Tips to be a better problem solver [Last lecture]
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Tips on problem-solving, with examples from geometry, trig, and probability. Past episodes with integrated quizzes: https://itempool.com/c/3b1bFull playlist: https://www.youtube.com/playlist?list...Home page: https://www.3blue1brown.comBrought to you by you: https://3b1b.co/ldm-thanksHuge huge thanks to Ben Eater: https://www.youtube.com/user/eaterbcAnd Cam Christensen, creator of ItemPool: https://itempool.com/Notes by Ngân Vũ:https://twitter.com/ThuyNganVu/status...Mistakes
, there should be a dx in the integra
, if you notice the mistake here and are inclined to complain, keep watching ------------------ Video timeline (thanks to user "noonesperfect"
9-Problem Solving Principles/Tip
Question 1 (Probability
Who is Ben Eater?
Inscribed angle theorem, θL=2*θ
Tip-
Tip-
Question 
Answer 2
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Tip-
Question 
Answer 3 (Marked incorrectly, Answer: Option D)
Answer 
Explanation for Q1, Floor functio
Tip-
Tip-
Question 
Answer 
Question 
Answer 
Probability graph in Desmos
Revisiting an alternating series sum for ln(2
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Audience questions through tweet
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Python programming for various probability questio
Arts created using Desmos graph tool with mathematical expression
Thank you, appreciation to the team and all. ------------------ The live question setup with stats on-screen is powered by Itempool.https://itempool.com/Curious about other animations?https://www.3blue1brown.com/faq#manimMusic by Vincent Rubinetti. Download the music on Bandcamp:https://vincerubinetti.bandcamp.com/a...Stream the music on Spotify:https://open.spotify.com/album/1dVyjw...If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people. ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribeVarious social media stuffs: Website: https://www.3blue1brown.comTwitter: https://twitter.com/3blue1brownReddit: https://www.reddit.com/r/3blue1brownInstagram: https://www.instagram.com/3blue1brown...Patreon: https://patreon.com/3blue1brownFacebook: https://www.facebook.com/3blue1brown
SHOW LESS
00:00
welcome back everybody it's hard to
00:02
define exactly what mathematicians mean
00:03
when they use the phrase problem-solving
00:06
however you go about it it's going to
00:07
involve some notion of approaching
00:09
puzzles that you've never seen before
00:11
and still being able to systematically
00:13
and creatively find some solution to
00:16
them but that's a weird thing when you
00:18
think about it because it makes it a
00:19
very hard thing to teach I mean you can
00:21
teach someone how to solve one
00:22
particular problem maybe even a class of
00:24
problems and teach them how to solve
00:25
another problem but how do you teach
00:27
someone how to approach a problem that
00:29
they've never seen before and still make
00:31
progress on it well I honestly don't
00:33
know how but what I want to do for this
00:35
lecture is to talk through a couple
00:36
different problem-solving principles
00:38
I've enumerated nine of them in total
00:40
and each one we're gonna walk through in
00:42
the context of a specific example and I
00:44
think each one is kind of simple you'll
00:47
look at it and you'll nod along thinking
00:48
yeah yeah of course that's the thing
00:49
that you should do but I would argue
00:51
that each one is deceptively simple that
00:53
you would be shocked at how often you
00:55
can make very meaningful progress in
00:57
very hard problems just by keeping some
00:59
of these tips in the back of your mind
01:00
and to give you a little flavor for
01:02
where we're going to be going today I
01:03
want to ask not as a quiz that I expect
01:07
you to necessarily be able to solve here
01:08
on the spot I want to ask you a question
01:11
that we will be solving later on in the
01:13
lecture just so you can have it in the
01:15
back of your mind a little thing to mull
01:16
over any a hard problem that will be fun
01:20
to tackle when the time comes
01:23
so the question asks suppose that two
01:25
numbers are chosen at random from the
01:27
range 0 through 1 and it's done
01:30
according to a uniform distribution so
01:32
maybe you pick like 0.385 and point 5 8
01:35
9 6 2 or something like that each one is
01:37
chosen at random suppose P is the
01:39
probability that the ratio of the first
01:41
number to the second rounds down to an
01:44
even number so you know maybe it rounds
01:46
down to zero or rounds down to 2 or to 4
01:49
and basically it's asking you to guess
01:52
where is this probability you know we'll
01:55
solve it exactly we'll get an exact
01:56
expression but just intuitively as you
01:59
hear the problem when you think of
02:00
choosing two random numbers between 0 &
02:01
1
02:02
looking at their ratio what do you think
02:04
that probability is going to be well
02:06
return back to this later and one thing
02:08
I want to say is that this whole live
02:10
quizzing software it's something that's
02:12
built by some friends of mine Ben eater
02:15
who many of you may know because of his
02:16
YouTube fame and then another person who
02:19
used to work with us at Khan Academy
02:20
named kam Kristensen and this is just
02:24
one small little outcropping of what's
02:25
actually a much deeper product at play
02:27
and I want to give you a little preview
02:28
of some of the other things that they've
02:30
been working on that will be developing
02:32
so if any of you want to share some of
02:34
the lectures here or go back and kind of
02:36
go through the live quizzing experience
02:38
if you go to the link that it's in the
02:40
description but it's at item pool comm
02:42
/c / 3 b 1b but you can follow the
02:44
description you can basically watch the
02:47
lectures in a way where you can do these
02:48
live quizzes along with it and your
02:50
progress is tracked and you get scores
02:52
and things like that so as you skip
02:54
ahead to various different questions it
02:56
will skip you to the right point in the
02:57
video and as the video plays forward
02:59
you'll also get to yeah there's just me
03:02
talking through some problem you'll also
03:04
get to whatever the appropriate part of
03:06
the problem is they often have
03:08
explanations associated with them and
03:10
hints and things like that and
03:12
ultimately if you look back in just a
03:15
couple days I'm going to fill out like
03:17
homework and challenge problems so if
03:19
you like challenging problem solving
03:20
contest math type stuff I'm gonna put
03:22
things that are relevant to the lectures
03:23
in there which I think should be fun
03:25
it's not not quite there yet but
03:27
definitely check back and I'm sure just
03:29
over the next couple months there will
03:31
be more item portion Anakin's going on
03:33
if any of you want to use this maybe to
03:36
do some live polling in your own live
03:38
classes for teachers who are dealing
03:40
with the whole remote landscape or if
03:42
you have your own streams anything like
03:44
that
03:44
they are looking for beta users so feel
03:46
free to reach out to them it should be
03:47
available on the website now let's dive
03:51
into the actual content shall we some of
03:53
the nine deceptively simple
03:54
problem-solving tricks and before we do
03:56
I want to specify that at some point
03:59
today in the next hour I'm going to
04:01
purposefully make a mistake okay and I
04:03
tell you that for two reasons one so
04:05
that you keep your eye out and you you
04:06
know you're a little bit skeptical of
04:08
each of the claims that I make and two
04:09
so that when I make that mistake and you
04:11
notice it and you're just you know
04:13
throwing things at the screen you're
04:14
getting angry you're clicking that
04:15
unsubscribe button you can at least
04:17
quell a little bit of what you're
04:18
thinking so just keep that in mind there
04:20
will be one very purposeful mistake now
04:22
before we get to the problem solving tip
04:24
I want to talk about
04:25
geometry this is one thing I was hoping
04:28
to do a little bit more of in this
04:29
series but this would be just a fun time
04:31
to give a little example of it and in
04:34
particular I want to talk about one of
04:35
my favorite little bits of geometry it's
04:38
not too simple but it's also not too
04:40
hard it's called the inscribed angle
04:42
theorem it comes up way more than is
04:44
reasonable for a simple little theorem
04:45
like this to actually come up when
04:47
you're solving problems okay and
04:49
basically an inscribed angle of a circle
04:52
refers to if you have two lines that
04:55
meet at a point of that circle and what
05:00
we care about is this little angle in
05:02
here okay and if you were just studying
05:05
this if you're an early mathematician
05:07
trying to make sense out of this you
05:08
might make a guess that something about
05:10
this angle is going to be related to the
05:12
arc of the circle that those lines hit I
05:15
mean that arc can also be described as
05:18
an angle of some kind if we draw lines
05:20
from the center of the circle it makes
05:23
it a little bit clearer so I'm just
05:25
gonna draw a couple green lines here and
05:27
Mark that we have a different angle here
05:29
and maybe I give them names I'll call
05:32
this one theta L for like the large
05:34
angle and this one I'm gonna call theta
05:37
s it's the small angle and you might
05:41
wonder is there a relationship between
05:42
these two angles and if so what is it
05:45
and for our purposes today the question
05:47
is how do you systematically approach a
05:49
puzzle like this find a relationship
05:52
between these two and prove prove its
05:54
existence so problems of problem solving
05:58
tip it's not problems if problem solving
06:00
tip number one which is just just so
06:04
useful in a way that again is kind of
06:06
deceptively simple make sure you're
06:07
always using the defining features of
06:09
whatever your setup is I swear a good
06:11
70% of the problem sets that I did as an
06:14
