4 ms·

Well, as far as I know, cosinus and sinus have relatively little to do with exponential and imaginary numbers, just like imaginary numbers have intrinsically li

Well, as far as I know, cosinus and sinus have relatively little to do with exponential and imaginary numbers, just like imaginary numbers have intrinsically little to do with the 2D plane. I just think of it as a convenient notation since those obey similar composition rules (that probably stem from their odd and even-ness, and taylor series decomposition from there). So you need at least that one, since it is by convention, and not really derivable[1]: exp(ix) = cos(x) + i*sin(x) I personally just use Euler's notation (using exp) as a tool to derive trig formulas, since I know power composition rules quite well already, and those tend to be more useful in general. [cos(x)]^2 + [sin(x)]^2 = 1 Is just the Pythagorean theorem on a unit circle, which also defines sin and cos :) [1] OK, you can look at the taylor series expansion, but you need to remember the derivatives for this.

dragontamer 6y ago
cos(x) and sin(x) are just the unit circles applied over the 2-dimesnional real + imaginary axis. Because x^2 + y^2 = 1 defines a unit circle, if you perform x^2 + i * y^2 = 1, you get a unit circle over the real/imaginary planes. Or to put it in a picture: https://upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/1024px-Euler%27s_formula.svg.png https://upload.wikimedia.org/wikipedia/commons/thumb/7/71/Eu... Remember the definition of sin / cos: https://upload.wikimedia.org/wikipedia/commons/b/bd/Sine_and_cosine_animation.gif https://upload.wikimedia.org/wikipedia/commons/b/bd/Sine_and... ------- Once you realize that "the imaginary axis" is just an arbitrary 2-dimension extension field (and that "imaginary" is a very bad name for it), it becomes way easier to see. ------- Now if you had a 3-dimension graph (real, imaginary, and time), and then you project the unit-circle over the real + time axis, you get a sin (or cos) respectively. ----------- Or to put it another way: there's a periodic nature of i: * i = i * i^2 = -1 * i^3 = -i * i^4 = 1 -- Cycle-length 4 * i^5 = i * i ^ x = i^(x mod 4) (In general) This period itself forms a circle, moving from real-to-imaginary and back. (e^i^x) = (e^(ix)) therefore is periodic as well. And the most natural periodic cycle is a circle (which is of course, just sin/cos). To make this "period 4" into a proper circle, multiply by pi/2. i^(pi/2)^x == period 3.141592.... or the "circumference" of the unit circle. e^i^(pi/2)^x == e^(i * pi/2 * x). Done.
gmadsen 6y ago
how can you say imaginary numbers have intrinsically little to do with the 2D plane? it is entirely possible and equally correct to define imaginary numbers geometrically. It is nice and convenient that this is equivalent to the algebraic definition.
- gmadsen 6y ago
  for anyone interested, a great text on this is https://www.amazon.com/Visual-Complex-Analysis-Tristan-Needham/dp/0198534469 https://www.amazon.com/Visual-Complex-Analysis-Tristan-Needh...
beervirus 6y ago
It's not just a convenient notation. Sine and cosine are intimately connected with exponentials and imaginary numbers.
smt1 6y ago
> So you need at least that one, since it is by convention, and not really derivable[1]: exp(ix) = cos(x) + i*sin(x) I think if you've read the cognitive science POV on this (I think I read Lakoff's books first), it helps rederive that. Keep in mind that Euler's notation is just notion (I also recommend visual complex analysis). It's better from a cognitive neuroscience to pretend to be Euler himself and invent it yourself (literally embody his cognitive process). It's just concept blending between stuff you already know and stuff you didn't know you know but hadn't put a link or thought to denote a name, but are quite natural things if you want to develop the intuition for understand complexity. From a computer engineering point of view, often a Fourier series expansion of an arbitrary periodic time signal. Like a sinc function. Usually in analog signal processing, it depends on approximation and either quadratic forms or hyperbolic transcendral forms are useful, and knowing in what dimension a "perfect" computational equivalent exists as function of a complex conjugate. Going back in time, this seems to be why the Shannon-Nyquist rate was incessantly in people's heads for about 30 years. It let us to the digital and discrete switchover in the '90s. It just caused a shift from the telecom companies, what "content" was worth, then a boom then bust, then boom cycle for the new internet companies. I think these days, with the advent of huge corpus of text and higher order content (whether it be source code, neural nets, etc) mined from the internet available for free, I think with all of the global issues taking place right now that probably require transnational mutually assured prevention of tragedy of the commons situations, but it seems like it needs much more interdisciplinary blurring of people's fields. Now, if you think about where you need to actively design controllers (traditionally this was the realm of control theory) - think about non-linear dynamics and modern robotics, actually it doesn't even needed to be robotics, it can be any cyber-physical system, which is /all systems/ in general, this is why data flow architectures are being shifted towards rather than classical control flow architectures in a computer science/computer engineering POV. You will have to develop the intuition of stopping time, etc. There are "deeper" intuitions involved with combinatorics and natural numbers, this is probably the nexus of computer, neuroscience, medicine, physics, philosophy, linguistics, math, logic, science, engineering, etc. Actually, you can disregard numbers (but still appreciate use number theory, I think it's best to realize you're basically sharing some cross cultural understanding), while still doing arithmetic. Sometimes this is very useful for physical (real) systems where you have to make computation tradeoffs between mathematical physics, stability, natural language, etc. If you have a deeper understanding, from my limited understanding of the biosciences, all of these are relatively translatable between mtRNA/RNA scales (where we understand the chicken/egg problem as simply codons, but proteins in effect are "simulating" what we call heat transference problems/radiative transfer problems. But a lot of genomic data isn't decoded yet.

