I finally understand imaginary numbers. It's actually simple, imaginaries count things that may exist, but don't exist yet. For example, if an apple tree is predicted to have 12 apples, and it currently has 8, then it has 4 imaginary apples.
Doesn't explain thw multiplication of imaginary numbers.Looks at it not as numbers, but coordinates. Afterall, they were created to make geometry easy.
>>16876796People are only confused because they're called imaginary. They might as well be called potato numbers and nobody would have a problem
>>16876887
imaginary numbers are perpendicular directions in the same dimensionsorry not sorry
>>16876887People of low iq hate imaginary numbers, dark matter etc. because the naming confuses them. It would have been better had we sticked to naming them orthogonal numbers and sterile matter because that makes it less interesting.
>>16876796They literally just describe the square roots of negative numbers. That's it. Anything else you can say about them is literally just downstream of that fact.
>>16876976>They literally just describe the square roots of negative numbers.So if you understand that so well, how does it help explain why e^i*pi is negative?
>>16877169Natural consequence of e^x being its own derivative.
>>16877172How so? That constant doesn't have anything to do with derivatives and none of the constants that compose it are negative.
>>16877176>That constant doesn't have anything to do with derivativesSkip to like 4 minutes in. At around 5:45 it gets much more explicit how i being the square root of -1 and e^x being it's own derivative are really the only necessary components here:https://m.youtube.com/watch?v=-j8PzkZ70Lg
>>16877169It's just a rotation by pi in the complex plane
>>16877343No, its a constant taken to the power of a constant times a constant that yields a constant, there is no motion or dynamic value involved, just a constant composed entirely of constants.
>>16877345e to the power of something is just notation. exp(i*pi) is a function that takes i*pi and outputs -1
>>16877351No, e is a constant number, not some variable function, its not inputting to a function, its taking the value of e to the power of i times pi which is equal to -1.
>>16877357>its taking the value of e to the power of iYou should try to understand what this means
>>16877359So what are some other examples of when three positive constants, I guess you call them functions, result in a negative number?
>>16877360i is not a positive constant
>>16877345>its a constant taken to the power of a constant times a constant that yields a constantYes.>there is no motion or dynamic value involvedA rotation is not necessarily a "motion or dynamic value" thoughbeit.
>>16877361Even though we are clearly talking about positive i rather than negative i, assuming you are correct, then what are some other examples of when three non-negative constants, I guess you call them functions, result in a negative number?
>>16877362So what are some examples of something that rotates without moving?
>>16877367>something that rotatesWho said anything about "something that rotates"?
>>16877368So its a type of rotation where nothing has actually rotated or is rotating?
>>16877373You're probably the kind of tard who thinks 0 represents nothingness. I bet you get extremely confused when someone says multiplying a real number by -1 corresponds to a flip on the real axis.
>>16877169>So if you understand that so well, how does it help explain why e^i*pi is negative?Unironically that has a simple explanation.
>>16877379Then what is the difference between adding 0 and adding nothing?
>>16877367>So what are some examples of something that rotates without moving?d=e^(it) is literally describing movement. At time t you have moved to some position d. You literally move around a circle over time. e^(it) is just a helix in time.
>>16877382>Then what is the difference between adding 0 and adding nothing?I see I nailed your type perfectly.
>>16877384>d=e^(it) is literally describing movement.Not in and of itself it doesn't.
>>16877169>i = sqrt(-1)>then explain why using i as an exponent yields negative numbers sometimes!Why is that even remotely surprising to you?
>>16876963Anon, I have been thinking about this a lot. Time is complex-valued: magnitude and phase. The fibroids usually run parallel, but at black hole levels of curvature, they begin to twist and form closed time-like loops.Basically. As you slip past the event horizon of a black hole you will experience the moment through multiple time paths within a narrow phase angle of each other.
>>16876887This is what did it for me. When I realized there was nothing that philosophical or deep about calling them "imaginary", it finally clicked. They are just a tool, it's like asking why use x y and z in algebra and not other letters.
Imaginary numbers are poorly named because the name is too broad. All numbers are imaginary. Once you realize this they fit right in with any other number.
>>16876887This desu, they should be orthogonal numbers.
>>16877606In GR, time isn’t “complex-valued,” and crossing an event horizon doesn’t imply closed timelike loops or multiple time paths. CTCs only appear in special, highly non-generic spacetimes (and often in regions thought to be unstable/unphysical). “Complex time” mostly shows up as a mathematical analytic-continuation trick in QFT.
