,

00:03:30.450 --> 00:03:37.082
and you get h squared is equal s
squared minus s squared over 4.

00:03:37.082 --> 00:03:38.970
Now, to do this I could
call it s squared.

00:03:38.970 --> 00:03:43.170
I could call this
4s squared over 4,

00:03:43.170 --> 00:03:45.120
just to be able to have
a common denominator.

00:03:45.120 --> 00:03:48.330
And 4s squared minus
s squared over 4

00:03:48.330 --> 00:03:51.435
is going to be equal
to 3s squared over 4.

00:03:51.435 --> 00:03:56.010
So we get h squared is
equal to 3s squared over 4.

00:03:56.010 --> 00:03:58.890
Now we can take the
principal root of both sides,

00:03:58.890 --> 00:04:00.790
and we get h is equal
to the square root

00:04:00.790 --> 00:04:04.230
of 3 times s over 2.

00:04:04.230 --> 00:04:07.140
So now we can just substitute
back right over here

00:04:07.140 --> 00:04:09.960
and we get our area.

00:04:09.960 --> 00:04:13.900
Our area is equal
to 1/2 s times h.

00:04:13.900 --> 00:04:15.020
Well h is this business.

00:04:15.020 --> 00:04:16.779
So it's s times this.

00:04:16.779 --> 00:04:23.590
So it's 1/2 times s times the
square root of 3s over 2, which

00:04:23.590 --> 00:04:26.260
is going to be equal to
s times s is s squared.

00:04:26.260 --> 00:04:28.230
So this is going to
be square root of 3

00:04:28.230 --> 00:04:32.630
s squared over 2 times 2 over 4.

00:04:32.630 --> 00:04:35.090
So this is the area of
an equilateral triangle

00:04:35.090 --> 00:04:38.371
as the function of the
length of its sides.

00:04:38.371 --> 00:04:40.370
So what's the area of
this business going to be?

00:04:40.370 --> 00:04:42.470
So the area of our little
equilateral triangle--

00:04:42.470 --> 00:04:44.110
let me write a combined area.

00:04:44.110 --> 00:04:46.100
And let me do that
in a neutral color.

00:04:46.100 --> 00:04:48.720
So let me do that in white.

00:04:48.720 --> 00:04:51.370
So the combined
area, I'll write it

00:04:51.370 --> 00:04:57.700
A sub c is going to be equal
to the area of my triangle

00:04:57.700 --> 00:05:03.910
A sub t plus the
area of my square.

00:05:03.910 --> 00:05:07.087
Well, the area of my triangle,
we know what it's going to be.

00:05:07.087 --> 00:05:09.170
It's going to be square
root of 3 times the length

00:05:09.170 --> 00:05:11.370
of a side squared divided by 4.

00:05:11.370 --> 00:05:13.150
So it's going to be
square root of 3.

00:05:13.150 --> 00:05:16.050
Let me do that in that
same yellow color.

00:05:16.050 --> 00:05:19.920
It's going to be-- make
sure I switch colors.

00:05:19.920 --> 00:05:24.900
It's going to be square root of
3 over 4 times a side squared,

00:05:24.900 --> 00:05:33.350
times x over 3 squared.

00:05:33.350 --> 00:05:36.356
All I did is the length
of a side is x over 3.

00:05:36.356 --> 00:05:37.730
We already know
what the area is.

00:05:37.730 --> 00:05:39.380
It's the square root of
3 over 4 times the length

00:05:39.380 --> 00:05:40.345
of a side squared.

00:05:43.590 --> 00:05:45.930
And then the area
of this square right

00:05:45.930 --> 00:05:47.880
over here, the
area of the square

00:05:47.880 --> 00:05:55.320
is just going to be 100
minus x over 4 squared.

00:05:55.320 --> 00:05:57.840
So our area, our combined
area-- and maybe I

00:05:57.840 --> 00:05:59.590
can write it like
this-- our combined area

00:05:59.590 --> 00:06:03.800
as a function of where we make
the cut is all of this business

00:06:03.800 --> 00:06:05.090
right over here.

00:06:05.090 --> 00:06:09.400
And this is what we
need to minimize.

00:06:09.400 --> 00:06:15.670
So we need to minimize
that right over there.

00:06:15.670 --> 00:06:18.620
And I will do that
in the next video.


Minimizing combined area

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:02.570
Where we left off
in the last video,

00:00:02.570 --> 00:00:04.470
we had come up
with an expression

00:00:04.470 --> 00:00:06.470
as a function of x
of our combined area

00:00:06.470 --> 00:00:08.254
based on where we make the cut.