undergraduate math major essentially
06:17
came down to looking at what was given
06:18
asking very critically what is the
06:20
definition of each term involved here
06:22
unraveling those definitions and then
06:24
just seeing how they pieced together so
06:25
use whatever's the defining feature in
06:27
our context what is defining the various
06:31
points and intersections that we have
06:33
here well it's the idea that they're on
06:35
a circle a circle is by definition all
06:38
of the points
06:39
that are a common distance from the
06:40
center so in particular we know that
06:42
this length is a radius of the circle
06:44
and this length is a radius of the
06:46
circle we should use the fact that those
06:47
are the same but moreover this other
06:49
point P that was not just chosen at
06:51
random it's defined to be on the circle
06:54
and unraveling what that means it means
06:57
it is a common distance away and it's
07:00
the same distance away from the center
07:01
as these other points so I and then
07:04
inspired to draw a line to add something
07:06
to the picture and to note that it's the
07:07
same there you know quite often when you
07:09
see people solve hard geometry problems
07:11
it comes down to adding something to the
07:13
picture and the most beautiful geometry
07:15
involves adding something that seems out
07:17
of left field and it just illuminates
07:18
everything you shift your perspective
07:19
and sometimes you look at him you wonder
07:21
like how would you have known what to
07:22
add like I get I'm supposed to draw an
07:24
extra circle here or put a you know put
07:28
a rectangle around this triangle
07:29
whatever it is you're doing but how do
07:30
you how do you systematically know what
07:32
you should add so in this context if it
07:34
relates to the definition of your
07:36
objects probably a good idea and that
07:37
line actually will be helpful to us we
07:41
can start giving a couple things names
07:42
that's something that a hint maybe it
07:45
shouldn't even be described as a tip but
07:47
I've put it down as number two that when
07:48
you give things meaningful names that
07:50
actually helps you move forward in your
07:52
problem and in this context it might
07:54
seem like an innocuous thing I'll call
07:56
this little angle that we formed with
07:57
our radius alpha and this little angle
08:00
beta and just to see if that helps us
08:04
move forward you know recognizing if
08:05
alpha and beta show up elsewhere rather
08:07
than me telling you I would like you to
08:09
tell me in the context of our next quiz
08:11
question so before I answer that
08:13
probability one for you I'm gonna pull
08:15
up question number two for today we've
08:18
got a diagram essentially what we just
08:20
drew the dot in the middle is the center
08:23
of the circle and I've labeled seven
08:25
different angles I've just labeled them
08:26
a through G and what I want you to do is
08:28
tell me which of these equations is true
08:31
one possibility is that four of the
08:34
angles a B D and E have a sum that's the
08:38
same as C plus F plus G which just
08:41
looking at C F and G that would be 360
08:43
degrees right the other is that a is
08:45
equal to B and D is equal to e another
08:48
possibility is that C is equal to
08:50
F and then maybe it's all of those or
08:52
maybe it's none of those and it looks
08:54
like we already have strong strong
08:56
consensus on this one with 360 400 just
09:00
a strong large number of people coming
09:02
in agreeing what they believe the right
09:04
answer is and you know if you if you
09:07
want to be involved the place to go 3 B
09:09
1 B Co /live that redirects you to an
09:11
item pool page so that you can follow
09:13
along and for those who are watching in
09:15
the future maybe you're watching it in
09:16
the embedded page where the problem is
09:17
just sitting right below you and even if
09:19
you're not technically participating in
09:21
the data contributing to the live
09:23
statistics what I was going through it
09:25
it's honestly pretty fun to just kind of
09:26
looking like I know I'm not a part of
09:28
this but it kind of feels like I am a
09:29
part of what was happening so I just
09:31
love it
09:31
alright so because there's such strong
09:33
consistence consensus
09:35
I feel comfortable grading this I
09:36
thought I got it right on one one one
09:38
one so the correct answer is that a is
09:41
equal to B D is equal to E and then none
09:44
of the others are necessarily true and
09:46
let's let's walk through why that's the
09:48
case yeah it comes down to exactly what
09:51
we just highlighted that these radii are
09:53
common which means you know let me just
09:56
give more things names let's call this
09:57
point a and this point B maybe that's
10:01
confusing cuz I was just naming angles
10:03
and B but separate contacts separate
10:05
picture you know what I mean the
10:07
triangle a P and then the center of the
10:10
circle that's isosceles you know there's
10:12
this symmetry about I could even draw
10:15
the little axis of symmetry and that
10:17
tells us that this is also alpha
10:19
likewise the triangle up here is
10:23
isosceles and that tells us that this
10:25
has an angle of beta and what we just
10:29
did in effect is leverage a little bit
10:31
of symmetry in this case it was
10:33
innocuous but quite often looking for
10:35
symmetry in much harder setups it's
10:38
super generalizable and it definitely
10:39
will help you move forward if you
10:41
recognize that there's something
10:42
symmetric use that symmetry in some way
10:44
which is effectively what we've just
10:46
done giving things a couple more names I
10:48
might want to call this angle I'll call
10:51
it something related to the alphas so
10:52
maybe I just call it alpha Prime and
10:54
similarly this angle over here is beta
10:57
Prime and by noting those facts I have
10:59
everything I need to do to draw a
11:01
connection between
11:02
this small angle and this large angle we
11:05
essentially just write down the facts
11:07
that we have based on the triangles
11:08
we're looking at so the triangle with
11:10
all of the Alpha angles the sum of those
11:12
angles has to be a hundred eighty
11:13
degrees and there's two different alphas
11:15
in there and then there's an alpha Prime
11:17
and instead of writing a hundred eighty
11:19
degrees of course we like radians so I'm
11:21
going to say that equals PI radians
11:23
similarly the one with all of the betas
11:26
tells us that if we take two times beta
11:28
and we add beta prime that's also PI and
11:32
then the other fact that we have that
11:34
just popping right out from the image is
11:36
that alpha prime beta Prime and theta L
11:38
add up to 360 degrees or two pi so just
11:45
writing down all of the relevant facts
11:46
that we have now we have some objects
11:48
that we can manipulate and work with to
11:51
draw some kind of conclusion which again
11:53
we're looking for a connection between
11:54
theta small and theta large and I know a
11:57
lot of you you're three blue and brown
11:58
audience members you know about the
12:00
inscribed angle theorem but I really
12:02
want you to think about this from a
12:03
beginner's mind right if you were just
12:05
approaching this and you didn't
12:06
necessarily already know about it what
12:08
would you have done to find that
12:09
solution and what principles can you
12:10
take away as you do that because that
12:12
helps us as we start to get to harder
12:14
and harder geometry setups so in this
12:16
context once I have these three
12:17
equations recognizing that there's an
12:19
alpha prime here and one here there's a
12:21
beta prime here and one here I might
12:23
think about cancelling them out so I'm
12:25
gonna you know add this top equation and
12:27
maybe I've subtract off the other two
12:29
equations and to subtract these off and
12:32
what that means is the Alpha prime gets
12:34
cancelled so does the beta prime I'm
12:36
left with my large angle I'm subtracting
12:39
off two times alpha plus beta you know
12:44
each of those get subtracted off with
12:45
the coefficient two and then we have two
12:47
pi minus two copies of Pi so that's all
12:49
equal to zero which is saying the same
12:51
thing as theta L is equal to two times
12:55
well rather than writing alpha plus beta
12:57
I'll just recognize that that is the
12:59
small little angle that we had it's two
13:01
times the small angle again I can't
13:04
emphasize enough just what a weirdly
13:05
useful fact this turns out to be in
13:07
various geometry puzzles you might do
13:09
it's definitely come up on the channel a
13:11
number of times in circumstance
13:13
regarding you know complex numbers or
13:16
pure geometry situations of course just
13:19
anytime you want to relate an angle to
13:21
two times that angle realizing them in
13:24
the context of a circle like this can be
13:26
strangely useful so this is just an
13:28
image to have burned in your mind as
13:30
you're solving problems and to give you
13:32
one example of a problem that you can
13:34
solve once this is sitting there burned
13:35
in the back of your mind I want you to
13:37
remember back to the lecture that we did
13:39
on trigonometry which conveniently it's
13:41
actually the example that I had pulled
13:42
up here so one of the central things we
13:44
were talking about was how just playing
13:46
with graphs you can get this
13:47
bizarre-looking fact that if you square
13:50
the cosine function you get something
13:52
that looks again just like a cosine
13:54
graph and you can get more exact about
13:56
that where if we start with an initial
13:57
cosine graph and you manipulate it a
13:59
little you know we shift it up we scale
14:01
it down we say we need to double the
14:03
frequency you get the exact same graph
14:05
so we have two different expressions for
14:08
the same thing but it's not at all
14:09
obvious why these would be related one
14:11
of them involves doubling the angle it's
14:14
a reference to the other involves
14:15
squaring the output okay so it's an
14:17
obvious fact we proved it using complex
14:20
numbers but what I'd like to do is try
