Eddie is the only maths teacher with neat handwriting lol
@kolokayla4899
3 жыл бұрын
True lol
@reynep
7 ай бұрын
He could have been a really fkin great artist
@kiaruna
3 жыл бұрын
You're AMAZING thank you so much !! as a med student I must review my basic knowledge from high school, it's so refreshing to learn that way !
@xxphoenixx8398
3 жыл бұрын
I've a friend who loves talking about euler's formula but I don't understand anything about mathematics, so seeing this video is making me wanna see>:)))
@lgl_137noname6
3 жыл бұрын
Say Eddie, will this 4-part be its own playlist ? Good reference. Thanks !!!
@ahusky4498
3 жыл бұрын
oooh this is a sick video!
@Qermaq
3 жыл бұрын
Can you link to the videos in the description please?
@bhobba
Жыл бұрын
Another great video Eddi; Kudos. However, there is a much simpler way of doing it. You switch your view of i to it being an operator that rotates whatever is after it by 90%. If nothing is after it, take it as 1 follows it. So i means rotating the number 1 by 90% anticlockwise (by convention). Hence i^2 = -1. Now the question is, what is the operator f(x) that rotates whatever is after it by the angle x. Well f(x) = f(n*x/n) = f(x/n)^n. But if n is large, the rotation by angle x/n = 1+(i*x/n) to a good approximation, getting better as n becomes larger. So to good approximation f(a) = f(a/n)^n = (1+ i*a/n)^n and we expect this to be exact as n goes to infinity. But from calculus, we know the e^x = (1+x/n)^n as n goes to infinity. It is only reasonable to define e^ix = (1+ ix/n)^n as n goes to infinity. So you get e^ix as an operator that rotates by an angle x. Doing this early on makes proving the trig identities, derivatives etc a snap compared to what is usually done.
@zizo-ve8ib
3 жыл бұрын
Amazing video as usual Out of topic though, I was wondering what program you're using to write here, and the tablet type too if you don't mind of course, out of curiosity ☺️
@georgedoran9299
3 жыл бұрын
I believe that’s notability, I know it’s on an iPad not sure about others
@PauxloE
3 жыл бұрын
What I'm missing here a bit is the motivation. What do we want to do - we want to *define* e^(i·x) in a way that's compatible to the existing differentiation rules for e^x for real numbers and the power laws (so we can later define powers with arbitrary complex exponents, and then also other bases, using the logarithm). It's not immediately clear that there is a function at all which works like this for complex numbers. So we *assume* there is some function x → e^ix (for real x) which works that way, and then check how it could look like. This gives us Euler's formula - so we know how e^ix would look like, if it exists. We then can check that also works as a definition, using the properties of sin and cos, which completes the actual theorem: There is exactly one function exp : iℝ → ℂ with exp' = exp, exp(a + b) = exp(a)·exp(b), [not sure whether others criteria are needed] and it is given by exp(i·x) = cos(x) + i·sin(x).
@bhobba
Жыл бұрын
See the answer I wrote. Also, remember this is calculus, which by tradition, is usually done without being totally rigorous. Later you may do what is called analysis, where all the hand-wavy stuff used in calculus is rigorously done. You could start with analysis, but many application areas do not require rigorous development - so it is split into calculus followed by analysis. If you want rigour, a ton of free books on analysis are available on the internet. Colloquially studying analysis is called doing your epsilonics for reasons that will be clear if you study it. But a word of advice - only attempt it once you have done the calculus. Really good students could, and some honours courses at university do it that way, but although I love analysis, IMHO, you are simply complicating things unnecessarily.
@rodicabrudea923
3 жыл бұрын
you said there are numbers other than complex (and the previous ones). which are they?
@caruccio
3 жыл бұрын
noooooooooo.. where is the other video??? i need to sleep
@lukaskrause6022
3 жыл бұрын
I don’t understand why the field of complex numbers is algebraically closed. Cant you use roots to create complex numbers from real numbers? For example, when given X^2+1=0 complex numbers naturally arise.
@ProjectDeathSpeakers
3 жыл бұрын
I wonder what kind of students this is aimed at. Is this for students of a (high) school or for students of a university? In Germany, we don't cover complex numbers in school.
@firstname9845
3 жыл бұрын
It’s highschool
@Shinoken53
3 жыл бұрын
What program are you using to present?
@kalechips22
3 жыл бұрын
He uses notability
@jeremypnet
3 жыл бұрын
ℕ and ℤ aren't fields.
@OriginalSuschi
3 жыл бұрын
Why aren’t they?
@OriginalSuschi
3 жыл бұрын
Why aren’t they?
@jeremypnet
3 жыл бұрын
@@OriginalSuschi Because the elements of the field have to have an additive inverse and all the elements except the additive identity (zero) must have a multiplicative inverse.
@OriginalSuschi
3 жыл бұрын
@@jeremypnet so the natural numbers aren’t a field, because they don’t have an additive inverse (nor a multiplicative inverse) *, while the integers do have additive inverses, but no multiplicative inverse *contained in the set itself. Ok, thanks for explaining!
@jeremypnet
3 жыл бұрын
@@OriginalSuschi Yep. Correct.
@PauxloE
3 жыл бұрын
Please don't call the natural numbers or integers a "field" - that name has a defined meaning, and it's not "set with operation". (ℕ is a half-ring and ℤ a ring, but certainly they are not fields (which needs two operations with their inverses and distributive laws).) The irrational numbers also is not even a "set with operations" (even less a field) (there are sums and products of irrational numbers which are rational), it's just the name for the set of real numbers which are not rational. An actual field between rational and real numbers might be the algebraic numbers (which include things like sqrt(2), but not pi or e, which are transcendental (non-algebraic)).
@naazir2011
3 жыл бұрын
TUSM dude
@OriginalSuschi
3 жыл бұрын
Quaternions are really sad now :((
@johanna9108
3 жыл бұрын
FIRST!! I LOVE YOUR VIDEOS EDDIE!
@user-bv8ix4fy2t
3 жыл бұрын
It's like the vigorous thing
@user-bv8ix4fy2t
3 жыл бұрын
I'm so ashamed
@user-bv8ix4fy2t
3 жыл бұрын
It's alright we come out the same
@frankteunissen6118
3 жыл бұрын
But, Eddie, you should have introduced imaginary numbers before complex numbers. The concept of i = SQRT(-1) is one hell of a mind jump for students.
@MCLooyverse
3 жыл бұрын
It really messes with me when he talks about the progression of numbers, then shoves the irrationals into this nice sequence of supersets.
@learnenglishwithjojo
3 жыл бұрын
Yeah aren't the real numbers just a combination of the rationals and irrationals or am I missing something
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