00:00:08.254 --> 00:00:09.670
And now we just
need to figure out

00:00:09.670 --> 00:00:12.190
where this hits a minimum value.

00:00:12.190 --> 00:00:15.359
And to do that, we
just have to take

00:00:15.359 --> 00:00:17.150
the derivative of this
business, figure out

00:00:17.150 --> 00:00:20.300
where our derivative is
either undefined or 0,

00:00:20.300 --> 00:00:22.520
and then just make sure that
that is a minimum value,

00:00:22.520 --> 00:00:24.380
and then we'll be all set.

00:00:24.380 --> 00:00:25.700
So let me rewrite this.

00:00:25.700 --> 00:00:29.434
So our combined area
as a function of x,

00:00:29.434 --> 00:00:31.600
let me just rewrite this
so it's a little bit easier

00:00:31.600 --> 00:00:32.560
to take the derivative.

00:00:32.560 --> 00:00:36.500
So this is going to be
the square root of 3 times

00:00:36.500 --> 00:00:40.380
x squared over-- let's
see, this is 4 times 9.

00:00:40.380 --> 00:00:42.370
This is x squared over 9.

00:00:42.370 --> 00:00:46.200
So this is going to
be 4 times 9 is 36.

00:00:46.200 --> 00:00:47.990
And then over here
in blue, this is

00:00:47.990 --> 00:00:55.470
going to be plus 100
minus x squared over 16.

00:00:55.470 --> 00:00:57.350
Now let's take the
derivative of this.

00:00:57.350 --> 00:01:02.590
So A prime, the derivative
of our combined area

00:01:02.590 --> 00:01:04.414
as a function of
x, is going to be

00:01:04.414 --> 00:01:06.580
equal to-- well, the
derivative of this with respect

00:01:06.580 --> 00:01:12.670
to x is just going to be
square root of 3x over 18.

00:01:12.670 --> 00:01:14.900
The derivative of this
with respect to x,

00:01:14.900 --> 00:01:17.840
well, it's the derivative
of something squared over 16

00:01:17.840 --> 00:01:19.940
with respect to that something.

00:01:19.940 --> 00:01:25.090
So that's going to be that
something to the first power

00:01:25.090 --> 00:01:28.947
times 2/16, which
is just over 8.

00:01:28.947 --> 00:01:31.030
And then times-- we're
just doing the chain rule--

00:01:31.030 --> 00:01:33.380
times the derivative of the
something with respect to x.

00:01:33.380 --> 00:01:37.030
The derivative of 100
minus x with respect to x

00:01:37.030 --> 00:01:41.680
is just negative 1,
so times negative 1.

00:01:41.680 --> 00:01:43.950
So we'll multiply negative
1 right over here.

00:01:43.950 --> 00:01:47.960
And so we can rewrite all
of that as-- this is going

00:01:47.960 --> 00:01:56.390
to be equal to the square root
of 3/18 x plus-- let's see,

00:01:56.390 --> 00:01:59.050
I could write this
as positive x/8.

00:01:59.050 --> 00:02:01.720
So I could write
this as 1/8 x, right?

00:02:01.720 --> 00:02:05.460
Because a negative 1 times a
negative x is positive x/8.

00:02:05.460 --> 00:02:13.570
And then minus 100/8, which
is negative 12.5-- minus 12.5.

00:02:13.570 --> 00:02:17.030
And we want to figure out an
x that minimizes this area.

00:02:17.030 --> 00:02:19.951
So this derivative right over
here is defined for any x.

00:02:19.951 --> 00:02:22.450
So we're not going to get our
critical point by figuring out

00:02:22.450 --> 00:02:24.006
where the derivative
is undefined.

00:02:24.006 --> 00:02:25.380
But we might get
a critical point

00:02:25.380 --> 00:02:27.560
by setting this derivative
equal to 0 to figure out

00:02:27.560 --> 00:02:29.940
what x-values make
our derivative 0.

00:02:29.940 --> 00:02:32.912
When do we have a 0 slope
for our original function?

00:02:32.912 --> 00:02:34.620
And then we just have
to verify that this

00:02:34.620 --> 00:02:37.330
is going to be a minimum
point if we can find an x that

00:02:37.330 --> 00:02:38.970
makes this thing equal to 0.

00:02:38.970 --> 00:02:41.280
So let's try to solve for x.

00:02:41.280 --> 00:02:43.550
So if we add 12.5
to both sides, we

00:02:43.550 --> 00:02:47.660
get 12.5 is equal to--
if you add the x terms,

00:02:47.660 --> 00:02:54.190
you get square root
of 3/18 plus 1/8 x.

00:02:54.190 --> 00:02:57.790
To solve for x, divide both
sides by this business.

