Think deeply about simple things

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what is this about this is really

important because in some ways this is

kind of the heart of the course you need

to understand you need to understand it

now at the beginning there's the idea of

the zone what it means to cultivate your

ability to think in mathematically

abstract terms is to be able to do this

how this is not original to me I have

this explained to me by a really

excellent university lecturer and I'd

love you to write it down it is to think

deeply about simple things something

deeply about simple things here's what I

need okay

we're used to seeing you know problems

that start simple and then they get more

complicated and we are used to thinking

of more complicated things as harder

classic example if I said to you this is

you have to understand if I said to you


what's um what's 1 plus 1 we can manage


that's ok you're like 1 plus 1/2 I can

manage that - right this is slightly

more complicated because you know and

this purpose 7 point in time he knew

what numbers were but you didn't know

what fractions or how to work with them

but I can keep on making these more

complex I could say well what about this

won't this be equal to now you can still

do it I might take a bit more time what

if I kept on going like you know how do

my series before right so I take a

simple idea and I just make it more

complicated I'll make it harder or I'll

give you another one

what's the first word that comes to mind

either five or Pythagoras private ballot

okay I guess yeah I can work that out

but I can make this harder I could do

something like this

I know I know I still do it but it's

more complex so it's hard right

I want to or not okay now here's the ID

here's what this is about

the hardest things in maths are not as

they get more complicated the hardest

things are when they get simpler give me

a tip this is the simplest geometric

shape that exists you might disagree

with that but actually there's a big

hundreds of pages long that make proof

for why it is it's cold and you should

be son you can write this down and look

nothing up it's called that Poincare

conjecture French guy okay

this was one of those difficult gnarly

knotty unsolved problems of the 20th

century right to prove that a circle is

the simplest shape you're like what's so

hot about it it's such a simple shape

well what happens when you think about

me this is how many dimensions is this

picture that I've drawn it's two

dimensions right what would it look like

if it was in three dimensions I'll try

this looked at ionosphere okay right

same shape more dimensions it's still

the simplest kind of shape that exists

in three dimensions what a

four-dimensional sphere look like okay

now they're having trouble long as we

don't leave in four dimensions but the

Poincare conjecture says no matter what

number of dimensions you throw at it

four five six 27 if you're interested in

the string theory it's still the

simplest shape it's a simple thing but

if you think deeply about it they're a

profound difficult problem in it there

are other things right like you know

they are the distance that defines this

is the diameter

right but then of course there's the

other distance which is a point to this

not the radius but the circumference

right okay but we all know what the

relationship between the circumference

of a circle and its diameter is right

what's the relationship it's if we call

it PI it's a number and you can recite

what the number it is right but it's a

it's strange number

it's a weird number it breaks all the

rules that other numbers follow like

it's a simple thing but strange things

emerge from a when you think deeply in

it you don't just say oh it's a circle

of cool don't just do that think deep

for that now here's how you do it let me

give you two guides I had a big diamond

this is what you want to keep with you

and refer back to you everyone's game

for how to do this because you're not

used to it number one you want to say

why is this the case why is it true that

this number is so unusual okay and then

you might come up with an answer or some

kind and then after you get an answer to

that you need to ask this question well

why is that true all right and you know

you know they were children if any of

you have young heads up younger siblings

any family yeah how old any any like

four or five four or five yeah okay so

you guys will know right they hit the

spot and maybe you remember this

interesting wings up hold it now where

they they'll just follow you around and

they'll be like hey hey why why this why

that right and you're like man you're so

annoying I've got things to do

I've got I've got you know YouTube music

videos to watch it I don't want to

answer questions okay but they're very

important they valuable right why is the

sky blue your noise wise guys because

because it's it's it reflects light in

some way why is my true friend

you get my point

ask it and don't stop

we're used to either not asking or after

a little while sort of shoving up the

boys showerhead try to make our little

siblings be quiet

but don't need to listen to again okay

and pursue the answers here's your first

technique for thinking deeply about

simple things the other thing is what if

okay now there's lots of things which

people will tell you you cannot do okay

you cannot do this you cannot do that

and so it's like well what would happen

if that happens but you don't know and

you will never know but in maths you can

always know right what if you've got

numbers right numbers like Sam here we

were over here what is this equal to

actually no to the action of how do you

work it out it's a square root of 103

points you squared right plus 4.8 squid

and you can chuck that in the calculator

I should really brave you can try and

doing you add I just made the numbers so

I assume they're pretty bad numbers okay

so now if you go right that's gonna be a

nice number what happens if this number

underneath here is not positive but

negative now some people said you can't

do that right yeah you can't you can't

take a like what what number what number

can you say that's equal to that will

give you back a negative number when you

square it yeah one two three fractions

negative notice doesn't work you can't

do it but what if you could write what

if you took a number like that and you

gave it a name and then saw what

happened right now we call these

imagining numbers and then you put them

together with other numbers the numbers

you want familiar with like like 2 and

we call these the complex numbers right

now it just so happens but it took

decades used to come out it just so

happens that imaginary numbers and

complex them is despite the name they're

just as real as any others and they

follow the same kinds of math

medical rules but it came from some

crazy guy saying well stuff for you guys

you said you can't do it

what if you can now this is a sideman

somebody picked up there are some there

are some inconsistencies that come out

when you think about this thing right

for instance one of the lines in my

proof one of the early ones was this and

then we thought what was on the right

hand side you remember this right now

people said you can't do this you can't

do it you can't just we sort of whipped

them around like they were normal

numbers and you can't cuz they're not

normal numbers they go on forever and

they're weird and crazy don't follow the

rules but what if they did right and

then these kinds of things emerge right

these two questions or are these two

sets of questions right they would guide

you through thinking deeply about simple

things okay and that's what's going to

help you cultivate your ability to think

about mathematically outside okay