Think deeply about simple things

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 it you need to understand it now at the beginning okay here's the idea really what it means um to cultivate your ability to think in mathematically abstract terms is to be able to do this i had this this is not original to me i had 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 so think deeply about simple things here's what i mean 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 write this down um if i said to you okay what's um what's one plus one we can manage that that's okay you're like one plus a half i can manage that two right this is slightly more complicated because you know at a certain point in time you knew what numbers were but you didn't know what fractions were or how to work with them but i can keep on making this more complex right i could say well what about this what would this be equal to now you can still do it it might take a bit more time what if i kept on going like you know how i did 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 your mind either five or pythagoras fourth vowel okay see look at this i can work that out but i can make this harder i could do something like this and you're like oh i can still do it but it's more complex so it's harder right and on and on and on okay now here's the idea here's what this is about right the hardest things in maths are not as they get more complicated the hardest things are when they get simpler give an example i've done better this is the simplest geometric shape that exists you might disagree with that but actually there's a big hundreds of pages long mathematical proof for why it is it's called and you should go look this up you can write this down and look it up later it's called the puankari conjecture french guide okay this is one of the most difficult gnarly naughty unsolved problems of the 20th century right to prove that a circle is the simplest shape you're like what's so hard about it it's such a simple shape well what happens when you think about i mean 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 i mean you know something like this look did i have to sphere okay right same shape more dimensions it's still the simplest kind of shape that exists in three dimensions what would a four-dimensional sphere look like okay now we're having trouble because we don't leave in four dimensions but the concrete conjecture says in no matter what number of dimensions you throw at it 4 5 6 27 if you're interested in string theory it's still the simplest shape it's a simple thing but if you think deeply about it there are profound difficult problems in it there are other things right like you know the um the distance that defines this is the diameter right but then of course there's the other distance which is important to this not the radius but the so it comes to the comforts 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 is right but it's a it's a strange number it's a weird number it breaks all the rules that other numbers follow right it's a simple thing but strange things emerge from it when you think deeply when you don't just say oh it's a circle cool don't just do that think deep for that now here's how you do it let me give you two guides for how to think deeply this is what you want to keep with you and refer back to you over and over again for how to do this because you're not used to it number one you want to say why 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 of some kind and then after you get an answer to that you need to ask this question well why is that true right and then your little children if any of you have young hands up younger siblings in your family yeah how old how are you talking about um 11. yeah young 10 to 10 any like four or five four or fives yeah okay so you guys will know right they hit this spot and maybe you remember this if your siblings are older now where they they'll just follow you around and they'll be like hey hey hey and then just tug in your pants 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 and i don't want to answer questions okay but they're very important and they're valuable right why is the sky blue who knows why this guy's blue because because it's it's it refracts light in some way well why does light refract and then why don't you get my point ask it and don't stop asking we're used to either not asking or after a little while sort of shutting up the voice in our head like we try to make our little siblings be quiet but don't you need to listen to it again okay and pursue the answers here's your first technique for thinking deeply about simple things the other thing is think about 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 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 sound here we were here over here okay what is this equal to actually no don't tell me the actual number how do you work it out it's the square root of what points of 3.2 squared right plus 4.8 squared and you can chuck that in the calculator if you're really brave you can try and do in your head i just made the numbers so i assume they're pretty bad numbers okay and off you go right that's going to be a nice number what happens if this number underneath here is not positive but negative right now some people will say you can't do that right 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 you know one two three fractions negative numbers doesn't work you can't do it but what if you could right what if you took a number like that and you gave it a name and then saw what happened right now we call these um imaginary numbers and when you put them together with other numbers the numbers you're more familiar with like like 3.2 and we call these the complex numbers right now it just so happens but it took decades for this to come out it just so happens that um imaginary numbers and complex numbers despite the name they're just as real as any others and they follow the same kinds of mathematical rules but it came from some crazy guy saying well stuff you guys who say you can't do it what if you can right now just as a side note some of you picked up there are some um 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 okay and then we got 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 we sort of ripped them around like they were normal numbers and you can't because they're not normal numbers they're going forever and they're weird and crazy they 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 will guide you to thinking deeply about simple things okay and that's what's going to help you cultivate your ability to think about mathematically advocating okay

Channel: Eddie Woo

Views: 1,394,232

Keywords:

Id: -YJSDJGyIaU

Length: 9min 1sec (541 seconds)

Published: Fri Jul 18 2014

Reddit Comments

[u/brinkbart]

I got nothing from this whatsoever.

[u/poggy39]

He would have kept me interested in class instead of having to hire a tutor after class explaining what the professor was trying to convey in our class. Math professors have a way of putting students to sleep quickly without drugs.

[u/[deleted]]

hes a really good teacher, duper engaging.

EDIT: when he said "whats 1 + 1 and a half. I can mannage that too." ....I would have been shouting out nahh bro it,s 1.5