00:02:57.790 --> 00:03:05.550
You get x is equal to 12.5
over square root of 3 over 18

00:03:05.550 --> 00:03:07.860
plus 1/8.

00:03:07.860 --> 00:03:09.990
And we are done.

00:03:09.990 --> 00:03:13.959
At x equals this, our
derivative is equal to 0.

00:03:13.959 --> 00:03:15.250
I shouldn't say we're done yet.

00:03:15.250 --> 00:03:17.120
We don't know whether
this is a minimum point.

00:03:17.120 --> 00:03:19.370
In order to figure out whether
this is a minimum point

00:03:19.370 --> 00:03:23.090
we have to figure out whether
our function is concave upward

00:03:23.090 --> 00:03:25.877
or concave downward when x
is equal to this business.

00:03:25.877 --> 00:03:28.210
And to figure that out, let's
take the second derivative

00:03:28.210 --> 00:03:30.470
here.

00:03:30.470 --> 00:03:33.540
So let me rewrite
the second derivative

00:03:33.540 --> 00:03:35.680
of all of this business.

00:03:35.680 --> 00:03:37.144
The second
derivative, well, this

00:03:37.144 --> 00:03:39.060
was the same function
as this right over here.

00:03:39.060 --> 00:03:40.810
So let me rewrite it.

00:03:40.810 --> 00:03:43.540
So A prime, the derivative
of our combined area,

00:03:43.540 --> 00:03:51.110
was equal to the square root of
3/18 x plus 1/8 x minus 12.5.

00:03:51.110 --> 00:03:54.460
The second derivative
is going to be

00:03:54.460 --> 00:03:59.050
square root of 3/18 plus 1/8.

00:03:59.050 --> 00:04:01.960
So this thing right over
here is greater than 0,

00:04:01.960 --> 00:04:06.790
which means we're concave
upward for all x's.

00:04:06.790 --> 00:04:10.260
Concave upwards, which
means for all x's.

00:04:10.260 --> 00:04:12.380
We're kind of doing a
situation like this.

00:04:12.380 --> 00:04:14.815
So if we find an x
where the slope is 0,

00:04:14.815 --> 00:04:16.190
it's going to be
over an interval

00:04:16.190 --> 00:04:17.315
where it's concave upwards.

00:04:17.315 --> 00:04:19.250
This is concave
upwards for all x.

00:04:19.250 --> 00:04:22.470
So we're going to be
at a minimum point.

00:04:22.470 --> 00:04:23.880
The slope is 0 right over here.

00:04:23.880 --> 00:04:26.820
This right over here
will be a minimum point.

00:04:26.820 --> 00:04:29.470
So once again, this is
going to be a minimum point.

00:04:29.470 --> 00:04:33.040
Now, if we actually
had a 100-meter wire,

00:04:33.040 --> 00:04:34.690
this expression
isn't too valuable.

00:04:34.690 --> 00:04:36.644
We'd want to get a pretty
close approximation

00:04:36.644 --> 00:04:38.060
in terms of where
to actually make

00:04:38.060 --> 00:04:39.880
the cut, an actual
decimal number.

00:04:39.880 --> 00:04:43.040
So let's use a
calculator to get that.

00:04:43.040 --> 00:04:52.600
So we have 12.5 divided by
square root of 3 divided

00:04:52.600 --> 00:05:00.310
by 18 plus 1 divided
by 8 gives us--

00:05:00.310 --> 00:05:02.520
and now we deserve
our drum roll.

00:05:02.520 --> 00:05:05.050
This is 56.5.

00:05:05.050 --> 00:05:14.090
So this is approximately
equal to 56.5 meters.

00:05:14.090 --> 00:05:19.830
So you make this
cut roughly 56.5--

00:05:19.830 --> 00:05:24.500
I'll write roughly-- 56.5
meters from the left-hand side.

00:05:24.500 --> 00:05:27.190
And you will minimize
the combined area

00:05:27.190 --> 00:05:29.733
of both of these figures.


Formula for arithmetic series

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:04.700
Let's write an arithmetic
sequence in general terms.

00:00:08.230 --> 00:00:11.460
So we can start
with some number a.

00:00:11.460 --> 00:00:13.280
And then we can
keep adding d to it.

00:00:13.280 --> 00:00:14.900
And that number
that we keep adding,

00:00:14.900 --> 00:00:16.950
which could be a positive
or a negative number,

00:00:16.950 --> 00:00:18.650
we call our common difference.

00:00:18.650 --> 00:00:22.100
So the second term in our
sequence will be a plus d.

00:00:22.100 --> 00:00:25.670
The third term in our
sequence will be a plus 2d.

