Think of what school does to pupils. Most of them got scared away from mathematics by their teachers. So am I. After school everyone find his way to avoid mathematics in his regular live. Due to this nearly noone is watching these fabulous videos. But it' s like in the Navy. You have to challenge your fear.
crazy edo mathematical videos are some of the most consistently well performing videos on this platform. Seeing as the quality of his videos are genuinely amazing it’ll only be a matter of time before he snowballs into the same popularity as say 3b1b
Absolutely amazing video! I found this channel because of the video explaining quantum computers and decided to watch through the other videos in order to develop more intuition for the math in that video. I absolutely loved the intuition at the beginning of the video with the animations of rotating and scaling numbers. The only part I didn't like was introducing exp(z) via the Taylor series. While that is obviously a mathematically correct definition it didn't give any intuition for the properties described. After putting the words e defined by d/dx exp(x) = exp(x) in the video you have all the tools necessary to introduce what I find by far the best intuition for the described property: Take the function exp(i*y) at some point. Incrementing the function by some small step dy increment the result by i*exp(i*y)dy (yes I know infinitesimals arent formally defined like this but I find the intuition is great). This means that any output of this function the derivative is that point rotated by i (90 degrees). Or in other words, the derivative of the function is orthogonal to the line from zero to the output at every point which is one way to define a circle. Again I loved the rest of the video and am not trying to detract from it, just trying to add a different geometric intuition because it always annoyed me in school when things like this were introduced as a Taylor series because while I could see that it was correct I didn't know why it was correct.
This channel reminds me of both Viascience and 3Blue1Brown. Keep it up. Great work! Subscribed and liked. I rarely "like" videos, I do not know if the YT algorithm takes into account how often I "like" a video, but if it does, this channel should get a boost. This is really highest quality content.
This video is the only good thing that I watched on youtube this year. I am grateful of the person who not only invested so much time in it but did it with love and perfection. Thanks for making this beautiful video. Now, I know what complex numbers are and how they merge sine and cosine into a single beautiful equation of which Euler's identity is a special case. Shame on RUclips for not showing this video to more people!
i found your channel from your recent video about bits and abstraction and I’m really glad I did (but i did see the kurzgesagt one lol). I’m really interested in maths and computer science and physics and your channel has the best explanations ive ever heard in my life- really appreciate your hard work on this channel, you’re in for a lot of recognition soon if you keep going for sure😊
I watched this after watching your quantum computing video and what you do here is fantastic. I literally said "That's so cool!" out loud at the end lol Great visualization, great sound design and great explanation of course!
Great as always. Had the privilege of finding your channel and your videos are amazing. Thank you so much for contributing so beautifully to our lives.
I love your content, please continue and you eventualy will get the recognition that you deserves. You make me think that complex numbers are the natural extension of the concept of real numbers not an arbritary thing. Thanks so much for this!
I really enjoyed this visual representation but I will admit that I stopped and reran the video, at certain points, umpteen times to clearly understand. It is like learning a new language's grammar. I think if I spent another hour that I could explain it to a high school student .. that is amazing! :-)
Visual makes this much easier to digest. This feels akin to 3 brown 1 blue. I subed great work. I am floored at the sub count. Keep up the wonderful things
Este video es una obra de arte, en verdad la forma en la que explcas y como lo complementas con las animaciones es simplemente maravillosa, gracias por este video buen hombre
Have you heard of geometric algebra? It explores non-real ("geometric") numbers that square to either one of 3 values: -1, 0 or +1 In fact, in GA unit "i" is represented as product of two +1 units, - e2 and e3, - which form a bivector e23 (simply an oriented plane with unit area lying in yz axes). GA is a much cleaner framework for dealing with rotations, reflections and translations within any dimensions, as well as working with quantum mechanics, relativity and many other mathematical fields. It's equations are extremely short, logical, uniform and concise. But it is sadly not widely known by general public and is neglected by oldschool professors.