14:22
to prove this using geometry to kind of
14:25
viscerally see the fact pop out right in
14:27
front of us and to do that let's go
14:30
ahead and write down what the fact is
14:32
again so that we can start thinking
14:34
about it we want to find that the cosine
14:38
squared of theta which is just saying
14:40
take the cosine of theta and square it
14:41
it's that awkward notation is equal to
14:44
1/2 of 1 plus cosine of 2 theta
14:50
you've got the strained relationship
14:51
between squaring things and doubling the
14:54
angle which as we've talked about in the
14:55
whole series is really a reflection of
14:58
the fact that a cosine is a shadow of an
14:59
exponential function but let's say you
15:01
didn't know that we want to see if this
15:03
relating geometrically so you don't
15:05
already know the double angle identities
15:06
or anything like that you just want a
15:07
very direct understanding of this
15:09
particular equation
15:10
well one common thing that comes up is
15:13
that a way to show the two non obvious
15:15
things are related or even equal is to
15:17
see if you can find one object that you
15:20
can describe in two different ways so
15:22
we're going to look for one object that
15:23
we have two different descriptions of
15:26
and this often can give you nice
15:27
equations in this context it might mean
15:30
relating the left and right hand side
15:32
this comes up in combinatorics all the
15:34
time when you have a counting puzzle
15:35
where you know you do something like
15:37
count how many ways you can have a
15:39
string of five bits that are either
15:41
zeros or ones and on the one hand you
15:43
can count it multiplicatively kind of
15:44
going through each one and saying well
15:45
you're multiplying the possibilities by
15:47
two but on the other hand you can go
15:49
iteratively and say well how many of
15:50
them have no ones how many of them have
15:52
one one two ones
15:53
something that kind of seems harder and
15:55
a more awkward way to count but by
15:56
describing the same thing twice you end
15:57
up with this really not obvious fact
15:59
from an algebraic standpoint and we're
16:01
going to do the same thing more
16:02
geometrically here but again just gotta
16:04
emphasize how like how general this ends
16:07
up being so what we want is some kind of
16:10
object that each of these describes and
16:12
to do that maybe we just think okay
16:14
let's let's draw a unit circle which is
16:18
where something like a cosine typically
16:20
comes up well for the first time in my
16:23
life I drew a quarter circle arc that
16:25
wasn't terrible it's not great but
16:27
usually that comes out much more
16:29
disastrous how pleasing it should not be
16:33
so pleased with a terrible quarter
16:35
circle great okay so that's our angle
16:40
theta right and what does cosine mean in
16:43
this context not cosine squared but just
16:45
plain vanila cosine well it tells us if
16:48
we look at the x-coordinate of this
16:50
point
16:51
that's the cosine of our angle and so
16:54
now I want you to think about how can we
16:55
represent the square of the cosine as
16:57
some kind of object some geometric thing
17:00
that we can point to in this image and
17:02
the first instinct might be something
17:04
like well let's draw a square off the
17:07
side of this you know something with
17:09
area to it and interpret it like that
17:11
but then there's going to be a problem
17:12
if the principle that we're trying to
17:14
apply is describing the same object in
17:16
two different ways because if we
17:18
describe this left-hand side as some
17:20
kind of area that would mean that we
17:22
have to find another description of that
17:24
same area that same object with the
17:26
right-hand side but that's going to be
17:28
weird because this doesn't involve
17:29
squaring or anything like that if it's
17:31
just a plain vanila cosine term it seems
17:33
much more natural to describe that as
17:35
some kind of ratio or maybe some kind of
17:37
length
17:38
it's just that we need the two theta to
17:41
pop up somehow so instead let's seek a
17:43
way to describe cosine squared that does
17:46
not involve area but is instead
17:47
something more of the flavor of a ratio
17:49
or a length and in this context the key
17:52
comes down to leveraging symmetry again
17:54
and it's a sneaky bit of symmetry it's
17:56
something we didn't talk about in the
17:57
trek lecture but I love it so much I'll
17:58
talk about it again if we think of this
18:01
angle theta on the one hand it's telling
18:03
us the angle between the x-axis and the
18:05
line but on the other hand it's also
18:07
telling us the angle between the line
18:08
and the x-axis and I know that see it
18:10
sounds like the same thing but it means
18:12
when I say project down the point at the
18:15
end of our radius which was length one
18:18
perpendicularly onto the x-axis that
18:22
length got scaled down by cosine theta
18:23
but now what if I do things the other
18:25
way what if I say I want to project down
18:27
in a perpendicular fashion onto this
18:32
line
18:32
well again I just have two lines
18:34
separated by an angle theta I'm doing a
18:36
projection which means it gets scaled
18:38
down by the cosine of that angle so now
18:40
I'm taking the cosine of theta scaled
18:42
down by the cosine of theta and it gets
18:44
me cosine squared of theta so cosine
18:47
squared refers to this length a portion
18:50
of the hypotenuse of our right triangle
18:53
if that hypotenuse had a length of 1
18:55
which in our unit circle it always does
18:57
and incidentally you can show very
19:00
similar reasoning that the sine of theta
19:02
is this other other portion of that
19:06
hypotenuse and this gives you a nice
19:07
sort of a clever proof of the
19:09
Pythagorean theorem baby of double
19:11
projecting based on asking you know is
19:14
is this line theta degrees away from
19:16
that or is that line theta degrees away
19:18
from this it sounds like you're saying
19:19
the same thing but it gets you something
19:20
of mathematical substance to recognize
19:22
that symmetry now why do I say this well
19:25
we have a representation of cosine
19:27
squared what we want now is to do
19:29
something in terms of two theta we just
19:31
were thinking about a context where
19:32
we're able to relate an angle to two
19:35
theta two to two times that angle so
19:38
somehow we want to realize this this
19:41
angle as an inscribed angle of some kind
19:43
of triangle and I'll show you okay how
19:46
you can do that but before I do just to
19:48
mention another fact that if you
19:50
been puzzlingly around with these sorts
19:51
of things might be burning in your mind
19:53
is a specific instance of the inscribed
19:57
angle theorem called Thales theorem so
20:00
let's say that that large angle we had
20:02
two theta was actually a hundred eighty
20:04
degrees and it was pi radians what would
20:07
that mean in terms of the inscribed
20:09
angle theorem it means that if we take
20:11
that diameter of the circle it's 180
20:13
degrees it's just drawing out a diameter
20:16
and we have an inscribed angle with
20:19
lines that hit either end of that
20:21
diameter then this angle is necessarily
20:24
half of that so what it means is that we
20:27
can we can put a right triangle inside a
20:29
circle and whenever you do that the
20:31
hypotenuse of the right triangle is
20:32
exactly the diameter of that circle it's
20:35
a very cute fact if you wanted another
20:37
proof of it that's not just via the the
20:40
inscribed angle theorem there's another
20:41
very wonderful leveraging of symmetry
20:43
that you can do where basically you take
20:45
this point and you say I'm going to
20:47
reflect it through the origin reflect it
20:51
through the center of my circle and see
20:53
where I get and recognize that
20:55
reflecting to the center is the same as
20:56
rotating the whole image 90 degrees so
20:59
if I have rotated the image 90 180
21:02
degrees my other vertex would end up at
21:05
that same point but now what we have is
21:08
a quadrilateral and one of the diagonals
21:11
is a diameter of the circle but the
21:13
other diagonal is also a diameter of the
21:15
circle which in particular means they
21:16
have the same midpoint under the same
21:18
distance apart and it can convince
21:21
yourself a little that implies it must
21:22
be a rectangle that could also be a
21:23
little side homework problem if you
21:25
wanted to chase around the relevant
21:26
angles but I think that's a very
21:28
beautiful way to think about fales
21:29
theorem that you reflect everything 180
21:31
degrees and you necessarily conclude it
21:33
must be a rectangle which means that
21:35
this is a right angle now for our
21:37
purposes what does that mean well we've
21:39
got a right triangle sitting here that's
21:40
from zero we've got one of the points
21:42
here another point on the circle let's
21:44
inscribe that in a separate circle okay
21:47
so I'm going to take a copy of that
21:49
triangle but I'm going to instead of
21:51
making the hypotenuse a radius of the
21:53
circle
21:54
I'm gonna make that hypotenuse a
21:56
diameter of the circle so this is still
22:00
going to be an angle of theta sitting
22:02
right here
22:03
the basically I flipped it around so
22:06
previously I was here but I flipped it
22:08
around so that my ninety degree angle is
22:10
sitting up unto the right rather than
22:12
sitting down here the length that we
22:15
care about is what what happens when we
22:17
project from the point at that ninety
22:19
degree angle in a perpendicular fashion
22:21
down onto the hypotenuse which now looks
22:24
like this and what we care about is this
22:28
long length okay and actually let me ask
22:32
you as a is a live quiz to see if you
22:35
can come up with an expression for that
22:37
length in the context of the diagram
22:39
that we're now looking at so pulling up
22:41
our quiz again congratulations to
22:43
everyone who got this one correct let me
22:47
give you a little bit of time to think
22:48
about this one because this is this is
22:50
kind of a heart part of the heart of
