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It's a nice video but it goes so fast I don't think people who don't already understand this stuff will get it.
> goes so fast Especially goes too fast right when it becomes less obvious.
Essence of Linear Algebra https://www.youtube.com/watch?v=kjBOesZCoqc&list=PLZHQObOWTQ...
It's a nice video but it goes so fast I don't think people who don't already understand this stuff will get it.
Earlier quoted context omitted.
> goes so fast Especially goes too fast right when it becomes less obvious.
Like when he presents equations that he said we wouldn't need...
https://www.youtube.com/watch?v=UcWsDwg1XwM&list=PLBE9407EA6...
At 2:37, right after finishing explaining adders and multipliers on the line it started going way too quick like where did e^x and some these infinite sums come from???
The infinite sum is not important. I believe he is showing it purely as a visual tool to indicate that e^x is just "some function".
More specifically, based on the video, the important feature for a adder -> multiplier converter to have is that f(x+y) = f(x)f(y). However, many functions satisfy this requirement. He states (without justification [0]) that e^x is the most "natural" choice for such a function, and provides the infinite sum as a visual aid to show that it is just some function.
[0] Rather, justification is provided by reference to another video.
For anyone that finds it beautiful isn't there a bit of humanisation and definition involved (for example the Sine function used in deriviation uses 'pi' instead of 90 degrees), not to mention Sine is a human created function. You could have e to pi (90 * -1) too.. or a different method to define angles instead of having 360 degrees (base 60)
e^x = an infinite series if you substitute x = iz you can split the even and odd terms of the infinite series into cos and sin e^iz = cos z + i sin z evaluating at z = pi yields cos pi + i sin pi -1 + 0i = -1 the exponential function maps the imaginary axis to the unit circle. pi gets mapped to -1. This shows its true, but the "why" and real understanding requires calculus and the first week of complex analysis. Othe…
This presentation does skip some nessasary legwork for complete rigor, but presents a valid non-standard construction of e^ipi.
Specifically, he defines two sets of objects: adders and multipliers, and a function (written e^x for historical reasons) that maps adders into multipliers.
He then generalizes this construction to work in 2 dimensions instead of 1 dimension.
I would add to this construction that e^x is the particular converter that maps pi -> -1. However, I think this requirement is implicit in him stating that e^x is the most "natural" of the converters, and that pi -> -1 is the most natural of mappings.
The only place where you might need calculus is to provide an explicit construction of e^x (and possibly to justify the notational choice of writing it like an exponential).
EDIT:
To make the point clearer. Under the construction presented by the video, e^x is not defined as an infinite some, but rather as a function satisfying certain properties. That this function is equal to a particular infinite sum is a statement that requires proof; and not a statement that is needed for many applications.
Or you might prefer Better Explained's article: https://betterexplained.com/articles/intuitive-understanding...
1) e^x is a function whose derivative is equal to its value. 2) e^ix is a function whose derivative is equal to its value rotated by 90 degrees (ie^ix). 3) As x goes from 0 to pi, the trajectory of e^ix always has a velocity vector perpendicular to its current position. For example, when x = 0, the current position is 1 and the velocity vector is i. 4) So the trajectory a circle arc of length pi, which ends at -1.
The complex plane can be thought of as being made up of two-dimensional geometric transformations consisting of rotation and scaling (Tristan Needham calls such transformations “amplitwists”), with 1 as the identity transformation, and i as a quarter turn anticlockwise, and i^2 = –1 as a half turn. To compose two such transformations, you multiply the scales and add the angle measures of the rotations. Because such operations are linear, you can also break them into a part parallel to 1 and a part perpendicular to 1 (some multiple of i), and multiply two such transformations component-wise, using the distributive law (a + bi)(c + di) = (ac – bd) + (ad + bc)i.
exp z is a complex function which maps (in an angle-preserving way, i.e. conformally) from an infinite two-ended cylinder to a whole plane minus one point (the “origin”). The exp function maps negative infinity on the cylinder to the origin on the plane, and it maps the zero point on the cylinder to a given “unit” point in the plane, and the “zero” circular slice through that point on the cylinder to the “unit circle” on the plane, containing the unit point and concentric with the origin. The coordinate system on the cylinder has 2πi measuring one loop around a circular slice, and 1 pointed along the cylinder axis. The coordinate system in the plane is the customary square grid. Addition of coordinates in the geometry of the cylinder (if you like, rotating and/or sliding the cylinder) corresponds to multiplication of complex numbers (composition of amplitwists) in the plane. That is, exp(w + z) = (exp w)(exp z).
In particular, exp iy for some real number y maps points at distance y along the zero circle in the cylinder to points on the unit circle in the complex plane at a proportional distance around the circle; that is, to rotation operators which correspond to the given angle measure in radians.
So πi is halfway around the zero slice in the cylinder, and exp maps it to the operator in the complex plane corresponding to a half-turn rotation, i.e. exp πi = –1.
The log function is the inverse map, from the plane to the cylinder; it is a multi-valued function because we can make multiple “straight” helical connections between arbitrary points on the cylinder, which wrap around different numbers of times.
Once we have this general concept for how we want the exp map to work, we can work out the details to find that the unique such function is the solution to a particular differential equation f'(z) = f(z), or alternately the Taylor series we are all familiar with, exp z = 1 + z + z^2/2 + z^3/6 + ...