Demystifying Euler's Formula

In this video, I attempt to demystify Euler's formula by showing you how to derive the 2x2 matrix form if Euler's formula, from first principles. This is a culmination of many sleepless nights. I hope it was worth it.

[@zlClutchy]

This channel is pure gold. Thank you for taking the time out of your day to make these videos, even though you don’t have to. You are more appreciated than you know. Cheers! 🎉

[@scolem2681]

Thanks! Again! I'm starting to see that I need to "always" approach a complex # as a 2D #

[@dr.rahulgupta7573]

Excellent presentation. Vow !!

[@mathsciencefancier]

You've worked for it, at least for 15yrs, based on the first video of this channel!

[@mathsciencefancier]

I feel very lucky to find this channel! I'm fan of fractal math.

[@FractalWoman]

Thanks. I feel I am very close to proving that the universe is "fractal" (in nature) beyond a reasonable doubt.

[@whig01]

Tau can also be seen as T for Turns. And the Pi symbol is like a Tau with an extra vertical stroke, so if the downstroke is the radius, it's twice the radius. Just visual things.

[@FractalWoman]

I like that.

[@beautheory6047]

Great primer Lori. I feel like this goes a long way to explaining all the non-critical zeros in the Reiman hypothesis that only show up at 1/2, and why the critical ones are only even negative numbers.

[@heterodoxagnostic8070]

very good video. could you make a series on quaternions (etc.), perhaps you could find a 4x4 matrix or something like that for it?

[@4v4t4rmusic]

Thank you!🙏

[@tootalldan5702]

Great video. The final 2x2 matrix video aid would be a spiral pattern for example the travel along the x axis as seen as 3d. You would always be 1 away from x axis. Sin waves are so flat and great for the look down effect. Spirals are more true for 2x2 matrix visually. :) Thanks for Tau representation too.

[@anthonywall5227]

Thank you

[@mcjgenius]

ty

[@pixbee3759]

Enjoyed this presentation. Thank you. Really curious what the 4x4 would look like wrt quaternions

[@FractalWoman]

I did make a video called Demystifying Quaternions. Here is what Euler's formula looks like in 4x4 matrix format: [cos sin sin sin] [-sin cos sin -sin] [-sin -sin cos sin] [-sin sin -sin cos] So it is cos on the forward diagonal, and sin everywhere else with some negatives and positives. This can be used to do rotations about an arbitrary axis, although you need the complex conjugate as well.

[@pixbee3759]

@@FractalWoman Amazing thanks. finding your work really fascinating :)

[@mathsciencefancier]

Exponential growth and decay is related to number e? I wondered this. I think I must watch this after my swim-class now.

[@giovannip.1433]

e =2.71.... Is there a 'base' in which the 'number' is a whole? e.g. base 3 = 0,1, 2, 1-0, 1-1, 1-2, 2-0... We are familiar with hexa-decimal, binary etc. All our calculations seem to be based on base 10. Degrees are in a different base...

[@lincolnuland5443]

"An irrational number is a type of real number which cannot be represented AS A SIMPLE FRACTION. Transcendental numbers are irrational". "e is transcendental." Going from that I would say not. But who's to say there isn't some odd way to make it work... At some point it's going to become obvious that singular bases are ridiculous and cannot actually indicate a number's 'address' effectively. Eventually it will become clear that the only base that is rational (english rational) is non-finite but that's a long ways off and we're going to have to throw away a lot of junk to get there. I'm willing to bet they'll be writing the same proofs in english and spanish just make sure they jive with relativity and that uncertainty asshole.

[@monoman4083]

I'm suitably demystified... thanks

[@FractalWoman]

Awesome.

[@xjuhox]

As you see, if we expand the term e^x into its real power series and then identify cos and sin series, then we can _define_ what it means when a real number is raised into a imaginary/complex power. It must have been a true heureka moment for Euler when he noticed that fact after playing around with power series with imaginary arguments.

[@FractalWoman]

The biggest revelation for me was when I realized that I could use the 2x2 matrix form of complex numbers to implement the Mandelbrot set using the equations Z = Z^2 + C DIRECTLY instead of having to break it up into it's algebraic components: Zreal = Zreal*Zreal- Zimag*Zimag + ConstantReal Zimag = 2 * Zreal*Zimag + ConsantImag The algebraic formalism doesn't look anything like Z = Z squared plus C.

