Thanks! I really needed this. I´m currently preparing for my bachelor thesis about NMR-Spectroscopy and was stuck on the representation of a oscillations with complex numbers. 7:40 blew my Mind!
If we keep the base as base e, then the rotation rate around the unit circle is such that we travel 1 radian for every 1 unit in the number multiplied by i. If we change the base, then we use the change-of-base rule, and get the following for a general base b: b^(i*theta) = e^(i*theta*ln(b)) This means that it would equal: b^(i*t) = cos(ln(b)*t) + i*sin(ln(b)*t) (Since it's not necessarily equal to an angle, I'll simply call it t) As you can see, this means we change the rotation rate by a factor of the natural log of the base. For 2^(i*t), we get cos(ln(2)*t) + i*sin(ln(2)*t). This means, when t = 1, we only travel about 69% of 1 radian around the unit circle, rather than 1 radian. This means instead of travelling 1 meter, if the unit of the unit circle were meters, that we'd travel 69 cm, and end up at an angle of 0.69 radians or 39.7 degrees. If we wanted to travel a quarter turn, instead of selecting t to equal pi/2, we'd have to select t to equal pi/(2*ln(2)), if we were using 2 as the base of the complex exponential instead of e.
@@damianflett6360 sqrt(-1) has no real roots. It makes it worst when you use ijk= -1 You can assign j as sqrt(-1) but then what is i and k? You see the insanity? Only matrix has the real i, j, and k values.
@@lukiepoole9254 you’re confusing quaternions and complex numbers. The two are completely different systems with different applications, which rarely interact, and both of them are perfectly well defined.
Complex numbers are fake invented math because (1) the definition of a complex number contradicts to the laws of formal logic, because this definition is the union of two contradictory concepts: the concept of a real number and the concept of a non-real (imaginary) number-an image. The concepts of a real number and a non-real (imaginary) number are in logical relation of contradiction: the essential feature of one concept completely negates the essential feature of another concept. These concepts have no common feature (i.e. these concepts have nothing in common with each other), therefore one cannot compare these concepts with each other. Consequently, the concepts of a real number and a non-real (imaginary) number cannot be united and contained in the definition of a complex number. The concept of a complex number is a gross formal-logical error; (2) the real part of a complex number is the result of a measurement. But the non-real (imaginary) part of a complex number is not the result of a measurement. The non-real (imaginary) part is a meaningless symbol, because the mathematical (quantitative) operation of multiplication of a real number by a meaningless symbol is a meaningless operation. This means that the theory of complex number is not a correct method of calculation. Consequently, mathematical (quantitative) operations on meaningless symbols are a gross formal-logical error; (3) a complex number cannot be represented (interpreted) in the Cartesian geometric coordinate system, because the Cartesian coordinate system is a system of two identical scales (rulers). The standard geometric representation (interpretation) of a complex number leads to the logical contradictions if the scales (rulers) are not identical. This means that the scale of non-real (imaginary) numbers cannot exist in the Cartesian geometric coordinate system.
I have lots of imaginary powers. The ability to stop bullets being one of them! Great presentation
Wow 😲
@@josephjatto6759imaginary powers.
@@josephjatto6759 he said that it was an *imaginary* power
@@yoylecake313 It's sarcasm bro
My God, had my lessons been this exciting, I'd have stayed for a PhD... ;(
LOVE what you do here!
That rotation is quite impressive.
Thanks! I really needed this. I´m currently preparing for my bachelor thesis about NMR-Spectroscopy and was stuck on the representation of a oscillations with complex numbers. 7:40 blew my Mind!
The rotation is also around the circumference of the unit circle
Platinum Content! Eddie you are a real Gent!
😃
Your lessons are just BRILLIANT!
Thank you.
this is so fascinating i love everything about this channel. love from india !!
7:56 every math student
You’re a great teacher Eddie
I need to know how old is this guy.
36
@@iiChosenOne Commented 36 minutes ago
@@thefoolishgmodcube2644 commented 36 ÷ 4.5 months ago
Why are student at a 90 degree angle to you?
Hey you, have a nice day tomorrow
Grazie
I will thank you very much stranger
why not today?
Thank you, You
Then to finish by saying that there’s not a unique value for 2^i (if we allow coterminal angles)
So cool. So no matter the base (e^i or 2^i) it always rotates around a unit circle? So we write e because it just looks the nicest?
The reason e is used as opposed to say 2 or any other number in general is so Euler’s formula can be used.