>>16877566>>16877617
>>16876881they make geometry stupid convolutedand you can do (algebraic) geometry without them
>>16877617>>16877621I'm going to go way out on a limb here, but not being able to reply correctly makes your opinions on quantum gravity and complex-valued time moot.
>>16877387>Not in and of itself it doesn't.No?
>>16877672Not any more than 2+x describes purchasing apples. Functions can be applied to many different contexts and it's meaningless to say "this function describes that" without everyone in the room agreeing on what that context happens to be.
>>16876796you got filtered by nomenclature, imaginary numbers, spin, that's it
>>16877699Counterpoint: "Movement" *in this context* referred to dynamic values in a space. If you're willing to backpedal on context, all things are meaningless. So *in this context* we have movement. Shut the fuck up.And this is setting aside the fact that I further narrowed the context to d(istance) and t(ime) to show how the movement in this context could be used to represent movement in a conventional sense more narrowly.
>>16877728>"Movement" *in this context* referred to dynamic values in a space.Then all continuous functions have that property.2+x describes "movement" by the same standard.
>>16877365e^ix=cos(x)+(i*sin(x)), nigga
>>16877365Nigga, you can replace "e" with literally any number you want and raise it to the power "it" and there will be infinitely many values for "t" that equal -1. It's all just downstream of the fact that raising i to the power of t yields -1 for all values of t that are odd multiples of 2.
>>16876887I feel like this is one of those easy-answers people just parrot because they heard someone else say it. Almost nobody goes "ohhhh that makes sense!" when you say that they should actually be called lateral numbers.What I think actually confuses people is why it makes sense or is even valid to ascribe two-dimensional values onto otherwise non-dimensional quantities.
>>16876887Potato number would make it even more confusing. People will wonder what it has to do with potatoes.But why was it originally called imaginary? Anyone smarter than me knows?
>>16876887Also, reminds me of the sonic hedgehog gene which confuses every lay person I talk to
>>16876796Absolutely not. They're called so because they're orthogonal in a way to the set called the "real" numbers, and we thought it'd be cute to continue calling them "imaginary" because they are independent from "real". It's how a lot of names end up sticking in technical fields like this.Their actual utility is more nuanced. IMO they have physical significance in how they bridge exponential and trig functions together, meaning they have heavy utility in anything described by a differential equation, which applies heavily in physics and engineering.
>>16877385So there is 0 difference because nothing is different between those things?
>>16877450A constant taken to a negative number isn't negative number and neither is i, though.
>>16877863You got filtered by 0. Maybe go start a thread about that.
>>16877869I accept your concession, you can't articulate any difference because there is none, so you will have to devolve to your insult database loop to cope with getting BTFO.
>>16877875I concede that I can't teach 70 IQs what "additive identity" means and how a mathematical object can't be "nothing". If it's possible at all, teaching you anything would require specialized skills and superhuman patience. I lack the pro tard wrangler training.
>>16877878You mean the additive identity, x=x+0, that in no uncertain terms, unassailable mathematically proves adding nothing to (ie x) is exactly equal to adding 0 (ie x+0)?
>>16877887>You mean the additive identity, x=x+0No.
>>16877892So x doesn't yield x the same as x+0 yields x and x=x+0 isn't taught in the first grade?
>>16877900>so [thing no one said or implied]?What is it with this board and spam bots that act like delusional psychotic patients?
>>16877730>Then all continuous functions have that property.Yes, and? I'm not the side that brought up movement in that context. It was being juxtaposed with mathematical constants.The person asking didn't understand that e^(ipi)=-1 is just f(t)=e^(it), which describes a continuous helical path drawing a circle over t just, at t=pi.It is literally a continuous function.
>>16876887the irish would have a problem
>>16877901The formula, x=x+0 (generally regarded as the arithmetic additive identity by everyone but you) says it as long as you also consider the law of identity, do you just not understand mathematical formulas or something? What exactly do you think I am advertising besides laws of arithmetic and are you saying that arithmetic is inherently psychotic? Could you even solve for x if x-2x+13x = 24 or are you just worth absolute shit at any math discussion still just screeching because you are mad the new captcha is way too hard for you?