I think -1 is a quantum phase of +1 made for our convenience, so the real value of √-1 will be a 1 which is plus and minus at the same time , √(-1*1)=√(1*-1) so the real value of √-1 is an alternating plus minus 1, a quantum 1 which can be plus and minus at the same time
I feel like I'm learning maths backwards. So, I got a job where I had to do a lot of Linear Algebra. I learned about vectors, matrices, and quaternions. Eventually I got pretty decent at it, but I never really understood where the formulas came from. Then, I stumbled onto Geometric Algebra. It felt like I finally understood how some of these formula worked. It takes a different route. Vectors, Bivectors, Trivectors, etcvectors... But, it was really interesting learning about the "geometric numbers". I.e. x*x=-1, x*x=1, x*x=0. Now, I'm learning about complex numbers. Which is basically just 2D GA. They used Euler's Identity in some of their examples, and I'd already seen it when investigating the dot product. It's interesting coming at these things from so many different angles. Everyone is talking about the same thing, but in different ways.
Imaginary numbers being associated with phase makes them very applicable to the real world. Almost every radio reciever and transmitter these days use quadrature sampling, so they can sample with a 90 degree and regular phase, so that you can essentially sample above and below zero frequency. You can directly mix your target center frequency instead of needing to offset it in a way to negate the mirror images. With the imaginary component, you can then derive the difference between positive and negative frequency. It takes two dacs, one for the phase shifted "imaginary" component, and one for the real component. Both are mixed with the target frequency before getting sampled.
The real imaginary numbers are negative numbers, because, if you think about it, there's no such thing as a negative number. We just made them up out of convenience.
amazing video, great explanation, thanks for sharing your work. My question is why...why imaginary numbers work in that way, and not in another...I know is a stupid question, but maybe you can help me with the answer
Hi, If I could give a 1000 likes, I would. This is the most direct dans precise vidéo I found on similar topic. I watched all you videos. Please upload more!!!
In seismology with the help of the Hilbert transform, the imaginary component of seismic waves is obtained, and it is assumed that the real data express the potential energy, since when a pendulum reaches its maximum amplitude then its potential energy is maximum, and its energy kinetic is zero, hence the imaginary component expresses the kinetic energy of the wave, in the same way that when the amplitude of a pendulum is zero, then its kinetic energy is maximum. I don't know, but the number i probably represents that nature of kinetic energy, as opposed to potential energy.
I call them "spherical numbers" because i is the fundamental constant of spherical geometry. It also has a few cousins like j from the split-complex numbers, which are also often called "hyperbolic numbers," since j has exactly the same connection to hyperbolic geometry that i does to spherical geometry. (See if you can find what an equivalent would be for flat Euclidean geometry. Hint: it's not the ℝeal numbers.)
I am on 5:27 asking myself, how the hell golden ratio came out of multiplication of complex numbers, which were basically a suggestion after some weird observations from curves found in nature... Universe is either insane, or exists because of mathematics 😮
I think you're assuming that the phi symbol is representing the golden ratio, which, in the case of this video, it isn't. It's just representing a real number for the phase of the complex number.
Is the imaginary component of a complex number, when referenced in quantum computing, meant to represent the third dimension (x, y, i) or time (x, y, z, i)? Or is it just useful for the mathematics?
Neither. The imaginary component is really just a part of the single value. The better way to think of complex numbers in reference to quantum computing is not as a real and imaginary component but as an amplitude and a phase (rotation). The amplitude in quantum mechanics is what's actually relevant and is what determines the probability of something being true while the phase is the property that fundamentaly makes these computers quantum because it is what determines how two numbers interfere with each other (add). So two amplitude 1 numbers added together could be anywhere between 0 and 2 in amplitude depending on their relative phase.
I like to imagine leaving comments on channels I like helps boost its presence on the algorithm. At the same time it feels kind of cargo-culty, talking about it like that.