22:54
this particulate proof and I think it's
22:56
very very pleasing to see so it
22:58
specifies that the hypotenuse of the
23:00
large bright triangle above has a length
23:01
of 1 in our context B is because it came
23:04
from the hypotenuse of the triangle
23:06
drawn in a unit circle so that potenuse
23:08
has a length of 1 what is the length L
23:10
in terms of theta yeah so can you find
23:15
an expression for L in terms of theta
23:17
and I'll give you a little bit of a
23:19
little bit of time for that bring back
23:21
our pause and ponder music get myself a
23:23
chance to take a drink
23:48
once again it looks like we have some
23:50
strong consensus for today so while
23:53
answers are rolling in before I grade it
23:54
I'm just gonna go ahead and start
23:55
describing how it goes since it seems
23:57
like a lot of you are already well ahead
23:59
of me it's fun to have though so of
24:01
course we're going to use the inscribed
24:02
angle theorem that is the whole reason
24:04
and bringing it up here it's a way to
24:06
relate a single angle to twice that
24:08
angle so in this context I would draw
24:10
some lines from the centre of my of my
24:14
new smaller circle that has radius only
24:16
1/2 and recognize that this is 2 theta
24:19
and this is just lovely now isn't it
24:21
because what is the length we care about
24:23
part of it
24:24
excuse me part of it is the radius which
24:28
is 1/2 but then the projection down
24:32
according to 2 theta onto this remaining
24:36
leg ends up being that radius 1/2 times
24:39
the cosine of 2 theta and of course
24:44
that's exactly what we want it to be
24:45
we've got the whole radius which is 1/2
24:47
and then we're multiplying that by 1
24:50
plus the cosine of 2 theta
24:51
it's a radius of the circle times a
24:53
scaled down version of that radius so
24:55
that's two different ways of viewing the
24:57
same object which is what we get when we
24:59
take this right triangle and we project
25:00
down and look at what part of the
25:02
hypotenuse that cuts off and it gives us
25:04
this non-trivial relationship in
25:05
trigonometry between the cosine squared
25:07
and the cosine of 2 theta the fact that
25:09
otherwise we were going into like
25:11
complex numbers and Exponential's to
25:13
understand so I think that's quite
25:14
beautiful I think that's just one of the
25:17
many many instances of where the
25:18
inscribed angle theorem suspiciously
25:20
lotion
25:20
it's just suspiciously shows up and
25:23
again what I want you to take away is
25:26
this principle that if you can have one
25:28
object described in two different ways
25:29
very powerful in terms of showing non
25:32
obvious how to break relations or
25:34
anything that's kind of written down
25:35
symbolically without immediate intuition
25:37
on top of it so with all of that let's
25:40
actually turn to the probability
25:41
question that I asked at the beginning
25:43
of the lecture so going back to our live
25:48
quiz I will go ahead and grade what we
25:53
we all now know the correct answer so
25:56
for those of you in a minute oh it was
26:00
marked incorrectly
26:02
oh that's my bad no I just slipped this
26:04
one in like last minute before the
26:06
lesson today so I I might have like
26:08
swapped around what the answers were so
26:10
well it looks like 1380 if we were
26:12
absolutely wrong according to whatever
26:14
jerk wrote this quiz so I don't know how
26:19
that shows up on the user interface if
26:20
it's like shocking read like oh no but D
26:24
was the actual correct answer here so
26:25
congratulations to those of you who got
26:27
that now back to our probability
26:29
question I'm curious to see what people
26:32
said on this one just in terms of their
26:34
instincts so this one we had a little
26:36
bit more of a spread and ah interesting
26:39
here the actual correct answer does show
26:41
up quite a bit quite a bit lower than
26:44
and now I'm questioning myself to make
26:46
sure that I've actually written the
26:48
thing appropriately so just as a
26:51
reminder of what the question is we're
26:52
choosing two random numbers from the
26:53
range 0 through 1 each according to a
26:55
uniform distribution and we're we're
26:57
guessing what the probability that they
26:59
round down the ratio of these numbers
27:01
rounds down to an even number remember 0
27:03
is an even number so that it rounds down
27:05
to 0 where 2 or 4 or anything like that
27:07
now this is a tricky problem to think
27:09
about and definitely no no fault at all
27:13
for anyone who isn't immediately able to
27:17
see roughly where it should be but I
27:18
think with a with a little bit of
27:20
progress on our way even before we get
27:22
the exact solution we can get to a point
27:23
where you might be able to intuitively
27:25
give some kind of ballpark estimate so
27:28
what have we got here
27:29
choosing two random numbers between 0 &
27:32
1 I think that's a weird thing to think
27:34
about especially if you're not familiar
27:36
with probability that well or when the
27:39
phrase uniform distribution is thrown up
27:41
if it's not clear what that means but
27:43
essentially it's what you would expect
27:46
when you're choosing some random point
27:47
on this line and the idea is that each
27:49
point is as likely as another or more
27:51
specifically a given range of points of
27:53
a certain size should have a given
27:56
probability that's independent of where
27:58
that range showed up it's only dependent
27:59
on its size now so you might have in the
28:02
back of your mind the idea that we've
28:03
chosen two points there each somewhere
28:05
between 0 and 1 and just to give an
28:08
example of what we mean by uniform
28:09
distribution the probability that X sits
28:12
between
28:14
point three and like zero point five
28:17
because it's going to be somewhere
28:19
between zero and one and the length of
28:21
that range is zero point two about a
28:23
fifth of the entire length that could
28:26
have come from what it means to be
28:27
uniform is that that probability is
28:30
actually just the length of the segment
28:31
that it came from now one thing you
28:34
might ask is well what if what about the
28:36
probability that it's precisely 0.3 or
28:39
precisely 0.5 would it matter if we made
28:41
these less than or equal to signs and
28:43
the answer is it doesn't actually matter
28:44
because the probability of hitting any
28:46
specific value on a real number line
28:48
ends up being 0 this is a thing that I
28:51
find very confusing
28:52
how can a probability of an event that's
28:54
possible it's definitely possible to hit
28:56
0.3 have probability 0 made a whole
28:59
video about it trying to describe this
29:01
but really what it comes down to is that
29:03
the things that have probability you
29:06
should think of ranges those are the
29:07
fundamental objects and it doesn't
29:09
really matter how we treat the boundary
29:10
and just think in terms of ranges even
29:14
still though what we're asking is a very
29:15
bizarre question which is if we take the
29:18
ratio of x and y and we round that down
29:22
which sometimes no mathematicians write
29:25
using this floor function saying we find
29:27
the greatest integer smaller than that
29:29
how do we know if that's an even number
29:30
that's a weird thing to think about it's
29:32
a hard problem in that way so you know
29:34
if you think through the principles here
29:36
it's kind of like use the defining
29:38
features of the set up well not clear
29:40
how to use the fact that it's a uniform
29:42
distribution I guess we'll be using
29:43
lengths in some way to yield
29:45
probabilities so maybe that gives us
29:47
some geometry but that's not really
29:48
helpful give things meaningful names you
29:51
know maybe x and y are meaningful or
29:53
some something suggestive like that
29:55
symmetry ok maybe you know the idea that
29:58
choosing X and then Y is as likely as
30:00
choosing Y than X you could use that to
30:02
conclude that this ratio x over Y is as
30:05
likely to be above 1 as it is to be
30:07
below 1 and that actually does tell you
30:10
something because if we're wondering how
30:11
often do you round down to be 0 right
30:14
you can say well x over Y is as likely
30:17
to be x is as likely to be bigger than Y
30:19
as Y is likely to be bigger than X so
30:22
there's a 50/50 chance that this should
30:24
happen this would be probability of si
30:29
percent so let gets you somewhere which
30:31
is kind of nice but it's not clear how
30:33
you would apply that to things like the
30:34
even numbers same object two different
30:36
ways unclear so principle number five
30:39
here is where we're going to come in
30:42
again it seems simple it's something
30:44
that you cannot along be like yeah yeah
30:45
drawing pictures it helps to think
30:47
through what I'm looking at but really
30:49
when you find yourself struggling with
30:50
some setup that's not already visual or
30:53
pictorial you know it doesn't have to be
30:55
making a geometric but just having some
30:56
kind of sketch to give meaning to your
30:59
terms can be very helpful and as a more
31:01
specific problem-solving tip when you
31:03
have some numbers multiple different
31:05
numbers see if you can make them
31:07
coordinates in some space so rather than
31:09
thinking about x and y is separate
31:10
things here we'll want to think about a
31:12
single point with XY coordinates and
31:14
what that does for us is it actually
31:16
turns the whole problem two-dimensional
31:17
in a very helpful way so first of all
31:20
I've lost track of my straightedge which
31:23
what have you gone little straightedge I
31:25
can only throw you so far here we go
31:28
things they run away from you even your
31:32
object sometimes get tired of math class
31:34
and want to play true into now and then
31:36
but he has to stay whether he likes to
31:40
or not I'm sorry so let's say this is
31:48
our x coordinate X can fall anywhere
31:50
between 0 & 1 with uniform probability Y
31:53
can fall between 0 & 1 so when we have a
31:55
pair of numbers you know something like
