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! 🎉

[scolem]

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

[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.

[whig]

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.

[beautheory]

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.

[tootalldan]

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.

[heterodoxagnostic]

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

[memegazer]

You are one cool lady...thanks for making this vid...I usually hate math bc it is so formal...but imo this is straight up metaphysics bc you don't put the horse before the Descartes

[pixbee]

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

[mathsciencefancier]

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

[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.

[monoman]

I'm suitably demystified... thanks

[giovannip.]

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...

[memegazer]

omg...demystifying my butt...I am now even more confused and mystified than I was from where I started

talk about a janus point

[Kydo]

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

[zdayz]

:) 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.

[douglasmagowan]

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

[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.

[lukiepoole]

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