"Imaginary" and "complex" are bad names. I like "rotational" numbers. Multiplying by -1 is rotating by 180 degrees, and multiplying by the right angle unit i is a rotation by 90 degrees. If there's rotation involved then you know there will be circles somewhere so a relationship between i and pi is a lot less surprising. Though "rotational" doesn't tell you they also do scaling, so it's not the best name either.
There is nothing wrong with "complex." A complex is anything composed of more than one part. Complex numbers have two parts compared to the simple pure-real or pure-imaginary numbers, which only have one part.
Easiest would be to treat them like vectors: real numbers are one-dimensional (have one component). Complex numbers are two-dimensional (have two components). Purely imaginary numbers are also one-dimensional, but on an orthogonal axis, so I've heard them called "lateral numbers."
I am going to be annoying, forgive me for that. 4:42 aCtUaLly tHeRe CaN't bE a fUnCtIOn tHaT gIvEs yOu tHe aRgUmEnt oR pHaSe iN a cOnTiNuOuS wAy fRoM eVeRy nOn zErO cOmPlEx nUmBeR. Ps:I can't believe how difficult was to write it that way hahaha. Also I will give some details in a comment to this for those interested.
If there was such a function f:C~{0}-->R (a function from the non zero complex numbers to the real numbers, as we don't care about magnitude and just the phase or argument (however you want to call it) then we can argue this about the complex numbers of the unit circle. Having that said if there was a continuous function "f" from the unit circle to the real numbers such that for every z in the unit circle, f(z) is an argument of z, by the formula this video is about we will have that z = e^(i*f(z)) = cos(f(z)) + i*sin(f(z)) (the last equality is euler's formula) then we will have that this function is injective, and also surjective over its image, therefore bijective over its image. There are some basic results in a branch of math called topology that say that if have some type of space and a continuous function from this type of space to another different, then its image will have the same "type". For those interested the properties or types of spaces that are preserved by continuous functions are topological properties. Well there are two topological properties that are really important, one is connectedness and the other is compactness. It can be proved that the unit circle is connected and compact (in the usual topology) we have that its image has the same properties, but there are some results in topology that say that the subspaces of the real numbers that are connected are those that don't miss any point in between those numbers that are in the subspace (this means [a,c[ U ]c,b], i.e. the interval [a,c] but missing a point b in between, is not connected) otherwise the subspace is connected. For the compact subspaces of the real numbers we have that those are precisely the ones are bounded and closed ( this means that the extremes of the interval have to be finite and the points in the extremes have to be in the subspace). So a connected and compact subspace of the reals has to be an interval [a,b] where a and b are finite. The thing is that by reasoning above, we have that the image of that function f has to be interval [a,b] as we said before, but if we take out a point of the unit circle and take the image of that resulting set (something like cutting band or a ring), this set is still connected, but its image that is something like [a,c[ U ]c,b] (because f is injective) with c being the image of the missing point in the unit circle, but this set is not connected, so we have a contradiction, therefore the function that we started with cannot exist. Ps: I am sorry if something is not well written or I have missed something, but I had to omit some details, english is my third language, its almost 4am in my country and I am a bit drunk. I don't really why I am writing this. I hope it makes someone get some interest in this kind of things.
At first multiplication of negative and positive must be explained and prooved, why no other way was multiplication was possibble. Then - why 2D-numbers stays "numbers", though they lack some previous functions of numbers, like comparasion.
You can, you can think of equally big forces against each other, where one points in the positive direction and the other one in the negative direction. If you add these forces, you get x-x=0.
imaginary numbers are not 2nd dimension as you told but 4th probably or just a relation between dimensions. because 2nd and 3rd dimensions are real numbers also.
They are just a relation between 2 dimensions. If you have 3 dimensions, then there are 3 different relations between them. Both also need an extra component to describe how much to do nothing for a total of 2 and 4 components respectively. ... wait. That's just the quaternions!
I appreciate your work deeply. I know how hard it is to make these animations. Thanks for doing it.