31:57
zero point two what does that maybe like
32:00
zero point eight pair of numbers is just
32:03
a single point in this diagram and now
32:05
to choose both of those numbers
32:07
uniformly at random means that we're
32:08
choosing a random point inside a square
32:10
a square with side length one and now
32:14
maybe we can make a little bit of
32:16
progress because it's going to come down
32:17
to some view of what's going on in this
32:19
square now with this specific example if
32:22
we're thinking about the ratio x over y
32:24
and taking its floor taking just
32:28
rounding it down
32:29
well that's 0.2 over 0.8 that's going to
32:33
round down to zero so this would end up
32:35
being even and like I just said that
32:38
that happens with 50% probability but
32:40
let's see if we can try to find a way of
32:42
thinking about that
32:43
that generalizes up to other examples
32:45
and again one very useful thing if you
32:48
get stuck but I have enumerated down
32:51
here as principle number six is to ask a
32:55
simpler variant of the problem you're
32:57
solving something it's hard it's too
32:59
hard see if you can make it simpler in a
33:01
way that you actually can solve and get
33:03
some kind of foothold maybe that means
33:05
loosening the constraints of the problem
33:06
in some setups or maybe it means looking
33:09
at a sub-problem so in our context
33:11
rather than asking the probability that
33:12
it rounds down to be even let me just
33:14
ask the probability that it becomes one
33:16
but I want you to answer it in a
33:18
geometric way is something that will
33:21
generalize to rounding to other things
33:24
because the next simpler question might
33:25
be probability that arounds down to two
33:26
and things like that you know where this
33:28
is gonna go we're gonna do it as a live
33:30
quiz because rather than me answering
33:32
things I want you guys to answer things
33:34
for me so jumping up to question number
33:36
four at this point it's going to give us
33:39
a couple different diagrams yeah we've
33:41
got a B C and E and it's asking us which
33:44
of these four regions corresponds to
33:46
values of x and y where X divided taking
33:50
the floor of X divided by Y is equal to
33:52
zero which is to say you take the ratio
33:54
you round it down you get zero which
33:56
region corresponds to that fact
34:17
[Music]
34:20
you
34:21
[Music]
34:32
you
34:35
okay I'm gonna go ahead and lock in
34:37
answers but if you want to keep thinking
34:39
about it definitely feel free to pause
34:41
the video and do so I don't want to rush
34:43
anyone
34:43
so looks like twelve hundred and sixty
34:46
five of you 1273 always answers rolling
34:50
in at the end correctly answered that
34:52
it's C and let's take a moment to think
34:54
about why that's the case yeah you can
34:56
do so just with a pile of examples and
34:58
just see which ones seem to fall in the
34:59
region or not but let's see if we can
35:01
understand this in a way that lets us
35:05
make progress on to the other even
35:06
numbers so when we say that it rounds
35:08
down to zero what we're basically saying
35:10
is that that ratio sits somewhere
35:11
between zero and one and it's awkward to
35:14
think of X divided by Y we kind of like
35:16
to think of Y in terms of X so if I
35:18
multiply everything by Y which is okay
35:20
to do with these inequalities because Y
35:21
is always positive so that's not going
35:24
to affect whether the inequality flips
35:25
one way or another I multiply everything
35:27
by Y and we're basically asking when is
35:31
X less than Y and whenever you see an
35:34
inequality the boundary of that region
35:36
is going to be described by the equality
35:38
so we're going to wonder when is it the
35:40
case that y equals x well that's just a
35:43
straight line that goes to agonal if we
35:45
draw our line y equals x that's what we
35:53
get now that's the boundary of our
35:54
region and to know whether we should
35:56
look to the left of it or to the right
35:57
of it either you can think very directly
35:59
and say well you know why should be
36:01
greater than X so at a given point we
36:03
want to move upward in the positive y
36:05
direction you could look at a specific
36:06
example like this but however you do it
36:09
you'll draw the conclusion that
36:10
geometrically the region of points such
36:13
that x / y rounds down to zero is this
36:16
sort of grilled-cheese cut of our
36:18
diagram so with that maybe you can start
36:22
to think about the harder variant which
36:24
is when is it that x / y rounds down to
36:28
be - and again I don't want to answer it
36:32
I want you to answer it when is it that
36:35
x / y inside our unit square of points x
36:40
comma y rounds down to two and we've got
36:44
four possible geometric regions that
36:46
this could
36:47
correspond to a B C and D and really you
36:51
know rather than just think about which
36:52
of these is it really try to think
36:54
through why it's the case and how you
36:56
can prove that one of these boundaries
36:58
is actually what it's supposed to be for
37:00
example I wanted to be the case that if
37:02
I didn't show the correct answer here
37:04
let's say I was just trolling with
37:06
everyone and I didn't show the correct
37:07
answer you would be able to confidently
37:08
come and say like no I'm quite positive
37:10
that the correct answer is nothing that
37:12
you've shown here see if that's the
37:14
level of reasoning that you can put
37:15
behind it so again I'll give you give
37:17
you a little moment to think about that
37:23
[Music]
37:39
you
37:39
[Music]
38:03
okay
38:05
so once again I'm gonna lock in answers
38:06
potentially earlier than you want me to
38:08
but keep the lesson moving forward
38:10
no hard feelings if things haven't
38:12
clicked yet because hopefully the
38:14
explanation will make them do so so the
38:16
correct answer is C which it looks like
38:19
most of you got and let's go ahead and
38:20
think through why that's the case very
38:22
similar reasoning to what we were just
38:24
doing the idea is that rather than
38:26
thinking about this ratio and a floor
38:28
which is kind of a kind of an awkward
38:29
thing
38:30
let's explicitly write out the
38:31
inequality this is referring to it's
38:33
saying that X divided by Y is greater
38:37
than or equal to 2 if it's rounding down
38:39
to that but it's not greater than 3 so
38:41
it's less than 3 and you know again it's
38:45
a little bit awkward to think of this
38:47
ratio so let's write that as 2 times y
38:50
is less than equal to X which is less
38:53
than or equal to 3 times y now quite
38:56
often we don't think of Y is a function
38:59
of X we think of X as a function of Y if
39:01
that makes you feel more comfortable so
39:03
if you want in the back of your mind you
39:04
can kind of think 2 y less than or equal
39:06
to X well that's the same thing as
39:08
saying Y is less than or equal to X
39:10
halves and same deal 3 y being greater
39:15
than X that's the same thing as saying Y
39:18
is greater than X divided by 3 because
39:21
that way we can look at the equality's
39:23
associated with each of these the line y
39:25
equals x halves which has a slope of 1/2
39:27
you can think of it as intersecting at
39:30
the point where y equals 1/2 when x
39:32
equals 1 so it'll be a line like this
39:36
that describes part of the boundary of
39:38
our region and the other line is when y
39:41
is equal to X thirds so we know we
39:44
actually have to be above this line that
39:46
I'm about to draw or one of these
39:48
represents x over 2 and one of these
39:52
represents x over 3 and then part of the
39:56
pardon the intrusion into the space of
39:58
my last inequality all right so we want
40:00
to be above the x equals x over 3 below
40:03
the X divided by 2 this region here
40:06
shows us everything we're rounding down
40:09
gets to 2 now what you appreciate how
40:11
this is a kind of complicated thing to
40:13
think about if we hadn't gone into a
40:15
picture that involves two dimensions if
40:18
you were just thinking of x and y
40:19
varying along this line and wondering
40:22
when is it the case that X is more than
40:24
twice or Y is more than two times what X
40:27
is now yeah sorry X is more than two
40:29
times or Y is but it's not three times
40:30
more than what Y is it's not no no I
40:35
said it wrong I said it wrong when y is
40:38
more than two times what X is but it's
40:40
not three times more than X is it's very
40:43
easy to get confront fuddled in that way
40:45
and it's even harder to try to give some
40:47
sort of probability to that whereas in
40:49
our diagram it has a very clear meaning
40:50
it is the area of this region because
40:53
the full area of the possibilities
40:55
already is one so these the probability
40:59
of something happening should be one and
41:02
we just need to look at the area of that
41:03
and now maybe you can see where this is
41:05
going to go because for the next term
41:07
when we want to know when is X divided
41:10
by Y sit between four and five we're
41:15
gonna be drawing lines that intersect at
41:18
x over four and you'll go to a different
41:20
color for this one x over four and x
41:25
over five which is going to require a
41:29
very small handwriting at this point but
41:30
I'm gonna give it a try nevertheless x
41:33
fourths y equals x fifths and this
41:37
little sliver of area gives us all of
41:40
the times that our ratio x over y rounds
41:44
to be four and we're gonna have to add
41:46
infinitely many of these so that gives
41:48
us sort of another phase of challenge to
41:49
the problem I've drawn this all out in
41:51
desmos by the way if you want to just
41:53
sort of see what some of these regions
41:54
look like where we've got our top region
41:57
of places where it rounds to zero then
42:00
we've got another region corresponding
42:01
to rounding to two rounding to four
42:03
rounding to six and just on and on each
42:06
one of these regions rounding and
42:09
rounding rounding I only went out to
42:11
like a hundred or something like that
42:12
but that gives you a sense of what we're
42:14