00:00:25.670 --> 00:00:29.560
So we keep adding d all
the way to the n-th term

00:00:29.560 --> 00:00:30.820
in our sequence.

00:00:30.820 --> 00:00:33.200
And you already see here
that in our first term,

00:00:33.200 --> 00:00:36.210
we added d zero times.

00:00:36.210 --> 00:00:39.580
Our second term,
we added d once.

00:00:39.580 --> 00:00:42.230
In our third term,
we added d twice.

00:00:42.230 --> 00:00:44.470
So you see, whatever the
index of the term is,

00:00:44.470 --> 00:00:47.650
we're adding d one less
than that many times.

00:00:47.650 --> 00:00:51.770
So if we go all the
way to the n-th term,

00:00:51.770 --> 00:00:54.315
we're going to add d
one less than n times.

00:00:54.315 --> 00:00:58.620
So it's going to be
n minus 1 times d.

00:00:58.620 --> 00:00:59.124
Fair enough.

00:00:59.124 --> 00:01:00.040
And let me write that.

00:01:00.040 --> 00:01:03.622
This right over here
is our n-th term.

00:01:03.622 --> 00:01:05.330
Now what I want to do
is think about what

00:01:05.330 --> 00:01:07.750
the sum of this arithmetic
sequence would be.

00:01:07.750 --> 00:01:09.360
And the sum of an
arithmetic sequence

00:01:09.360 --> 00:01:11.590
we call an arithmetic series.

00:01:11.590 --> 00:01:12.905
So let me write that in yellow.

00:01:17.260 --> 00:01:19.390
Color changing is
sometimes difficult.

00:01:19.390 --> 00:01:26.310
So the arithmetic
series is just the sum

00:01:26.310 --> 00:01:28.420
of an arithmetic sequence.

00:01:28.420 --> 00:01:31.770
So let's call my
arithmetic series s sub n.

00:01:31.770 --> 00:01:34.030
And let's say it's going to
be the sum of these terms,

00:01:34.030 --> 00:01:43.780
so it's going to be a plus
d, plus a plus 2d, plus all

00:01:43.780 --> 00:01:51.150
the way to adding the n-th term,
which is a plus n minus 1 times

00:01:51.150 --> 00:01:52.274
d.

00:01:52.274 --> 00:01:53.690
Now I'm going to
do the same trick

00:01:53.690 --> 00:01:56.930
that I did when I did the most
basic arithmetic sequence.

00:01:56.930 --> 00:01:59.170
I'm going to add this
to itself, but I'm

00:01:59.170 --> 00:02:01.620
going to swap the order
in which I write this sum.

00:02:01.620 --> 00:02:04.812
So s sub n I can
write as this, but I'm

00:02:04.812 --> 00:02:06.270
going to write it
in reverse order.

00:02:06.270 --> 00:02:08.120
I'm going to write
the last term first.

00:02:08.120 --> 00:02:15.090
So the n-th term is a
plus n minus 1 times d.

00:02:15.090 --> 00:02:17.290
Then the second to
last term is going

00:02:17.290 --> 00:02:22.000
to be a plus n minus 2 times d.

00:02:22.000 --> 00:02:29.860
The third to last is going to
be a plus n minus 3 times d.

00:02:29.860 --> 00:02:31.830
And we're going to
go all the way down

00:02:31.830 --> 00:02:36.250
to the first term,
which is just a.

00:02:36.250 --> 00:02:38.280
Now let's add these
two equations.

00:02:38.280 --> 00:02:42.410
We are going to get, on the
left hand side, s sub n plus s

00:02:42.410 --> 00:02:43.230
sub n.

00:02:43.230 --> 00:02:48.330
You're going to get
2 times s sub n.

00:02:48.330 --> 00:02:52.890
Well, what's the sum of
these two first terms

00:02:52.890 --> 00:02:54.100
right over here?

00:02:54.100 --> 00:02:57.140
I'm going to have a plus
a plus n minus 1 times d.

00:02:57.140 --> 00:03:03.170
So it's going to be 2a
plus n minus 1 times d.

00:03:03.170 --> 00:03:06.070
Now let's add both of
these second terms.

00:03:06.070 --> 00:03:08.200
So if I were to add both
of these second terms,

00:03:08.200 --> 00:03:09.360
what do I get it?

00:03:09.360 --> 00:03:12.940
I'm going to get 2a plus 2a.

00:03:12.940 --> 00:03:16.610
And what's d plus
n minus 2 times d?

00:03:16.610 --> 00:03:19.004
So you could view
it several ways.

00:03:19.004 --> 00:03:20.170
Let me write this over here.