Imagine working countless hours and getting 600 views..... you're so underrated bro
Nice video...
Think of what school does to pupils. Most of them got scared away from mathematics by their teachers. So am I. After school everyone find his way to avoid mathematics in his regular live. Due to this nearly noone is watching these fabulous videos. But it' s like in the Navy. You have to challenge your fear.
and then you upload a 2 second clip of google translator failing at pronouncing "kurzgesagt" and get a million views.😐
@@VitaMoonshadowOLD 💔
crazy edo mathematical videos are some of the most consistently well performing videos on this platform. Seeing as the quality of his videos are genuinely amazing it’ll only be a matter of time before he snowballs into the same popularity as say 3b1b
He has 18k now
Absolutely amazing video! I found this channel because of the video explaining quantum computers and decided to watch through the other videos in order to develop more intuition for the math in that video. I absolutely loved the intuition at the beginning of the video with the animations of rotating and scaling numbers. The only part I didn't like was introducing exp(z) via the Taylor series. While that is obviously a mathematically correct definition it didn't give any intuition for the properties described. After putting the words e defined by d/dx exp(x) = exp(x) in the video you have all the tools necessary to introduce what I find by far the best intuition for the described property: Take the function exp(i*y) at some point. Incrementing the function by some small step dy increment the result by i*exp(i*y)dy (yes I know infinitesimals arent formally defined like this but I find the intuition is great). This means that any output of this function the derivative is that point rotated by i (90 degrees). Or in other words, the derivative of the function is orthogonal to the line from zero to the output at every point which is one way to define a circle.
Again I loved the rest of the video and am not trying to detract from it, just trying to add a different geometric intuition because it always annoyed me in school when things like this were introduced as a Taylor series because while I could see that it was correct I didn't know why it was correct.
This channel reminds me of both Viascience and 3Blue1Brown. Keep it up. Great work! Subscribed and liked. I rarely "like" videos, I do not know if the YT algorithm takes into account how often I "like" a video, but if it does, this channel should get a boost. This is really highest quality content.
It was so cool how phi was counting up to and landed on pi as the lead up for your explanation of Euler's Identity! Really awesome video!
This video is the only good thing that I watched on youtube this year. I am grateful of the person who not only invested so much time in it but did it with love and perfection. Thanks for making this beautiful video. Now, I know what complex numbers are and how they merge sine and cosine into a single beautiful equation of which Euler's identity is a special case.
Shame on RUclips for not showing this video to more people!
i found your channel from your recent video about bits and abstraction and I’m really glad I did (but i did see the kurzgesagt one lol). I’m really interested in maths and computer science and physics and your channel has the best explanations ive ever heard in my life- really appreciate your hard work on this channel, you’re in for a lot of recognition soon if you keep going for sure😊
I watched this after watching your quantum computing video and what you do here is fantastic. I literally said "That's so cool!" out loud at the end lol
Great visualization, great sound design and great explanation of course!
good work, I'll suggest your channel to my students.
we here bhaiya😬😂
The clicking sounds at 4:40 are so satisfying!
Cool video
Great as always. Had the privilege of finding your channel and your videos are amazing. Thank you so much for contributing so beautifully to our lives.
This is the very best video on the topic I have ever seen! Huge congrats to the author
I love your content, please continue and you eventualy will get the recognition that you deserves. You make me think that complex numbers are the natural extension of the concept of real numbers not an arbritary thing. Thanks so much for this!
Awesome value, thank you for your diligent and original work man!
I really enjoyed this visual representation but I will admit that I stopped and reran the video, at certain points, umpteen times to clearly understand. It is like learning a new language's grammar. I think if I spent another hour that I could explain it to a high school student .. that is amazing! :-)
Visual makes this much easier to digest. This feels akin to 3 brown 1 blue. I subed great work. I am floored at the sub count. Keep up the wonderful things
Wonderful!!!! Great video and very clear explanation . Keep on going!!!! ⭐⭐⭐
Excellent work! Love this.