trying to do so we've made progress but
42:17
this is still hard what is the area of
42:20
all of those triangles added together
42:21
that's not necessarily an obvious thing
42:24
so let's just start by writing it out
42:27
and seeing what help that can give us so
42:30
every one of these is a triangle it's
42:31
gonna look like 1/2 base times height
42:32
and in fact every one of them if we
42:34
think of the left right direction as
42:36
being their height every one of them has
42:37
a height of one so each one is going to
42:39
look like 1/2 times a base of some kind
42:42
so I'm gonna take don't write this out
42:45
on a different piece of paper actually
42:46
so I can keep it up close I'll think 1/2
42:49
times whatever the base of the triangle
42:51
is times the height
42:53
so our first triangle that bases that
42:56
base has a length 1 so that's going to
42:59
correspond to the 1/2 probability of
43:00
going to 0 the next triangle we have to
43:03
look at this length here between 1/3 and
43:06
1/2 what is that length well actually
43:09
I'm just gonna write it out as 1/2 minus
43:12
1/3 it equals 1/6 but writing it out
43:15
like that kind of reminds us where it
43:17
came from so we don't want to collapse
43:18
things too soon that could maybe be
43:20
another problem solving tip don't
43:21
collapse things too soon and try to let
43:23
your notation have a memory for where
43:25
things came from because sometimes that
43:27
helps see overall patterns this next one
43:30
what's the distance between these two
43:31
points
43:32
well it's 1/4 minus 1/5 that's the
43:35
distance between these given how they
43:36
were defined so we have 1/4 minus 1/5
43:40
and in general we have this kind of
43:42
oscillating some 1/6 minus 1/7 where we
43:46
have all the reciprocals of the natural
43:48
numbers but we're adding in that up
43:50
infinitely many different times and we
43:52
want to know what that sound happens to
43:54
be and from here the the problem-solving
43:59
tip associated with this will seem a
44:00
little bit strange but it might be the
44:04
case that you recognize this fact from
44:06
somewhere else you might recognize let's
44:09
say if you were watching a particular
44:10
lock-down math lecture week or two ago
44:13
this alternating sum one minus 1/2 plus
44:16
1/3 minus fourth plus 1/5 on and on
44:20
actually equals the natural log of 2
44:23
okay and the way this actually came
44:27
about it's such a weird procedure it's
44:29
worth just like walking through again
44:31
really quickly because it's a bizarre
44:32
thing that you're not gonna you're not
44:33
gonna be able to just stare at this
44:35
formula and then immediately see that
44:37
this
44:37
how you're gonna solve it unless it's
44:38
something that you've seen before which
44:40
can make it seem all the more opaque we
44:43
did the strange thing where we made it
44:45
seem like a harder question at first
44:47
where rather than asking about one
44:49
particular sum we turned it into a
44:51
function which is effectively asking
44:53
about infinitely many different sums
44:54
like this so X to the fourth over four
44:57
then we're adding X to the fifth over
45:00
five and the reason for doing this is
45:03
that this plays nicely in calculus land
45:04
because those denominators are now
45:07
related to the exponents in a way that
45:09
we can kind of cancel out by doing an
45:11
integration trick each one of those
45:13
terms that can nicely express as an
45:15
integral of a much more simple monomial
45:18
term X cubed minus now let's see plus X
45:24
to the fourth if I integrate this thing
45:26
each one of them has an exponent that
45:28
increases and then we divide by what the
45:30
exponent is and the reason that you
45:33
would want to do this is that this now
45:37
has a nice way to collapse itself so
45:40
just to make this may be more explicit
45:41
if we evaluate the integral from 0 to 1
45:43
that's the same as taking this whole
45:45
expression and evaluating it at 1 and
45:48
subtracting at 0 so that will give us
45:50
what happens when we plug in at 1 and
45:52
again this is just such a bizarre thing
45:54
that if you hadn't recognized the sum
45:56
seeing someone prove it to you like this
45:58
doesn't necessarily make it feel like
46:00
something that you could have found
46:01
which is frustrating in the context of
46:03
trying to develop problem-solving tips
46:05
that are generalizable but I'll keep
46:07
walking through it just to give a little
46:09
bit of closure to this we've got this
46:12
infinite sum that if you had been
46:13
familiar with geometric sums where each
46:17
term looks like a certain product from
46:19
the last you would be able to write this
46:21
as 1 over 1 plus X because we're always
46:24
multiplying by negative x so you always
46:26
take 1 over 1 minus the thing you're
46:27
multiplying by which again it's kind of
46:30
one of these things where it's relying
46:32
on you recognizing it in some way and
46:34
then the last bit of recognition is
46:36
knowing how to take integrals of 1
46:37
divided by a thing and in this context
46:40
it works out very nicely to just be the
46:42
natural log and we're evaluating this
46:46
between 0 & 1
46:48
which is to say we're taking the natural
46:49
log of one plus one or two minus the
46:52
natural log of one which is zero and
46:54
that's why all of these things are the
46:56
natural log of two and think about what
46:59
has to go on there in order to be able
47:01
to take this and then apply it to a
47:03
probability question you would have to
47:05
recognize the alternating sum as
47:06
something that you had seen from another
47:07
context or if you didn't you would have
47:10
to be aware of this trick to somehow
47:12
turn it into a polynomial that can be
47:14
nicely expressed as an integral that can
47:16
be collapsed because of geometric series
47:17
which can be integrated because of the
47:19
natural log of X and then thinking about
47:21
like what is what is the thing that you
47:23
can teach someone to take say come away
47:25
and be able to solve problems in the
47:27
same way I have what might seem like
47:30
kind of a facetious tip but I actually
47:33
think it's maybe the most potent one and
47:34
the most honest one the way that you can
47:37
get to this sort of point just read a
47:38
lot read as much as you can you know
47:41
watch youtube videos on math if they're
47:42
substantive things like that and think a
47:44
lot about problems which is maybe
47:46
frustrating because what you want is to
47:48
be able to say like well how could I
47:50
have come to this on my own without
47:51
having nearly recognized it but I think
47:53
the truth of the matter is a lot of what
47:55
looks like insight and ingenuity is
47:56
really just pattern recognition but
47:58
wearing a little bit of added clothing
48:00
and sometimes it's patterns not so much
48:02
that you're directly recognizing exactly
48:04
this but maybe you had seen a series
48:06
like this or a tactic like this before
48:08
like geometric series that comes up a
48:10
lot if you read a lot and if you think a
48:11
lot about problems you will recognize
48:13
geometric series even if it's in a
48:15
context that you've never actually seen
48:16
before or a specific geometric series
48:18
that you've never seen before
48:19
similarly if you read a lot and you
48:21
think a lot about calculus knowing that
48:23
you can have the natural log pop out of
48:24
an integral like this it becomes second
48:26
nature and I think recognizing the truth
48:29
of this as being the key to a lot of
48:30
problem solving it's actually pretty
48:32
inspiring because often times you find
48:35
yourself in a situation where somebody
48:37
is they're just faster they're just
48:39
better they're just recognize things
48:40
more so than you do and that can be a
48:44
little bit intimidating right to look at
48:45
one problem think of yourself as pretty
48:47
savvy with math and knowing what's going
48:48
on and then just having someone burn
48:50
through with a wonderful bit of
48:51
cleverness this super beautiful argument
48:53
that leaves you sitting there in the
48:55
dust wondering like why
48:57
I just you know I'm just not in the same
48:59
league at all right and you sometimes
49:01
even think well he just has a math gene
49:03
right that person they just have some
49:05
sort of innate instincts that makes them
49:07
really good at this stuff but I think
49:09
the truth of the matter is the the
49:10
people who are showing that kind of
49:11
ingenuity they Virts expose themselves
49:14
to a huge number of patterns and you too
49:17
could get there right there is a path
49:19
towards that which takes the form of
49:21
practice but not just practice practice
49:23
where you're taking each problem that
49:25
you're looking at and trying to digest
49:26
the deeper principles behind it maybe
49:28
some of the ones I'm trying to talk
49:29
through today like okay I can say
49:31
leverage symmetry but what does that
49:32
actually mean you know what does it mean
49:34
to look at a symmetry of a problem and
49:36
turn that into something that's
49:37
formulaic ly useful you just have to see
49:40
it a lot and then be pensive when you do
49:42
don't just be satisfied with the answer
49:43
see if you can understand why that
49:45
answer came out so in that way this
49:48
number seven like it's it's the most
49:49
frustrating but it's the the most real
49:51
of all the problem-solving tips that
49:53
there can be which is that true
49:55
problem-solving comes down to a kind of
49:57
pattern recognition and there's no two
50:00
ways around it you just have to do a lot
50:02
of practice and expose yourself to a lot
50:04
now I would bet that when we did talk
50:06
about this infinite series a couple
50:08
lectures back you wouldn't have thought
50:09
that that's a thing that you're going to
50:11
be using in a probability question one
50:12
day but that's just how these things go
50:14
they show up in unexpected places
50:16
so what you can do then and say well
50:18
we've got our whole expression that
50:20
involves this very alternating sum and
50:22