>>16877916>the additive identity is an equationLOL. I didn't bother reading 90% of your post but be sure to bother writing more than a sentence in your next reply.
>>16877929That is the arithmetic formula for the additive identity, though.https://grokipedia.com/page/Additive_identity>in the field of real numbers, 0 is the additive identity, satisfying x + 0 = x for any real x.https://math.arizona.edu/~rsims/ma425a/axioms.pdf>• The Additive Identity: There is a real number, denoted by 0 ∈ R,for which0 + x = x + 0 = x for all x ∈ R .90% is not the answer, you are too retarded for a third graders formula, you are just mad the captcha is too hard for you most of the time now.
>>16877945See >>16877901Either all mentally ill retards act the same or it's one spambot polluting multiple threads.
>>16877950Not going to bother reading retarded bullshit from a literal retard who can't solve a simple 3rd grade math problem and will say words like additive identity without having any idea how that translates to math.Feel free to keep screeching and lash out at everyone who proves you are retarded while blaming it on some scary boogeyman, though, its pretty funny to watch you writhe.Make sure to respond again without providing the answer to a simple formula while trying to cast all your mental disorders onto others through projections.
>>16877957LOL @ you losing your mind with rage. Write another paragraph for me.
>>16877958Only if you make sure to respond again with impotent rage, but without the ability to provide the answer to a very simple formula while trying to cast all your mental problems onto others through projections instead demonstrating a shred of intellect.
>>16877892Then what is the name of the identity described by the formula x=x+0, do you think it is true that x=x+0 or are you saying that isn't even true?
>Then what is the name of the identity described by the formula x=x+0Mentally ill retard board.
>>16877966That could be said of anywhere you patronize since you are too mentally retarded to even solve for x at a 3rd grade level and you don't know that x=x+0 is the arithmetic additive identity.
>>16877968>additive identity is an equationLOL.>you are too mentally retarded to even solve for x Well, why don't you go ahead and solve for x yourself, mentally ill retard? :^)
>>16877982Yes, the arithmetic additive identity is described by the arthmetic formula x=x+0 as shown in the sources provided >>16877945.You didn't provide an arithmetic formula to solve like I did >>16877916, so there is no x to solve for in your post like the post I provided that you will never be able to answer since you don't even have the math education of a grade schooler.
>>16877987>as shown in the sources providedRight. Let's see what your source says:>in the field of real numbers, 0 is the additive identityPretty funny to watch you play out the exact same sequence of psychotic behaviors in every thread. You're now at the preprogrammed stage of posting random links that directly contradict you but somehow failing to notice it.
>>16877995Then why are you the one who had to disingenuously cut off the sentence that I already quoted in >>16877945 because it supports my claim?>in the field of real numbers, 0 is the additive identity, satisfying x + 0 = x for any real x.
>>16877998>it supports my claimYour source:>in the field of real numbers, 0 is the additive identityYour claim:>x=x+0 is the arithmetic additive identity."x=x+0" =/= "0" AFAICT. Anyway, I lied. It's actually boring to watch you play out the same psychotic pattern in every thread. Closing this 70 IQ thread now but you WILL shit out another desperate post no one will ever read.
>>16878000Nope, you are still disingenuously cutting off the part where it says it has to satisfy x + 0 = x for any real x>>16878000>>x=x+0 is the arithmetic additive identity.Also the claim in the source since it says it must satisfy that condition to be 0.>"x=x+0" =/= "0" AFAICT.Not according to the sources I posted >>16877945 that explicit say the additive identity satisfies x+0=x.
>>16878000Even if you don't accept that it is called the additive identity, you still must accept that x=x+0 is true and that adding 0 to x is the same as adding nothing to x.
>>16877909>The person asking didn't understand that e^(ipi)=-1 is just f(t)=e^(it), which describes a continuous helical path drawing a circle over t just, at t=pi.Nothing they said indicates that they didn't understand that.
>>16877909>>16878131Btw, e^it does not necessarily describe a helix on its own. Which is more relevant to the core point of discussion.
>>16878131>Nothing they said indicates that they didn't understand that.>>16877345>No, its a constant taken to the power of a constant times a constant that yields a constant, there is no motion or dynamic value involved, just a constant composed entirely of constants.>>16878134>Btw, e^it does not necessarily describe a helix on its own.Of course not. d=e^(it) does.