Wowman, you are something else
Why is this lad not famous yet
the way this taught me so much in so little.....I loved your demonstrations definitely made it all "click"
Thank you verry much for helping me understand intiuitively my math lessons ! Great work !
The sound design is amazing here :)
Este video es una obra de arte, en verdad la forma en la que explcas y como lo complementas con las animaciones es simplemente maravillosa, gracias por este video buen hombre
Amazing Video! In my german engineering classes we call the amplitude and phase form the polar form
Have you heard of geometric algebra?
It explores non-real ("geometric") numbers that square to either one of 3 values: -1, 0 or +1
In fact, in GA unit "i" is represented as product of two +1 units, - e2 and e3, - which form a bivector e23 (simply an oriented plane with unit area lying in yz axes).
GA is a much cleaner framework for dealing with rotations, reflections and translations within any dimensions, as well as working with quantum mechanics, relativity and many other mathematical fields. It's equations are extremely short, logical, uniform and concise. But it is sadly not widely known by general public and is neglected by oldschool professors.
I think -1 is a quantum phase of +1 made for our convenience, so the real value of √-1 will be a 1 which is plus and minus at the same time , √(-1*1)=√(1*-1) so the real value of √-1 is an alternating plus minus 1, a quantum 1 which can be plus and minus at the same time
I feel like I'm learning maths backwards.
So, I got a job where I had to do a lot of Linear Algebra. I learned about vectors, matrices, and quaternions. Eventually I got pretty decent at it, but I never really understood where the formulas came from.
Then, I stumbled onto Geometric Algebra. It felt like I finally understood how some of these formula worked. It takes a different route. Vectors, Bivectors, Trivectors, etcvectors... But, it was really interesting learning about the "geometric numbers". I.e. x*x=-1, x*x=1, x*x=0.
Now, I'm learning about complex numbers. Which is basically just 2D GA. They used Euler's Identity in some of their examples, and I'd already seen it when investigating the dot product.
It's interesting coming at these things from so many different angles. Everyone is talking about the same thing, but in different ways.
Could you please give me a roadmap to learn maths without wasting time, I mean I don't want to learn it backwards as you unfortunately did
Awesome video. Congrats and thanks!
It's impossible to watch your videos and not fall in love with math
Such a great video thank you!!
Imaginary numbers being associated with phase makes them very applicable to the real world.
Almost every radio reciever and transmitter these days use quadrature sampling, so they can sample with a 90 degree and regular phase, so that you can essentially sample above and below zero frequency. You can directly mix your target center frequency instead of needing to offset it in a way to negate the mirror images. With the imaginary component, you can then derive the difference between positive and negative frequency. It takes two dacs, one for the phase shifted "imaginary" component, and one for the real component. Both are mixed with the target frequency before getting sampled.
Excellent work, thank you for sharing
Great video, intuitive explaination and mesmerizing I love it.
But I wanted to know what software do you use to produce this wonderful animation?
great video!
Holy smokes.. we learn this stuff in highschool here, i never understood it before watching this.
At 4:23 shouldn't the resulting vector be 0.62-0.73i?
I think that the numbers were rounded when they were displayed on the screen.
Also remember that √-1 = ±i
1+(x^2) = 0 and x = ±i
03:35 Why didn't you write \arctan in the code?
Just forgot ¯\_(ツ)_/¯
WE NEED MORE........
What program do you use to make these animations? They look fantastic!
The real imaginary numbers are negative numbers, because, if you think about it, there's no such thing as a negative number. We just made them up out of convenience.
amazing video, great explanation, thanks for sharing your work. My question is why...why imaginary numbers work in that way, and not in another...I know is a stupid question, but maybe you can help me with the answer
Hi,
If I could give a 1000 likes, I would. This is the most direct dans precise vidéo I found on similar topic. I watched all you videos. Please upload more!!!
Oh these number systems! Now I have one more in my tool belt, right between binary and decimal.