[@xjuhox]

@@FractalWoman When you compute Z^2+C you are still using algebra, i.e. _matrix algebra._ And in fact, the matrix multiplication does unnecessary many calculations when it computes every entry of the resulting 2x2 matrix.

[@FractalWoman]

@@xjuhox Yes. I know. However, the 2x2 matrix version of complex numbers gives two solutions. In most situations, you only need one solution. But what if that other solution has some physical meaning that we are ignoring because we want to be efficient? Just think about it.

[@xjuhox]

@@FractalWoman Hmm... But should that iteration converge to only one "solution" point in the complex plane? Or are you talking about a polynomial equation z = z^2 + C, that always has two complex solutions?

[@FractalWoman]

@@xjuhox Technically, yes, except that, in the 2x2 matrix formalism, I see it as two REAL solutions embedded into one complex number (one 2x2 matrix), not two complex solutions.

[@Kydo2003]

How do you use the 3x3 rotation matrix on a 2x2 matrix? Thank you 🙏

[@FractalWoman]

As I showed in the video, the 2x2 Euler matrix is embedded into a 3x3 matrix which in turn is embedded into a 4x4 matrix. Depending on where the 2x2 matrix is placed, you can rotate a 3D point about one of the three orthogonal axes, x, y or z. You can read morea bout that here: www.brainvoyager.com/bv/doc/UsersGuide/CoordsAndTransforms/SpatialTransformationMatrices.html You will see the three rotation matrices I showed in my video, only for y and z, they use the complex conjugate which changes the rotation direction. Otherwise, they are the same.

[@Kydo2003]

@@FractalWoman thank you for the response. I think I follow what you’re saying. I’m used to representing a 3D point as a 1x3 matrix. By taking the sumproduct with a 3x3 identity matrix I get another 1x3 matrix with the rotated coordinates. How do you represent a 3D point in a 2x2 matrix?

[@FractalWoman]

@@Kydo2003 "How do you represent a 3D point in a 2x2 matrix?" Of course that is impossible. You can only do one plane at a time with a 2x2 matrix which is why you have to do Pitch (y,z rotation about x), Yaw (x,z rotation about y) and Roll separately (x,y rotation about z). If you want to do all three dimensions at once, you need quaternions.

[@Kydo2003]

@@FractalWoman ok, many thanks. I kinda suspected that might be the case as the 2x2 represents a ‘simplex’ number in the 2D simplex plane. I just can’t figure out the details of representing a 3D point by 3, 2x2 matrices. Will look into quaternions. Thank you.

[@FractalWoman]

@@Kydo2003 "I just can’t figure out the details of representing a 3D point by 3, 2x2 matrices." A point (x,y,z) can be represented by 3 2x2 matrices: [x, y ; -y, x] , [y, z ; -z y] and [z, x ; x ,-z]. If you multiply the first one by [cos(t), sin(t) ; -sin(t), cos(t)] you will get a counter-clockwise rotation in the x,y plane about the z-axis. The rest you can probably figure out for yourself.

[@3zdayz]

:) I'm not trying to change you; but I do want to share an alternative view. turns is a way of looking at things, but it turns out to not be exactly as easy as quarter turns. If you have 1 = 90 degrees like 'i' = 90 degrees( or pi/2 radians) .. and 1i and 1 are pretty similar (there needs to be some sort of additional thing like 'i' associated with it somewhat). But then a sine wave's cycles are numbered 0 1 2 3, and 0 1 0 -1 ... 3 is -1 and -3 is 1... so it ends up being a (x+2)/mod 4)-2 . It would be nice if sin() and cos() took quarter turns too... then I could just go 0 1 2 3 for cos is 1 0 -1 0 (also for 4567 and every mod 4). Then when you get to describing how weight reacts on a balance scale with a fixed pivot fulcrum, the ratio is a ratio from -1 to 1 which is -90 and +90 degrees respectively for the tilt of the scale for a ratio of things. (things can be any thing; apples, pence, grams... grams are a good choice because every gram is like every other gram; and the requirement for things to be on a scale is that they all have the same size). You can then weigh samples of things too - a number of marbles that fall in a bowl vs another bowl. or +1 and -1 counters for win lose in games.

[@FractalWoman]

"It would be nice if sin() and cos() took quarter turns too". You can plug quarter turns into Sine and Cosine. You can plug in any "turn" or any angle I like. Sine and Cosine are not the problem. Sine and cosine are defined by ratios. Sine of an angle is opposite over hypotenuse and Cosine of the same angle is adjacent over hypotenuse when the angle is defined using a right angled triangle.