00:03:20.170 --> 00:03:24.675
What is d plus n
minus 2 times d?

00:03:24.675 --> 00:03:26.050
Well, this is just
the same thing

00:03:26.050 --> 00:03:28.559
as 1d plus n minus 2 times d.

00:03:28.559 --> 00:03:30.350
And so you could just
add the coefficients.

00:03:30.350 --> 00:03:35.100
So this is going to be n
minus 2 plus 1 times d, which

00:03:35.100 --> 00:03:39.910
is equal to n minus 1 times d.

00:03:39.910 --> 00:03:51.360
So the second term also becomes
2a plus n minus 1 times d.

00:03:51.360 --> 00:03:53.030
Now let's add the third term.

00:03:53.030 --> 00:03:53.910
I'll do it in green.

00:03:53.910 --> 00:03:55.782
The third terms, I should say.

00:03:55.782 --> 00:03:57.740
And I think you're going
to see a pattern here.

00:03:57.740 --> 00:04:01.970
It's 2a plus 2a.

00:04:01.970 --> 00:04:06.410
And if I have 2 plus n minus 3
of something and then I add 2,

00:04:06.410 --> 00:04:08.410
I'm going to have n minus
one of that something.

00:04:08.410 --> 00:04:12.130
So plus n minus 1 times d.

00:04:12.130 --> 00:04:14.430
And you're going to keep
doing that all the way

00:04:14.430 --> 00:04:16.890
until your n-th pair
of terms, all the way

00:04:16.890 --> 00:04:19.410
until you add these two
characters over here, which

00:04:19.410 --> 00:04:25.220
is just 2a plus n
minus 1 times d.

00:04:25.220 --> 00:04:28.310
So you have this
2a plus n minus 1 d

00:04:28.310 --> 00:04:30.437
being added over and over again.

00:04:30.437 --> 00:04:32.020
And how many times
are you doing that?

00:04:32.020 --> 00:04:34.110
Well, you had n
pairs of terms when

00:04:34.110 --> 00:04:35.610
you were adding
these two equations.

00:04:35.610 --> 00:04:37.700
In each of them,
you had n terms.

00:04:37.700 --> 00:04:39.750
This is the first term,
this is the second term,

00:04:39.750 --> 00:04:43.230
this is the third term, all
the way to the n-th term.

00:04:43.230 --> 00:04:48.430
So I can rewrite 2
times the sum 2 times

00:04:48.430 --> 00:04:51.770
s sub n is going to be
n times this quantity.

00:04:51.770 --> 00:05:03.452
It's going to be n times
2a plus n minus 1 times d.

00:05:03.452 --> 00:05:05.160
And then if we want
to solve for s sub n,

00:05:05.160 --> 00:05:07.340
you just divide both sides by 2.

00:05:07.340 --> 00:05:10.190
And you get s sub n is equal
to, and we get ourselves

00:05:10.190 --> 00:05:11.940
a little bit of
a drum roll here,

00:05:11.940 --> 00:05:17.750
n times 2a plus n
minus 1 times d.

00:05:17.750 --> 00:05:20.340
All of that over 2.

00:05:20.340 --> 00:05:23.520
Now, we've come up
with a general formula,

00:05:23.520 --> 00:05:26.169
just a function of
what our first term is,

00:05:26.169 --> 00:05:28.460
what our common difference
is, and how many terms we're

00:05:28.460 --> 00:05:29.310
adding up.

00:05:29.310 --> 00:05:33.980
And so this is the generalized
sum of an arithmetic sequence,

00:05:33.980 --> 00:05:36.089
which we call an
arithmetic series.

00:05:36.089 --> 00:05:37.880
But now, let's ask
ourselves this question.

00:05:37.880 --> 00:05:39.460
This is hard to remember.

00:05:39.460 --> 00:05:44.124
The n times 2a plus n
minus 1 times d over 2.

00:05:44.124 --> 00:05:46.540
But in the last video, when I
did a more concrete example,

00:05:46.540 --> 00:05:53.210
I said well, it looks like the
sum of an arithmetic sequence

00:05:53.210 --> 00:05:58.460
could be written as
perhaps the average

00:05:58.460 --> 00:06:01.490
of the first term a1 plus an.

00:06:01.490 --> 00:06:06.450
The average of the first
term and the last term

00:06:06.450 --> 00:06:08.930
times the number of
terms that you have.

00:06:08.930 --> 00:06:11.990
So is this actually the case?

00:06:11.990 --> 00:06:14.780
Do these two things gel?