you'd say okay that means that the
50:24
answer to our final question is you know
50:28
one half of the natural log of two it's
50:30
one half of what that alternating sum
50:31
came out to be which is very nice
50:33
you know it involves this natural log
50:35
expression and it's very clean and we've
50:37
got this picture for where it came from
50:38
adding up all of these all of these
50:42
areas now at this point there is one
50:44
thing that I think separates really good
50:47
problem solvers the ones who get like
50:48
nearly perfect scores all the time to
50:50
ones who are like merely good who you
50:52
know they aren't necessarily perfect
50:54
scores there's some like sealing
50:55
mistakes they come in here they're at
50:58
this point when you've done the problem
50:59
and you've got your nice elegant
51:01
solution you want to draw a box around
51:02
it you're not done just always always
51:06
principle number eight here always gut
51:09
check
51:09
your answer hey because there's gonna be
51:11
some little mistake that happens all the
51:13
time the great problem solvers aren't
51:14
the ones who just never make little
51:16
mistakes they're the ones who have some
51:17
way of recognizing what those mistakes
51:19
are so in this context let's say I I
51:23
just want to see numerically what my
51:25
answer turns out to be right we said
51:27
that it was 1/2 times the natural log of
51:29
2 so let's just see what does that end
51:32
up being and that 4 log of 2 is around
51:34
point 6 9 so maybe it's not too
51:35
surprising where around zero point 3 4 6
51:38
around 0.35 okay so if we write that
51:40
down as one of the is the answer that we
51:44
just got I told you I would purposefully
51:47
make a mistake so hopefully you're not
51:49
yelling too loud at this point does that
51:52
make sense does that pass a basic
51:54
reasonability test and if you look at
51:56
our picture well the probability has to
51:58
be at least 1/2 because that's the
52:00
region where it rounds down to 0 so
52:02
certainly it couldn't be the case that
52:04
the whole thing adds up to be only 0.35
52:06
so there must have been some mistake we
52:08
had along the way everybody makes silly
52:10
mistakes everybody drops a minus sign or
52:12
apply some rule that doesn't quite apply
52:14
in a circumstance you're not going to
52:16
make you're not going to approach
52:17
perfection by avoiding silly mistakes
52:20
the way to do it is to be able to
52:21
systematically know when you've when
52:24
you've made them so always go check your
52:26
answer have like 2 different
52:27
perspectives that can give you a reason
52:29
ability check in this context if it
52:31
inspired us to go and look a little bit
52:33
more carefully at how we were applying
52:34
things this sum isn't quite the
52:37
alternating sum that converges to
52:39
natural log of 2 in particular we added
52:42
the 1 but then we also add 1/2 and it's
52:44
only after that that we start
52:45
alternating it was plus plus than minus
52:47
plus minus plus on and on and you might
52:50
see this by recognizing we were
52:51
subtracting all the even numbers here
52:53
but we're subtracting all the odd
52:54
numbers here so it's similar but it's
52:56
not the same we will be able to use our
52:58
knowledge but let me just give this part
53:01
where things actually start alternating
53:02
a name something like s what this bottom
53:05
equation is telling us is that when we
53:07
take 1 minus s so that would mean we're
53:10
subtracting in 1/2 then we're adding the
53:12
1/3 then we're subtracting the 1/4 we're
53:14
flipping all of the signs of everything
53:15
beyond that 1 that is the thing that
53:18
equals the natural log of 2 which in
53:21
turn implies that
53:23
that remainder of the sum looks like one
53:25
minus the natural log of two okay so all
53:28
of that what does that tell us when we
53:31
plug it into our original expression
53:33
it's saying that the actual answer
53:35
should not be one half the natural log
53:37
of two that didn't even pay past our
53:39
basic reasonability test instead it'll
53:42
be one half of one plus one minus the
53:45
natural log of 2 which is just 2 minus
53:48
the natural log of 2 so does that pass
53:52
our reasonability check what does this
53:54
actually equal numerically you can get a
53:57
loose approximation in your head if you
53:58
want or if you want to see more
53:59
precisely we can pull up a calculator so
54:02
if we go over here and we're saying no
54:04
no we don't want the natural log of two
54:06
that was wrong too - that may be 0.65
54:10
three does that pass our reason our
54:12
basic reasonability test yeah I think so
54:15
all right 0.65 that looks like a
54:18
reasonable answer to what the area in
54:22
our in our diagram was because if we
54:24
looked at that diagram which was you
54:28
know we've got one wedge here that's
54:30
covering 0.5 and then this other one
54:32
covers well about one-sixth half of a
54:36
sixth basically because this length was
54:38
1/2 minus 1/3 which makes it a sixth and
54:40
then it's a triangle so 1/2 base times
54:42
height so that's about a twelfth or
54:44
point zero eight three and then the rest
54:46
of it you know it's not going to fill a
54:48
huge amount something around 0.65 seems
54:50
pretty reasonable and so that you know
54:53
we could call that good on our gut check
54:55
that this is probably the correct answer
54:56
but we could go one level further if we
54:59
wanted because probability questions
55:01
very often you can kind of cheat and
55:03
just see the answer by simulating it and
55:05
actually taking a bunch of samples and
55:06
seeing what happens so I want to do that
55:09
actually on this one just to give us a
55:10
little bit of confidence that our answer
55:12
of 0.65 that there wasn't some other
55:15
silly mistake that we made along the way
55:17
because everyone knows there certainly
55:19
are silly mistakes that we can make
55:21
before I jump to the programming though
55:22
I just want to see if there's any
55:24
questions from the audience and it
55:25
certainly looks like there are so let's
55:27
see if we can address some of these get
55:29
myself out of the way of the questions
55:32
this is like from the book of The Chosen
55:34
where the rabbi makes an intentional
55:35
mistake in his long
55:36
speech to test his son it is exactly
55:39
like that my children I just want to
55:41
test you every time I make a mistake it
55:43
was always purposeful and I was always
55:44
doing it just to test you I keep making
55:48
more episodes why am i stopping two
55:51
reasons
55:51
you know the main one honestly if I go
55:53
much longer in this lockdown without
55:55
getting a haircut we're gonna have to
55:57
start filming these in the style of like
55:58
a 1980s music video just to keep
56:00
stylistic consistency I'm not up for
56:03
that you know I think that's just gonna
56:04
be a little bit too much effort so I'll
56:06
have to wait until whenever it's
56:07
possible to get a haircut again then
56:09
then we can go really there I also just
56:12
missed my old content
56:13
I love visualizing stuff I've got a long
56:14
pile of things I want to get to and you
56:16
know the lectures take time I might spin
56:18
up something like this again I might do
56:21
it on a separate channel we'll see what
56:22
plays out but I would love to do like a
56:23
full course where it's a little bit more
56:25
clear exactly what we're going to talk
56:27
about from beginning to end
56:28
loosely I'm thinking like combinatorics
56:29
would be fun but don't hold me to that
56:31
and lastly can you help us understand
56:34
arrangements and the principles of
56:36
inclusion and exclusion just like
56:38
combinatory intuitive yeah boy how much
56:43
time do we have today tell you what
56:44
maybe just a separate video if I do the
56:47
combinatorics course actually that that
56:49
would be a perfect example for it it it
56:53
takes a little bit too much to describe
56:55
here but very naturally what you end up
56:56
with is taking 1 over 1 factorial minus
56:59
1 over 2 factorial plus 1 over 3
57:01
factorial minus and you're doing this
57:03
alternating sum with factorials which is
57:06
ye pops out and if you think of E is
57:08
fundamentally being the sum of the
57:10
reciprocals of factorials it doesn't
57:12
seem crazy surprising that it's related
57:13
to counting problems in that way but for
57:16
another day I'll say because I don't
57:17
know if we have time today because I
57:19
would love to just show how you could
57:21
maybe get check this programmatically if
57:23
you wanted to because the very last I
57:25
have principle I have for problem
57:26
solving for mathematical problem solving
57:28
is to learn at least a little bit of
57:30
programming okay and the reason here one
57:32
for what we're about to do you can
57:34
sometimes basically cheat and see what
57:36
an answer is numerically to see that
57:39
that verifies how you're thinking about
57:40
it analytically but also importantly it
57:43
forces you to think about things in two
57:45
separate ways sometimes you can go about
57:46
it mathematically
57:48
but then when you try to make it
57:49
computational you you run into certain
57:52
walls about like how are things actually
57:54
defined or I don't have infinity
57:56
available to me how can I do this in a
57:57
more approximate way very often when I'm
58:00
animating things for usual videos the
58:02
piece of math that I'm describing you
58:03
know it lines up with some way that I'm
58:05
programming it to give the illustrations
58:06
and windows mismatched that's actually
58:09
what's most interesting when it's not
58:10
computationally viable to give a perfect
58:13
illustration of what I'm describing and
58:14
I actually think that does make for
58:16
better problem solving in general where
58:17
you're coming at it from two different
58:18
angles so let's just try this out see if
58:21
we can understand the probability
58:23
question we were just looking at in
58:25
terms of just you know yeah what's the
58:30
word I'm looking for
58:30
cheating seeing seeing what it turns out
58:33