dragontamer 6y ago

cos(x) and sin(x) are just the unit circles applied over the 2-dimesnional real + imaginary axis. Because x^2 + y^2 = 1 defines a unit circle, if you perform x^2 + i * y^2 = 1, you get a unit circle over the real/imaginary planes. Or to put it in a picture: https://upload.wikimedia.org/wikipedia/commons/thumb/7/71/Euler%27s_formula.svg/1024px-Euler%27s_formula.svg.png https://upload.wikimedia.org/wikipedia/commons/thumb/7/71/Eu... Remember the definition of sin / cos: https://upload.wikimedia.org/wikipedia/commons/b/bd/Sine_and_cosine_animation.gif https://upload.wikimedia.org/wikipedia/commons/b/bd/Sine_and... ------- Once you realize that "the imaginary axis" is just an arbitrary 2-dimension extension field (and that "imaginary" is a very bad name for it), it becomes way easier to see. ------- Now if you had a 3-dimension graph (real, imaginary, and time), and then you project the unit-circle over the real + time axis, you get a sin (or cos) respectively. ----------- Or to put it another way: there's a periodic nature of i: * i = i * i^2 = -1 * i^3 = -i * i^4 = 1 -- Cycle-length 4 * i^5 = i * i ^ x = i^(x mod 4) (In general) This period itself forms a circle, moving from real-to-imaginary and back. (e^i^x) = (e^(ix)) therefore is periodic as well. And the most natural periodic cycle is a circle (which is of course, just sin/cos). To make this "period 4" into a proper circle, multiply by pi/2. i^(pi/2)^x == period 3.141592.... or the "circumference" of the unit circle. e^i^(pi/2)^x == e^(i * pi/2 * x). Done.

gmadsen 6y ago

how can you say imaginary numbers have intrinsically little to do with the 2D plane? it is entirely possible and equally correct to define imaginary numbers geometrically. It is nice and convenient that this is equivalent to the algebraic definition.

gmadsen 6y ago
for anyone interested, a great text on this is https://www.amazon.com/Visual-Complex-Analysis-Tristan-Needham/dp/0198534469 https://www.amazon.com/Visual-Complex-Analysis-Tristan-Needh...

beervirus 6y ago

It's not just a convenient notation. Sine and cosine are intimately connected with exponentials and imaginary numbers.

smt1 6y ago

> So you need at least that one, since it is by convention, and not really derivable[1]: exp(ix) = cos(x) + i*sin(x) I think if you've read the cognitive science POV on this (I think I read Lakoff's books first), it helps rederive that. Keep in mind that Euler's notation is just notion (I also recommend visual complex analysis). It's better from a cognitive neuroscience to pretend to be Euler himself and invent it yourself (literally embody his cognitive process). It's just concept blending between stuff you already know and stuff you didn't know you know but hadn't put a link or thought to denote a name, but are quite natural things if you want to develop the intuition for understand complexity. From a computer engineering point of view, often a Fourier series expansion of an arbitrary periodic time signal. Like a sinc function. Usually in analog signal processing, it depends on approximation and either quadratic forms or hyperbolic transcendral forms are useful, and knowing in what dimension a "perfect" computational equivalent exists as function of a complex conjugate. Going back in time, this seems to be why the Shannon-Nyquist rate was incessantly in people's heads for about 30 years. It let us to the digital and discrete switchover in the '90s. It just caused a shift from the telecom companies, what "content" was worth, then a boom then bust, then boom cycle for the new internet companies. I think these days, with the advent of huge corpus of text and higher order content (whether it be source code, neural nets, etc) mined from the internet available for free, I think with all of the global issues taking place right now that probably require transnational mutually assured prevention of tragedy of the commons situations, but it seems like it needs much more interdisciplinary blurring of people's fields. Now, if you think about where you need to actively design controllers (traditionally this was the realm of control theory) - think about non-linear dynamics and modern robotics, actually it doesn't even needed to be robotics, it can be any cyber-physical system, which is /all systems/ in general, this is why data flow architectures are being shifted towards rather than classical control flow architectures in a computer science/computer engineering POV. You will have to develop the intuition of stopping time, etc. There are "deeper" intuitions involved with combinatorics and natural numbers, this is probably the nexus of computer, neuroscience, medicine, physics, philosophy, linguistics, math, logic, science, engineering, etc. Actually, you can disregard numbers (but still appreciate use number theory, I think it's best to realize you're basically sharing some cross cultural understanding), while still doing arithmetic. Sometimes this is very useful for physical (real) systems where you have to make computation tradeoffs between mathematical physics, stability, natural language, etc. If you have a deeper understanding, from my limited understanding of the biosciences, all of these are relatively translatable between mtRNA/RNA scales (where we understand the chicken/egg problem as simply codons, but proteins in effect are "simulating" what we call heat transference problems/radiative transfer problems. But a lot of genomic data isn't decoded yet.