Nice vids, btw. Please, make more! }:-]
Nice name
I remember when i studied this stuff in highschool,
how do u make this animation manim?
In seismology with the help of the Hilbert transform, the imaginary component of seismic waves is obtained, and it is assumed that the real data express the potential energy, since when a pendulum reaches its maximum amplitude then its potential energy is maximum, and its energy kinetic is zero, hence the imaginary component expresses the kinetic energy of the wave, in the same way that when the amplitude of a pendulum is zero, then its kinetic energy is maximum. I don't know, but the number i probably represents that nature of kinetic energy, as opposed to potential energy.
Did you use manim to make this video?
why does the real and imaginary, and amplitude and phase sound like ways to express cartesian and polar coordinates
They are identical operations.
God tier
Thank you :)
Be careful saying "positive" when you mean "non-negative." Zero still exists.
And in higher algebras that lose ordering, and for decimals, the concept of polarity falls apart
I said the exact same thing and noone paid attention
bro it is a jem. I mean how can this be online and for free. Just bbrilliant work brother
What software do you use to create your videos??
Blender
They should have called real numbers "Normal Numbers", and Imaginary Numbers "Lateral Numbers" TBH.
I call them "spherical numbers" because i is the fundamental constant of spherical geometry. It also has a few cousins like j from the split-complex numbers, which are also often called "hyperbolic numbers," since j has exactly the same connection to hyperbolic geometry that i does to spherical geometry. (See if you can find what an equivalent would be for flat Euclidean geometry. Hint: it's not the ℝeal numbers.)
Subscribed❤
I am on 5:27 asking myself, how the hell golden ratio came out of multiplication of complex numbers, which were basically a suggestion after some weird observations from curves found in nature... Universe is either insane, or exists because of mathematics 😮
I think you're assuming that the phi symbol is representing the golden ratio, which, in the case of this video, it isn't. It's just representing a real number for the phase of the complex number.
Did you use the "manim" Python library to animate all this?
I actually used Blender, I wouldn't be able to make the fuzzy number things in Manim.
@@JoshsHandle Ah fair enough. So even the equations are Blender-made?
@@4sety I use a plugin that lets me type LaTeX in Blender's text editor and then click a button to convert it into a mesh.
YOU ARE HERE AFTER........THE PLATFORM 2...
ARE YOU???
impressive
Which software you used to make this video
It was probably manim
Mathematics is so beautiful..
But what does "the square root of negative one" mean. What root?
@@evgtro8727 it means a number which has an amplitude as well as phase, so when you multiply it with itself, you get a negative result
de locos
Is the imaginary component of a complex number, when referenced in quantum computing, meant to represent the third dimension (x, y, i) or time (x, y, z, i)? Or is it just useful for the mathematics?
Neither. The imaginary component is really just a part of the single value. The better way to think of complex numbers in reference to quantum computing is not as a real and imaginary component but as an amplitude and a phase (rotation). The amplitude in quantum mechanics is what's actually relevant and is what determines the probability of something being true while the phase is the property that fundamentaly makes these computers quantum because it is what determines how two numbers interfere with each other (add). So two amplitude 1 numbers added together could be anywhere between 0 and 2 in amplitude depending on their relative phase.
I like to imagine leaving comments on channels I like helps boost its presence on the algorithm. At the same time it feels kind of cargo-culty, talking about it like that.
"Imaginary" and "complex" are bad names. I like "rotational" numbers. Multiplying by -1 is rotating by 180 degrees, and multiplying by the right angle unit i is a rotation by 90 degrees. If there's rotation involved then you know there will be circles somewhere so a relationship between i and pi is a lot less surprising. Though "rotational" doesn't tell you they also do scaling, so it's not the best name either.
There is nothing wrong with "complex." A complex is anything composed of more than one part.
Complex numbers have two parts compared to the simple pure-real or pure-imaginary numbers, which only have one part.