>>16877909No, f(t) is defined for all of t, so it is definitely in motion since its basically a video, but e^i*pi is just one single static snapshot, there isn't a set variable like t, so it literally is not a continuous function, it is a static constant.
>>16878408i is a constantboth pi and e represent recursive functions which calculate pi and e to arbitrary precision
ITT: retards who don't understand how values, parameters, functions and plots relate to each other are try to argue about Euler's identity using moronic, made up terms like "dynamic value" and "static snapshot". One of them even thinks pi is a function.
>>16878422pi represents a ratio. It is a constant.e represents a coefficient of continuous exponential growth. It, too, is a constant.e^it is a function that outputs a complex value. pi is just one of infinitely many inputs for the variable t which happens to have an output of -1.
>>16878425pi is really the leibniz series, a function that spits out digits of pi with each iteration
>>16876796Theyre called imaginary because we pretend to be retarded and say that -1×-1 = -1.
>>16878432A function which approximates a value is not the value itself. Especially when there are other functions which do the same thing (see: Chudnovsky algorithm).
>>16878451and I am sure you can show that there exists a number which is output by both the Chudnovsky and Leibniz functionsoh wait, you actually cannot
>>16878458What are you even asking here? They both converge on the same value.
>>16878463>What are you even asking here?what are natural numbers m and n such that Leib(m)=Chud(n)
>>16878473I do not know and it is not relevant to the point. Your claim that pi *is* the Leibniz series is exactly as valid as saying pi *is* the Chudnovsky algorithm.
>*...*Reddit-trained spambot getting filtered by basic math again.
>>16878479so, neither claim is valid?there is no function that outputs pi?
>>16878486>there is no function that outputs pi?Wrong. f(x) = pi outputs pi.
>>16878486There are infinitely many functions that output pi at certain values.That is not the same as saying pi *is* those functions. 1 is the limit of (x-1)/x as x approaches infinity. Are you gonna try and tell me 1 is a function?
>>16878489>pi at certain valueswhat does that even mean you mongoloid lol>1 is the limit of (x-1)/x as x approaches infinity(x-1)/x!=1 for any x>>16878487you can't really write that because pi isn't a number, sorry
>>16878489>Are you gonna try and tell me 1 is a function?f(x)=x/x outputs exactly 1 for any xso you can treat 1 as a function if you feel like it
>>16878492You're a retard arguing with a bot.
>>16878489>Are you gonna try and tell me 1 is a function?Is this a veiled bait to compel a 1=.999... post?
>>16878408>No, f(t) is defined for all of t, so it is definitely in motion since its basically a video, but e^i*pi is just one single static snapshot, there isn't a set variable like t, so it literally is not a continuous function, it is a static constant.Is f(t)=-1 when t=pi for f(t)=e^(it)? You are literally claiming a function has nothing to do with the points that make it up.The points making up a function are obviously "involved" with it.
>>16878492>what does that even mean you mongoloidIt means you need to work on your reading comprehension.Arccos(x) = pi when x = -1.>(x-1)/x!=1 for any xNeither does the Leibniz series. >>16878493By that same logic, i cand be treates as a function which still undermines the other anon's original point.>>16878497No
>>16878492>>16878639>Neither does the Leibniz series.My bad, I meant to say Leibniz series != pi at any x. You get my point.
>>16878639arccos(-1)=NaN! because there is no triangle at -1, it's just a line segment
>>16878773The calculator begs to differ.
>>16878793well that's obviously a lieit is trying to peddle a mere rational number as pi
>>16877169Complex unit circle begins at 1, whene^(i*0) = e^0 = 1With pi radians, you've traveled 180 degrees, arriving at the opposite side of the 0-origin, at negative 1If you add another pi-radians, you've come full circle and you're back at 1, because the equation has periodicity of 2pi or tau
>>16876796Anon, you are still filtered.
>>16876887Present it right away as a "bidimensional number," focus on its geometry and carrefully, more rigorously, polish your way from there. I believe this way is much more effective to teach the set of complex numbers because it dissipates the annoying philosophical questions of its counter-intuitive name and what is it even used for.I mean, it sounds better than just saying I now define the existence of square roots of negative numbers which will be called imaginary, and let [math]i=\sqrt{-1}[/math]. I helped my friends with complex numbers essentially doing trigonometry on the complex plane.