00:06:14.780 --> 00:06:16.500
Because this is very
easy to remember--

00:06:16.500 --> 00:06:19.379
the average of the first and
the last terms multiplied

00:06:19.379 --> 00:06:21.420
by the number of terms
you had and actually makes

00:06:21.420 --> 00:06:23.130
intuitive sense,
because you're just

00:06:23.130 --> 00:06:25.280
increasing by the same
amount every time.

00:06:25.280 --> 00:06:30.580
So let's just average the
first and the last term

00:06:30.580 --> 00:06:33.600
and then multiply times the
number of terms we have.

00:06:33.600 --> 00:06:36.290
Well, all we have to do is
rewrite this a little bit

00:06:36.290 --> 00:06:38.700
to see that it is indeed
the exact same thing as this

00:06:38.700 --> 00:06:39.860
over here.

00:06:39.860 --> 00:06:41.810
So all we have to do
is break out the a.

00:06:41.810 --> 00:06:43.270
So let me rewrite it.

00:06:43.270 --> 00:06:46.620
So, this could be
rewritten as s sub

00:06:46.620 --> 00:06:57.460
n is equal to n times a plus
a plus n minus 1 times d.

00:06:57.460 --> 00:07:00.030
I just broke up this
2a into an a plus a.

00:07:00.030 --> 00:07:04.030
All of that over 2.

00:07:04.030 --> 00:07:06.690
And you see, based on how
we defined this thing,

00:07:06.690 --> 00:07:12.180
our first term a1 is a.

00:07:12.180 --> 00:07:19.930
And then our last term, a sub
n, is a plus n minus 1 times d.

00:07:19.930 --> 00:07:26.380
So this whole business
right over here

00:07:26.380 --> 00:07:35.040
really is the average of
the first and last terms.

00:07:35.040 --> 00:07:37.120
I got my first term,
adding it to my last term,

00:07:37.120 --> 00:07:38.410
dividing it by 2.

00:07:38.410 --> 00:07:41.190
And then I'm multiplying by
the number of terms we have.

00:07:41.190 --> 00:07:43.810
And that's true for any
arithmetic sequence,

00:07:43.810 --> 00:07:46.260
as we've just shown here.


Arithmetic series intro

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.374
Let's say we have the simplest

00:00:02.374 --> 00:00:03.769
of arithmetic sequences

00:00:03.769 --> 00:00:05.436
and probably the simplest of sequences

00:00:05.436 --> 00:00:06.667
one, two...

00:00:06.667 --> 00:00:07.773
we're going to start at one

00:00:07.773 --> 00:00:08.660
and just increment by one

00:00:08.660 --> 00:00:09.659
one, two, three

00:00:09.659 --> 00:00:13.643
and we're going to go all the way to n

00:00:13.643 --> 00:00:14.905
And what I want to think about is

00:00:14.905 --> 00:00:17.907
what is the sum of this sequence going to be?

00:00:17.907 --> 00:00:20.249
And the sum of a sequence, we already know

00:00:20.249 --> 00:00:21.842
we call a series

00:00:21.842 --> 00:00:23.433
so what is the sum

00:00:23.433 --> 00:00:25.075
and I will just call it Sn

00:00:25.075 --> 00:00:26.436
what is this going to be equal to

00:00:26.436 --> 00:00:30.905
one plus two plus three plus

00:00:30.905 --> 00:00:34.909
going all the way to n

00:00:34.909 --> 00:00:37.041
well, we're going to do a neat little trick here

00:00:37.041 --> 00:00:38.661
where I'm going to rewrite this sum

00:00:38.661 --> 00:00:42.799
so I will write it again as Sn

00:00:42.799 --> 00:00:44.436
but now I'm just going to write it in reverse order

00:00:44.436 --> 00:00:47.910
I'm going to write it as n plus n minus one

00:00:47.910 --> 00:00:51.107
plus n minus two

00:00:51.107 --> 00:00:53.904
all the way to one

00:00:53.904 --> 00:00:56.572
and now I'm going to add these two equations

00:00:56.572 --> 00:00:58.409
So we know that Sn is equal to this

00:00:58.409 --> 00:01:01.506
so we are adding the same thing to both sides of this equation up here

00:01:01.506 --> 00:01:05.902
So on the left hand side we're going to have Sn plus Sn

00:01:05.902 --> 00:01:08.441
is just two times Sn

00:01:08.441 --> 00:01:10.036
and on the right hand side

00:01:10.036 --> 00:01:12.835
and this is where we start to see something kind of cool