If we keep the base as base e, then the rotation rate around the unit circle is such that we travel 1 radian for every 1 unit in the number multiplied by i. If we change the base, then we use the change-of-base rule, and get the following for a general base b:
b^(i*theta) = e^(i*theta*ln(b))
This means that it would equal:
b^(i*t) = cos(ln(b)*t) + i*sin(ln(b)*t)
(Since it's not necessarily equal to an angle, I'll simply call it t)
As you can see, this means we change the rotation rate by a factor of the natural log of the base. For 2^(i*t), we get cos(ln(2)*t) + i*sin(ln(2)*t). This means, when t = 1, we only travel about 69% of 1 radian around the unit circle, rather than 1 radian. This means instead of travelling 1 meter, if the unit of the unit circle were meters, that we'd travel 69 cm, and end up at an angle of 0.69 radians or 39.7 degrees. If we wanted to travel a quarter turn, instead of selecting t to equal pi/2, we'd have to select t to equal pi/(2*ln(2)), if we were using 2 as the base of the complex exponential instead of e.
The complex plane is incredible. Even when I use it in my work I can't fully grasp it or even believe it.
"i" doesn't exist and sqrt(-1) doesn't exist. There is only ONE real "sqrt(-1)" and that is sqrt(2x2 Matrix[-1])
@@lukiepoole9254 can’t tell if crank or really good meme
@@damianflett6360 sqrt(-1) has no real roots. It makes it worst when you use ijk= -1 You can assign j as sqrt(-1) but then what is i and k? You see the insanity? Only matrix has the real i, j, and k values.
@@lukiepoole9254 you’re confusing quaternions and complex numbers. The two are completely different systems with different applications, which rarely interact, and both of them are perfectly well defined.
@@damianflett6360 What I am saying isn't that.
ijk= -1
i^2 = -1 j^2 = -1 k^2 = -1
if j is sqrt(-1)
wtf is i and k?
I hope Mr. Woo goes over i^i - wait that thing is REAL???
me before watching: how is it even possible for these calculators to compute that stuff?
me after watching:I AM UNSTOPPABLEEEEEEEEEE
Beautiful
Something bothers me at 7:41 , when the dot gets near x = -1.... why isn't the value of a = pi....
the base of the complex exponent in this case is 2, not e
Potential.
WOW. i am actually loving math now
Woohoo! I have a new imaginary power!
Can it be a sophism?
Why are the students turned to other way?
Such a sick teacher.
An imaginary power is your power to impress any member of the opposite sex after your eighth pint.
Hello sir i want to ask that (1/e)^e and (e)^1/e are same or not
They aren’t. Why would they be?
No-one reacting there is litterly kid learning complex analysis ?
0:01 WHEN THE IMPOSTER IS SUS
jerma invades every bits of my personal space
Eeeeeh a new vid
Are they Learning complex analysis at 9?!?!
I thought this was a college course
@@forever_put_at_ease they look don't like college students at all
@@Crackkka they're Asian(joke)
Complex numbers are fake invented math because (1) the definition of a complex number contradicts to the laws of formal logic, because this definition is the union of two contradictory concepts: the concept of a real number and the concept of a non-real (imaginary) number-an image. The concepts of a real number and a non-real (imaginary) number are in logical relation of contradiction: the essential feature of one concept completely negates the essential feature of another concept. These concepts have no common feature (i.e. these concepts have nothing in common with each other), therefore one cannot compare these concepts with each other. Consequently, the concepts of a real number and a non-real (imaginary) number cannot be united and contained in the definition of a complex number. The concept of a complex number is a gross formal-logical error; (2) the real part of a complex number is the result of a measurement. But the non-real (imaginary) part of a complex number is not the result of a measurement. The non-real (imaginary) part is a meaningless symbol, because the mathematical (quantitative) operation of multiplication of a real number by a meaningless symbol is a meaningless operation. This means that the theory of complex number is not a correct method of calculation. Consequently, mathematical (quantitative) operations on meaningless symbols are a gross formal-logical error; (3) a complex number cannot be represented (interpreted) in the Cartesian geometric coordinate system, because the Cartesian coordinate system is a system of two identical scales (rulers). The standard geometric representation (interpretation) of a complex number leads to the logical contradictions if the scales (rulers) are not identical. This means that the scale of non-real (imaginary) numbers cannot exist in the Cartesian geometric coordinate system.