>>16877382same as you vs your aborted sibling
>>16879313Anything to the power 0 is 1, there is no "traveling", its not a function, there is not some infinite mapping from input to output, its just one number composed of three constants.
this is the ultimate brainlet thread. there is absolutely nothing trivial about imaginary/complex numbers. which is why there are ZERO (0) trivial proofs of the fundamental theorem of algebra. The simplest proof we have to this day is the topological proof in Gauss's PhD dissertation If your explanation of complex numbers involves the complex plane in any capacity whatsoever (muh they're just (x,y) = x+iy!!!), you are a brainlet fullstop. Please, for a polynomial with REAL coefficients, graph it and please show me on its real-valued graph where the information about its complex roots are encoded exactly and how they can be graphically derived in the general case using that non-complex graph alone, and then explain what this has to do with the general physical meaning of complex numbers (energy sinks/sources).
>>16880136i ain't doing your homework nigga
>>16880147Since you clearly didn't even understand what anon was saying, you could have just asked them to clarify.
>>16880136>there is absolutely nothing trivial about imaginary/complex numbers.Square root of nagative numbers. That's it.>which is why there are ZERO (0) trivial proofs of the fundamental theorem of algebra.Non-sequitur.>If your explanation of complex numbers involves the complex plane in any capacity whatsoever (muh they're just (x,y) = x+iy!!!), you are a brainlet fullstopI agree. The complex plane is a system invented utilizing them and not a fundamental component of imaginary numbers themselves.>>16880136>for a polynomial with REAL coefficients, graph it and please show me on its real-valued graph where the information about its complex roots are encoded exactly and how they can be graphically derived in the general case using that non-complex graph alone"Demonstrate blue without referencing color!"Snide rebuttal aside, this is also a non sequitur even understanding that using the complex plane as a source of complex numbers is dumb.>explain what this has to do with the general physical meaning of complex numbersSquare root of a negative number.>(energy sinks/sources).Imaginary numbers can be useful in describing some properties of these but are not, themselves, intrinsically tied to them.>>16880148Obvious samefag is obvious.
>>16880169>Square root of nagative numbers. That's it.Nope, that stopped being it after euler discovered how to combine i with e and pi.>"Demonstrate blue without referencing color!"0x0000FF>Obvious samefag is obvious.The only thing that is obvious is that you had trouble understanding what anon was saying, but you are clearly too self conscious to ask question instead of constantly making poor assertions, then acting snide when they get refuted.
>>16880171>that stopped being it after euler discovered how to combine i with e and pi.Excruciatingly low IQ take. See >>16877758>0x0000FFMeaningless without the concept of hexadecimal representation of color as implicit context.>The only thing that is obvious is that you had trouble understanding what anon was sayingI understand clearly what he was saying and have demonstrated why it is dumb.To summarize, see: >>16876976
>>16880176>Excruciatingly low IQ take. Nope the excruciating low IQ thing is to just name call without explaining anything of your own.>Meaningless without the concept of hexadecimal representation of color as implicit context.Nope, hexadecimal gradients exist for many other things besides color, I didn't even reference color, you did.
>>16880180>the excruciating low IQ thing is to just name call without explaining anything of your own.The explanation was given in the post linked.>hexadecimal gradients exist for many other things besides colorWhich is exactly my point. That value does not intrinsically denote "blue" unless it is assigned to a color.
>>16880184Except in the linked post you didn't use three different constants, you used two of the same constants and one that is just cyclical function of pi.>That value does not intrinsically denote "blue" unless it is assigned to a color.Which is exactly what you asked for.
>>16880187>Except in the linked post you didn't use three different constants, you used two of the same constants and one that is just cyclical function of pi.Incorrect.You don't need e or pi to get -1. i times some coefficient as an exponent is all that's necessary.Any number can be a constant.>Which is exactly what you asked for.I asked you to describe blue without referencing color and you responded with 255 in base 16. You did not complete the assignment.
>>16880194>i times some coefficient as an exponent is all that's necessary.So now you are down to 2 constants with one of them still being the same constant as before?>You did not complete the assignment.I did, though, and you know I did because you know exactly what I was describing without a single direct reference to color that you yourself didn't infer.