Easiest would be to treat them like vectors: real numbers are one-dimensional (have one component). Complex numbers are two-dimensional (have two components).
Purely imaginary numbers are also one-dimensional, but on an orthogonal axis, so I've heard them called "lateral numbers."
💜
♥️
I am going to be annoying, forgive me for that.
4:42 aCtUaLly tHeRe CaN't bE a fUnCtIOn tHaT gIvEs yOu tHe aRgUmEnt oR pHaSe iN a cOnTiNuOuS wAy fRoM eVeRy nOn zErO cOmPlEx nUmBeR.
Ps:I can't believe how difficult was to write it that way hahaha. Also I will give some details in a comment to this for those interested.
If there was such a function f:C~{0}-->R (a function from the non zero complex numbers to the real numbers, as we don't care about magnitude and just the phase or argument (however you want to call it) then we can argue this about the complex numbers of the unit circle. Having that said if there was a continuous function "f" from the unit circle to the real numbers such that for every z in the unit circle, f(z) is an argument of z, by the formula this video is about we will have that z = e^(i*f(z)) = cos(f(z)) + i*sin(f(z)) (the last equality is euler's formula) then we will have that this function is injective, and also surjective over its image, therefore bijective over its image. There are some basic results in a branch of math called topology that say that if have some type of space and a continuous function from this type of space to another different, then its image will have the same "type". For those interested the properties or types of spaces that are preserved by continuous functions are topological properties. Well there are two topological properties that are really important, one is connectedness and the other is compactness. It can be proved that the unit circle is connected and compact (in the usual topology) we have that its image has the same properties, but there are some results in topology that say that the subspaces of the real numbers that are connected are those that don't miss any point in between those numbers that are in the subspace (this means [a,c[ U ]c,b], i.e. the interval [a,c] but missing a point b in between, is not connected) otherwise the subspace is connected. For the compact subspaces of the real numbers we have that those are precisely the ones are bounded and closed ( this means that the extremes of the interval have to be finite and the points in the extremes have to be in the subspace). So a connected and compact subspace of the reals has to be an interval [a,b] where a and b are finite. The thing is that by reasoning above, we have that the image of that function f has to be interval [a,b] as we said before, but if we take out a point of the unit circle and take the image of that resulting set (something like cutting band or a ring), this set is still connected, but its image that is something like [a,c[ U ]c,b] (because f is injective) with c being the image of the missing point in the unit circle, but this set is not connected, so we have a contradiction, therefore the function that we started with cannot exist.
Ps: I am sorry if something is not well written or I have missed something, but I had to omit some details, english is my third language, its almost 4am in my country and I am a bit drunk. I don't really why I am writing this. I hope it makes someone get some interest in this kind of things.
0:19 the result won't be negative
why
@@FeynVisa will always be positive and will never be negative are completely different statements
1) "Both real and imaginary numbers are necessary to describe the very real effects of quantum mechanics".
Question: Why?
Because quantum mechanics likes to go spinny, and complex numbers are _the_ way to represent spinny.
I know some of these words.
Great but how do I pronounce kurzgesagt
Court-gas-sight maybe???
curts-geh-saght
At first multiplication of negative and positive must be explained and prooved, why no other way was multiplication was possibble. Then - why 2D-numbers stays "numbers", though they lack some previous functions of numbers, like comparasion.
You can't find negative thing in the universe
Antimatter: 💀
You can, you can think of equally big forces against each other, where one points in the positive direction and the other one in the negative direction. If you add these forces, you get x-x=0.
imaginary numbers are not 2nd dimension as you told but 4th probably or just a relation between dimensions. because 2nd and 3rd dimensions are real numbers also.
They are just a relation between 2 dimensions. If you have 3 dimensions, then there are 3 different relations between them. Both also need an extra component to describe how much to do nothing for a total of 2 and 4 components respectively.
... wait. That's just the quaternions!
You are mixing up different uses of the word "dimension."
So why is i^i ∈ R?
@@trappedcosmos
No.
💜