00:01:12.835 --> 00:01:15.916
you have a one plus an n

00:01:15.916 --> 00:01:18.769
which is just going to be n plus one

00:01:18.769 --> 00:01:22.501
you have a two plus n minus one

00:01:22.501 --> 00:01:25.332
which is... well two plus n minus one

00:01:25.332 --> 00:01:27.108
is going to be n plus one again

00:01:27.108 --> 00:01:29.913
plus n plus one

00:01:29.913 --> 00:01:32.976
you have a three plus n minus two

00:01:32.976 --> 00:01:36.706
well that's going to be n plus one again

00:01:36.706 --> 00:01:38.437
I think you see what is going on here

00:01:38.437 --> 00:01:42.990
And we're going to go all the way to this last pair

00:01:42.990 --> 00:01:47.664
I guess you could say, or call it these last two terms

00:01:47.664 --> 00:01:50.970
and we have an n plus one again

00:01:50.970 --> 00:01:52.580
plus n plus one

00:01:52.580 --> 00:01:56.168
So how many of these n plus one's do we have?

00:01:56.168 --> 00:01:57.700
well we have n of them

00:01:57.700 --> 00:02:00.836
there were n of these terms in each of these equations

00:02:00.836 --> 00:02:03.077
one two three... all the way to n

00:02:03.077 --> 00:02:06.909
So, we can rewrite this thing as two times Sn

00:02:06.909 --> 00:02:10.577
is equal to, and you have n (n+1) terms

00:02:10.577 --> 00:02:14.642
so we could write it as n times n plus one

00:02:14.642 --> 00:02:17.702
n times n plus one

00:02:17.702 --> 00:02:21.991
and now to solve for Sn to solve for our sum

00:02:21.991 --> 00:02:26.835
we can just divide both sides by two

00:02:26.835 --> 00:02:31.640
And so we are going to be left with the sum from one to n

00:02:31.640 --> 00:02:34.582
this arithmetic sequence where we're just incrementing by one

00:02:34.582 --> 00:02:41.416
starting at one, is going to be equal to n times n plus one over two

00:02:41.416 --> 00:02:45.993
And this is neat because now you can quickly find the sum

00:02:45.993 --> 00:02:47.747
let's say from one to a hundred

00:02:47.747 --> 00:02:51.769
it will be 100 times 101 over 2

00:02:51.769 --> 00:02:53.903
So very quickly you can find these sums

00:02:53.903 --> 00:02:56.835
And what I'm curious about and what we will explore in the next video

00:02:56.835 --> 00:02:59.768
is can we generalize this for any arithmetic sequence

00:02:59.768 --> 00:03:01.299
we started with a very simple one

00:03:01.299 --> 00:03:03.749
we started at one, we just incremented by one

00:03:03.749 --> 00:03:06.799
And it looks like

00:03:06.799 --> 00:03:08.170
so if I were to write it this way

00:03:08.170 --> 00:03:13.969
this is n times (n + 1) over 2

00:03:13.969 --> 00:03:17.709
So this right over here, this n

00:03:17.709 --> 00:03:20.102
this is the nth term in our sequence

00:03:20.102 --> 00:03:21.768
and this right over here, this one

00:03:21.768 --> 00:03:23.836
was the first term in our sequence

00:03:23.836 --> 00:03:26.769
So at least in this case it looks like I took the average

00:03:26.769 --> 00:03:29.168
of the first term and the nth term

00:03:29.168 --> 00:03:31.769
so this right over here

00:03:31.769 --> 00:03:33.911
this is the average

00:03:33.911 --> 00:03:39.502
this right over here is the average

00:03:39.502 --> 00:03:42.909
of a1 and an

00:03:42.909 --> 00:03:45.663
and then I'm multiplying that times n

00:03:45.663 --> 00:03:48.041
and what I'm curious about is whether this is going to be true

00:03:48.041 --> 00:03:49.325
for any arithmetic sequence

00:03:49.325 --> 00:03:51.659
that the sum of it is going to be the average of the first

00:03:51.659 --> 00:03:55.659
and the last term times the number of terms


Series as sum of sequence

WEBVTT
Kind: captions
Language: en

00:00:00.580 --> 00:00:04.880
Let's say that I have
a geometric series.

00:00:04.880 --> 00:00:06.960
A geometric sequence,
I should say.

00:00:06.960 --> 00:00:08.680
We'll talk about
series in a second.

00:00:08.680 --> 00:00:12.160
So a geometric series,
let's say it starts at 1,

00:00:12.160 --> 00:00:13.875
and then our common
ratio is 1/2.

00:00:13.875 --> 00:00:15.250
So the common
ratio is the number

00:00:15.250 --> 00:00:16.460
that we keep multiplying by.