to be so I'm gonna import numpy which
58:37
evidently some heathens refer to as
58:39
numpy which just I just can't
58:41
that seems awful to me so we can find
58:45
random numbers if you just call random
58:48
this first random refers to the library
58:50
the second one is a function it will
58:53
return a number between 0 & 1 according
58:54
to a uniform distribution so it is as
58:57
likely to choose something between you
58:59
know zero point one and zero point two
59:00
as it is to choose something between
59:02
zero point eight and zero point nine and
59:03
what's nice is I can get a list of them
59:06
so in this case I get a list of ten
59:07
random numbers and maybe I call that
59:09
something like X and maybe I create
59:11
another list of Y so X some random
59:13
numbers Y some random numbers and if I
59:16
take X divided by Y it does it term by
59:19
term so for example this this first
59:22
number that we see in here that's point
59:24
seven three nine
59:25
that's taking the point five two divided
59:27
by the point seven this next one is
59:29
taking point six six divided by point
59:31
five six on and on so we can see all of
59:32
our ratios like that
59:33
and in general I might just define the
59:37
ratios list where I'm gonna take my a
59:40
bunch of random numbers of size N and
59:44
then divide it by another bunch of
59:46
random numbers of the same size and is
59:48
not too fine but I'll give it a
59:49
definition some nice and big number like
59:51
a million okay
59:54
and now I'm holding on to a million
59:57
ratios a million examples of X divided
59:59
by y where x and y are
60:01
chosen according to our constraints
60:02
which is kind of cool if you think about
60:04
it so you know we can see some examples
60:06
here some that have come up to 0 and
60:09
some that would round down to zero sum
60:10
that would round down to to some that
60:12
would round down to one so that's nice
60:13
and now I can start asking questions
60:15
like you know when is it that I take
60:18
these ratios I take the floor function
60:20
of them not the floor the floor and I
60:24
want to know when is that equal to zero
60:25
this gives me a list of trues and falses
60:27
basically saying it is or it isn't zero
60:29
and if I take the mean of that which is
60:33
treating the trues and falses as ones
60:35
and zeros this tells me the proportion
60:36
in total that will actually be zeros so
60:39
we expect it to be about a half and we
60:41
can verify yeah okay it's about a half
60:43
we could have also asked when is it
60:46
about to okay and it looks like 0.8 3
60:50
and remember from our diagram what we
60:52
were looking for is when when it's in
60:56
this green region here when it's in that
60:58
green triangle which has an area that
61:00
was 1/2 minus 1/3 but all times 1/2
61:03
because it's 1/2 base times height for a
61:05
triangle so if we pop back over to our
61:08
terminal and say ok if we were looking
61:11
for 1/2 times the base of that triangle
61:15
which was 1/2 minus 1/3
61:17
we would expect that proportion to have
61:19
been point 8 3 and yeah it looks like it
61:21
was about that and we could even answer
61:24
our actual question which is to take the
61:26
floor and then if I say I want to divide
61:28
by 2 and ask when the remainder is 0
61:30
that's a way of asking when it's even
61:31
and then taking that whole list and
61:33
taking a mean of it is a way of asking
61:36
how often that ends up looking like true
61:39
what proportion of them give me true and
61:40
yet point 6 5 which is about the answer
61:43
that we were looking for you know we
61:44
were looking for something that was 2
61:46
half of 2 minus the natural log natural
61:50
log of 2 so we have this this wonderful
61:53
way to kind of empirically verify and of
61:56
course in most of the times with in
61:57
lockdown map that I pulled a Python I'm
61:59
just doing relatively simple things
62:01
numerically if you wanted to you could
62:03
try to visualize stuff so something like
62:05
matplotlib is definitely a great pie
62:09
plot I'll import it as PLT this is a
62:12
very good library for
62:14
just like simple data visualization that
62:16
you can pull up pretty quickly so in our
62:19
context I can pull up a histogram so I'm
62:22
going to take a histogram of all of my
62:24
data and I have to specify a range so
62:27
maybe we just want to look at when the
62:29
when those values get bucketed in
62:32
between like zero and 20 so my number of
62:35
bins is 20 that was about to sneeze but
62:40
I held it in don't you love when that
62:42
happens when you're like a sneeze is
62:43
coming and you don't have to catch it it
62:45
just recedes into the darkness because
62:47
it knows that you're better than this
62:49
needs right again I'm more pleased with
62:52
myself than I should be there and I
62:55
always you always have to add a relative
62:57
width on these sorts of histograms
62:58
because they just look ugly if they're
63:00
all excited by side I think sometimes so
63:02
if I do this and then I show what the
63:05
plot ends up being I have something that
63:08
is shown up on my screen but not your
63:10
screen so let's pop it on over yeah
63:12
there we go so you can get a sense of
63:14
you know this bar represents all of the
63:15
ones that rounded down to zero and you
63:17
see it's about half of them about half a
63:19
million all the ones that round it down
63:20
to one rounded down to two and you can
63:23
get this nice nice sense for what all of
63:25
your data is again just adding that kind
63:27
of empirical validation on top of
63:29
whatever you find more analytically and
63:31
that back and forth I really do think is
63:32
helpful for more like pure mathematical
63:35
problem solving so with that that's
63:37
actually all that I have for the lesson
63:39
today just two quick things that I want
63:41
to go over before we end things here
63:44
I've been using decimals a lot to like
63:47
show graphs of things because I just
63:48
love desmos and I'm actually friends
63:51
with some of the people who work there
63:52
because quite often accompanies people
63:56
are as delightful as its products and
63:58
the CEO Eli
64:00
shared with me the fact that they were
64:01
doing like an art contest among some
64:03
students and I just wanted to showcase
64:05
some of I'm not sure if these are the
64:07
winners of the finalists of the art
64:08
contest but basically in various
64:10
different categories of students who
64:11
were I think it was like 12 to 14 15 to
64:14
17 or something like that
64:16
using like mathematical graphs to try to
64:19
draw pictures okay so keep in mind what
64:21
I'm about to show you our mathematical
64:23
graphs did someone wrote
64:25
just with an analytic description and
64:27
they were they were just prompted to
64:30
create something artistic from that okay
64:32
so one of my favorites and I think this
64:36
one was from someone named Carrie is two
64:39
giraffes that just from an artistic
64:41
standpoint it's actually quite lovely
64:42
and then to think through like actually
64:44
mathematically describing everything
64:46
involved here it's such a beautiful
64:47
blend of well like the creative side of
64:50
things the artistic side with aesthetics
64:52
and everything and the analytic side
64:54
another that was this is just genuinely
64:57
insane this is by a cat simi is a
65:01
Bayesian night so I definitely wanted to
65:03
highlight this on behalf of the team
65:05
over at desmos where each one of the
65:07
curves here is described according to
65:10
something that's called a Bezier curve
65:12
very useful for computer graphics it's a
65:14
kind of cubic parametric term and they
65:17
just recreated a starry night in a way
65:19
that's completely beautiful I think and
65:22
then the very last one which is
65:25
genuinely shocking that you can do with
65:26
mathematical graphs in any way was a
65:28
self-portrait by Jared that's just like
65:32
genuinely insane I mean I remember
65:34
playing around with like a ti-84 and
65:36
trying to come up with little pictures
65:38
of like a smiley face and things like
65:39
that so the amount that things have
65:42
changed in terms of when someone's
65:43
noodling off with their graphing
65:45
calculator in class and what they can do
65:47
truly next level and then at the very
65:50
end I just want to say again like a
65:51
highlighted thank you to Ben eater and
65:53
to cam who've been extremely helpful
65:56
with the whole series in ways that's
65:57
like hard to even articulate properly
65:59
eater in particular I mean he's let me
66:02
borrow a lot of his equipment and helped
66:04
out with like figuring out live footage
66:06
type stuff because that's not something
66:07
I usually do which is not even to
66:09
mention the work on the live stats and
66:11
live quizzes so if you aren't already
66:14
familiar with his channel
66:15
that's simply named Ben eater like 100%
66:17
check it out definitely subscribe to it
66:20
try some of the projects he has a very
66:22
project oriented way of teaching I think
66:24
it's absolutely great so cannot
66:26
emphasize enough how how grateful I am
66:28
in the direction of both of those two
66:30
and check out the the item pool site
66:33
where we are going to let you kind of
66:34
relive the lockdown math experience and
66:36
have like homework
66:39
all challenges associated with each one
66:41
of them so stay tuned on that and if
66:43
you're interested in being a beta user
66:44
of it there will be forms for how you
66:46
can how you can reach out to both of
66:47
them thank you everyone for joining in
66:50
the whole series this has been very fun
66:51
for me very different for me and I will
66:54
shortly get back to the more usual
66:55
videos which I'm very excited about
66:57
there's a couple topics that I just
67:00
think you're thoroughly gonna enjoy
67:01
especially if you like problem-solving
67:02
stuff like this like really getting into
67:04
good meaty problem-solving that's some
67:06
of what's on the horizon and with that I
67:09
will simply say keep loving math and
67:11
enjoy the rest of your day
67:13
[Music]
67:26
[Music]