>>16880196>So now you are down to 2 constants with one of them still being the same constant as before?Are you stupid or something?Yes, i is the subject of discussion so any relevant function will include it.The point is there isn't really anything special to e^(pi*i). e^(3*pi*i) yeilds the same result, even. 5^(log_5(-1)) trivially yields -1. >>16880196>you know exactly what I was describing without a single direct reference to color that you yourself didn't infer.The fact that it was the response you gave when I asked you for said description is implicit context. Without said context, it is meaningless.You could have just as easily said 0x00FF00 and I would have responded in exactly the same manner since I don't know hexadecimal color codes off the top of my head.
>>16880201>Yes, i is the subject of discussion so any relevant function will include it.No, the whole point of the question was 3 different non negative constants, so picking the same one over and over or picking -1 as one of them is definitively contrary to the point you are attempting to make.> Without said context, it is meaningless.Its not though, you knew what it referred to without me referencing color.>I would have responded in exactly the same manner since I don't know hexadecimal color codes off the top of my head.You would have still been wrong and you still could have looked it up to fill in the information that I didn't reference myself.
>>16877169Expand sine, cosine and e^x into their taylor series and everything will be clear
>>16880208>the whole point of the question was 3 different non negative constantsNo. The point was explicitly laid out here: >>16876976The question was supposed to be a rebuttal to that simple principle.I guess we could get into quaternions to more thoroughly refute this silly idea but most calculators do not support them and I honestly can't be fucked.>>16880208>you knew what it referred to without me referencing color.Because I asked about color.If I asked "how many apples do you have" and you responded "3," would you then conclude there's some intrinsic relationship between the number 3 and "apples" on the grounds that I knew what you were referring to in that moment?>you still could have looked it up to fill in the information that I didn't reference myself.Hence why that value is meaningless without provided context.0x0000FF only refers to blue because those are standards set forward by tech bureaucracy. There is no intrinsic reason that the order goes RGB and not GBR other than someone arbitrarily decided it one day. That's the whole point.
>>16880216No, I am talking about this point >>16877365.And once again, taking e to a negative exponent, wouldn't even result in a negative number, so your point makes no sense in this context.>The question was supposed to be a rebuttal to that simple principle.Because raising e to a negative wouldn't result in a negative number.>Because I asked about color.Which is why you knew exactly what I was talking about when I gave the hexcode.>intrinsicYour problem wasn't about intrinsic relationships it was about overt references.>without provided context.The context was provided in your question without me making any further references.>There is no intrinsic reason that the order goes RGB and not GBR other than someone arbitrarily decided it one day.Which is exactly why I didn't need to reference color for you to understand I was talking about blue.
>>16880234>taking e to a negative exponent, wouldn't even result in a negative number, so your point makes no sense in this context.Not relevant. I honestly do not follow why you think is is relevant.>Because raising e to a negative wouldn't result in a negative number.But raising literally any number to the power of "it" yields negative numbers for infinitely many values of t.>Which is why you knew exactly what I was talking about when I gave the hexcode.Therefore color was being implicitly referenced and you did not complete the assignment.>context was provided in your question without me making any further references.Which is the only reason your hex number was recognized as a color. That's my point.>Which is exactly why I didn't need to reference color for you to understand I was talking about blue.The point went over your head. This is the last time I'll address the retarded color thing because it was just a snide comment about you asking for complex values of real functions thinking you were making a coherent point.