00:00:16.460 --> 00:00:22.510
So 1 times 1/2 is 1/2, 1/2
times 1/2 is 1/4, 1/4 times 1/2

00:00:22.510 --> 00:00:26.510
is 1/8, and we can keep going
on and on and on forever.

00:00:26.510 --> 00:00:29.190
This is an infinite
geometric sequence.

00:00:29.190 --> 00:00:30.980
And we can denote this.

00:00:30.980 --> 00:00:35.570
We can say that this is equal to
the sequence of a sub n from n

00:00:35.570 --> 00:00:43.525
equals 1 to infinity, with
a sub n equaling 1 times

00:00:43.525 --> 00:00:45.560
our common ratio
to the n minus 1.

00:00:45.560 --> 00:00:48.770
So it's going to be our
first term, which is just 1,

00:00:48.770 --> 00:00:51.570
times our common
ratio, which is 1/2.

00:00:51.570 --> 00:00:53.834
1/2 to the n minus 1.

00:00:53.834 --> 00:00:54.750
And you can verify it.

00:00:54.750 --> 00:00:57.980
This right over here you can
view as 1/2 to the 0 power.

00:00:57.980 --> 00:01:01.200
This is 1/2 half to the first
power, this is 1/2 squared.

00:01:01.200 --> 00:01:04.319
1/2 to the first,
this is 1/2 squared.

00:01:04.319 --> 00:01:06.020
So the first term
is 1/2 to the 0.

00:01:06.020 --> 00:01:08.090
The second term is 1/2 to the 1.

00:01:08.090 --> 00:01:09.710
The third term is 1/2 squared.

00:01:09.710 --> 00:01:13.470
So the nth term is going
to be 1/2 to the n minus 1.

00:01:13.470 --> 00:01:18.100
So this is just really 1/2
to the n minus 1 power.

00:01:18.100 --> 00:01:19.406
Fair enough.

00:01:19.406 --> 00:01:20.780
Now, let's say we
don't just care

00:01:20.780 --> 00:01:22.360
about looking at the sequence.

00:01:22.360 --> 00:01:24.830
We actually care about
the sum of the sequence.

00:01:24.830 --> 00:01:26.882
So we actually
care about not just

00:01:26.882 --> 00:01:28.340
looking at each of
these terms, see

00:01:28.340 --> 00:01:30.660
what happens as I keep
multiplying by 1/2,

00:01:30.660 --> 00:01:34.710
but I actually care
about summing 1 plus 1/2

00:01:34.710 --> 00:01:40.710
plus 1/4 plus 1/8, and keep
going on and on and on forever.

00:01:40.710 --> 00:01:44.760
So this we would now
call a geometric series.

00:01:44.760 --> 00:01:47.110
And because I keep adding
an infinite number of terms,

00:01:47.110 --> 00:01:49.590
this is an infinite
geometric series.

00:01:49.590 --> 00:01:55.200
So this right over here would be
the infinite geometric series.

00:01:55.200 --> 00:01:59.310
A series you can just view
as the sum of a sequence.

00:01:59.310 --> 00:02:00.890
Now, how would we denote this?

00:02:00.890 --> 00:02:02.950
Well, we can use
summing notation.

00:02:02.950 --> 00:02:05.880
We could say that this
is equal to the sum.

00:02:05.880 --> 00:02:08.060
We could say that this
is equal to the sum.

00:02:08.060 --> 00:02:10.851
Let me make sure I'm not
falling off the page.

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Let me just scroll
over to the left a bit.

00:02:12.600 --> 00:02:20.060
The sum from n equals 1
to infinity of a sub n.

00:02:20.060 --> 00:02:22.530
And a sub n is just
1/2 to the n minus 1.

00:02:22.530 --> 00:02:25.480
1/2 to the n minus 1 power.

00:02:25.480 --> 00:02:26.980
So you just say OK,
when n equals 1,

00:02:26.980 --> 00:02:28.804
it's 1/2 to the 0, which is 1.

00:02:28.804 --> 00:02:30.220
Then I'm going to
sum that to when

00:02:30.220 --> 00:02:33.690
n equals 2, which is 1/2,
when n equals 3, it's 1/4.

00:02:33.690 --> 00:02:36.730
On and on, and on, and on.

00:02:36.730 --> 00:02:39.240
So all I want to do in this
video is to really clarify

00:02:39.240 --> 00:02:41.780
differences between
sequences and series,

00:02:41.780 --> 00:02:44.410
and make you a little bit
comfortable with the notation.

00:02:44.410 --> 00:02:46.120
In the next few
videos, we'll actually

00:02:46.120 --> 00:02:48.820
try to take sums
of geometric series

00:02:48.820 --> 00:02:52.139
and see if we actually
get a finite value.