[@3zdayz]

@@FractalWoman I mean as a native mode like they take radians which aren't turns or quarter-turns.

[@FractalWoman]

@@3zdayz Radians are the natural units to use when using Sine and Cosine. I can't think of one good reason to change that. In a calculator, you can switch to degrees since that works better for some people. Units in general are arbitrary. The only unit that is not arbitrary is the unit of the cycle which I refer to as Delta in Modified Unit Analysis. One cycle is one cycle not matter how you spin it (pun intended).

[@3zdayz]

@@FractalWoman :) Already gave reasons that quarters are more natural full cycles. Doesn't matter in the end you just keep a bunch of 1/4 multipliers you don't need. 0/4, 1/4, 2/4 and 3/4 are just as good, just you don't really need the 4, and can simply count the quarters as wholes. But then I guess you've not worked out weights on a balance scale; which is another case that works better with 1=pi/2 .

[@3zdayz]

@@FractalWoman In binary 2 bits..00 is 0 01 is 1 10 is 2 or -2 11 is 3 or -1 and then you get back to 0. there's a sign bit for sin, and a 1 or 0 bit for sin too that overlaps the value of the curve.

[@alanx4121]

you rule

[@FractalWoman]

you so kind. thank you.

[@douglasmagowan2709]

@6:51 1 is the identity of both the real numbers and the imaginary numbers.

[@FractalWoman]

Actually, this is not correct in my opinion. The identity of a number system is that which IDENTIFIES said system. '1" times any ponderable quantity or value gives you a real number and "i" times any ponderable quantity or value gives you an imaginary number. They is why they are called identities. The identity "1" IDENTIFIES a value as a real number. This is also referred to as a unit. So the real value of 2 should be written 2 x 1, but the x 1 is usually hidden. Technically every real value has a hidden x1 attached to it. The unit "i" IDENTIFIES a value as an imaginary. So 2 x i is an imaginary number. This is what distinguishes real numbers from imaginary numbers. In the real number system, we have gotten into the habit of not writing the x1 after every value. But in the real number system, we have to write x i (2 x i for example). Hope this helps.

[@douglasmagowan2709]

I might prefer the term “unit” as “identity” has well defined meaning in algebra - that is the element of the algebraic system that has no affect under a binary operation. e.g. AI = IA = A in linear algebra. That aside, I have a rather different intuition about Euler’s formula as I default to thinking about a complex number as a rotation in the plane, where by this video you it appears that you default to thinking about them as matrices. Rotation matrices so still fundamentally the same thing geometrically.

[@FractalWoman]

@@douglasmagowan2709 Euler's formula in 2x2 matrix form is a rotation in a plane. It is a rotation matrix. Euler's formula itself can be written in many ways. 1) e to the power of i times theta 3) cos(theta) + i sin(theta) and 3) the 2x2 matrix cos(theta) sin(theta) -sin(theta) cos(theta) These are different ways of saying the same thing. My way is much easier to implement in a computer program. I can implement it directly without having to do any algebra.

[@mathsciencefancier]

I have to watch this, but serial distraction made me not to do, sorry. I'll do. Anyway, have you ever studied bio-information-theory which is studied by Prof. David Sinclair, he said at his book, which combines Shannon's information theory and genetics. The reason why I ask this to you, is because, as like you, I'm also interested in Mandelbrot's Set and fractal math, and they remind me of cleveage and growing of an embryo, and it functions based on genes, as far as I know.

[@FractalWoman]

I am very interested in the relationship between fractals and biology.

[@mathsciencefancier]

@@FractalWoman Wow, then I'll comment whenever I find wonderful information. I'm reading his book right now.

[@lukiepoole9254]

Tau/2 is not pi in any ways. Sqrt(16/golden ratio) is pi.

[@FractalWoman]

Tau = 2PI 2PI/2 = PI therefore Tau/2 = PI.

[@lukiepoole9254]

@@FractalWoman It would be much better to reconstruct trigonometric functions to fit the true value of constant pi. It would completely change mathematics.

[@whig01]

​@@lukiepoole9254 If you wanted to work with rational trigonometry you should check out Professor Wildberger's series. kzbin.info/www/bejne/mofOZHamn62Fars

[@FractalWoman]

@@lukiepoole9254 Show me the money and I will believe you.