>>16880493>because it was just a snide comment about you asking for complex values of real functions thinking you were making a coherent point.Please kill yourself brainlet. You are talking to a different anon. I (>>16880136) haven't responded to you once because you didn't make any substantive points whatsoever ("non-sequitur", "<quote something I never said and don't explain anything further>!", "muh square root of negative number", "square root of a negative number!", etc.). The fundamental theorem of algebra is not a non-sequitur to the conversation whatsoever. Complex roots can be graphically derived on a real-valued plane and any physical phenomena that maps to such an equation is taking part in some sort of generalizable behavior that is substantively different (exp. growth/decay) from when the roots are "real" (0i). Repeating over and over as if by rote (brainlet behavior btw) that complex numbers are just the square root of negative numbers is, as you put it, "meaningless w/o the concept of [...] as implicit"), just like Blue = 0x0000FF. All your response tells me is that you are able to mindlessly regurgitate the formula your professor told you. There is a reason Gauss disregarded quadratic integers, and every other mathematician, whose minds intellectually circumscribe yours in every possible way, explicitly regarded imaginary numbers with apprehension. Saying "they are the square root of negative one" would not have been news to any of them, and not one of them would have responded to your brainlet regurgitations with "Oooooh, I get it now!" Just end your own life asap please
>>16880651Sir, are you having a schizophrenic fit?All your crying aside...>>>16880651>Saying "they are the square root of negative one" would not have been news to any of them, and not one of them would have responded to your brainlet regurgitations with "Oooooh, I get it now!"Correct. Because that is the definition of of i. The reason there was so much apprehension towards the concept is that accepting them leads to logical contradictions within the mathematical systems we have laid out by that point. Once their utility was thoroughly established, exceptions to many commonly held rules had to be covered by stating "for all real numbers" as a caveat. There was never an "aha!" moment to be had in the first place. The only people confused by this are retards. Which is especially evident by the constant regurgitation of euler's identity by people who clearly don't understand where it comes from
>>16880493>Not relevantIts is though, you are saying that it is obvious that it would be negative because the exponent is negative when squared, but a negative exponent doesn't result in a negative number, it results in a smaller number.>But raising literally any number to the power of "it" yields negative numbers for infinitely many values of t.No, you only get a negative number if e is taken to a factor of i times pi.1^-1 equals 1 and 2^-1 equals .5.>implicitlyImplicit is not a reference, it is you deducing something from the context, not me making a reference.>Which is the only reason your hex number was recognized as a color.Yes you recognized it as the hexcode for blue without it being specifically referenced as a color just like you asked.>The point went over your head.No, you are just eternally buttmad that it wasn't the unsolvable gotcha you that you assumed when you said it.>you asking for complex values of real functionsNo, you obviously missed the point since I was asking how entirely positive complex and transcendental values could collapse into a negative real unit constant.
it's just sqrt(-1)what's so hard to understand?
>>16880833>what's so hard to understand?How exactly 2.718281828...^(sqrt(-1)*3.141592654..) equals -1.
>>16880814>you are saying that it is obvious that it would be negativeAt no point did I say that. Wbat I said is that it shouldn't be as surprising as some anons ITT make it out to be.>>16880814>you only get a negative number if e is taken to a factor of i times pi.log_5(-1) = ~1.95iSo taking 5 and raising it to the power, ~1.95i, you get -1.
>>16880922>What I said is that it shouldn't be as surprising as some anons ITT make it out to be.But producing a negative number by raising a number to a negative power is pretty surprising since it only happens in the case of e raised to a power of i*pi.>shouldn't be as surprisingWhy are you surprised that things that are not very obviously tend to be surprising?>~No you are clearly approximating so you can justify dropping off the complex portion that remains to make it seem to work.
>>16880936>it only happens in the case of e raised to a power of i*pi.*any number raised to the power i*[some transcendental number]>Why are you surprised that things that are not very obviously tend to be surprising?Complex functions tend to be cyclical in nature. See: i^t>>16880936>No you are clearly approximating so you can justify dropping off the complex portion that remains to make it seem to work.No. I was rounding to 3 significant figures because log_5(-1) is transcendental just like ln(-1) is.
>>16880940Yes so its still a surprising property of i which is what surprises people in the first place and why you can't produce a negative out of constants that don't invoking imaginary numbers whose special property of being neither positive nor negative results in those non-obvious values.
>>16880945>surprising property of iIt's a natural result of it being the square root of a negative number. The i, -1, -i, 1 cycle comes out quite trivially when you just start repeatedly multiplying by i. It should not be all that baffling when you see similar cycles appear when you start playing with powers of i.The real issue here is people get exposed to euler's identity with such emphasis on its ellegance that it's generalization (that being [literally any real number]^[some imaginary number] will also give -1) now requires convincing because all these students now think there's something special about e and pi that make this identity work.
>>16880955ln is specifically defined according to e, so you are still effectively using e in your formula.
>>16880963Log_5 is not ln.ln(-1), which is essentially a restatement of euler's identity, shows up in the usual computation for log_x(-1) but this is a matter of convenience rather than anything intrinsically to do with that particular identity.
>>16880814nigga i ain't either positive or negative
>>16880837e^(ix)=cos(x)+i*sin(x)
>>16881613see>>16877450The discussion was a result of another anon originally implying that being the square root of a negative should make it obvious that a negative would result.
>>16876796It's just 2 dimensional vectors with a specific multiplication rule. >>16876881The multiplication rule just comes from [math] i^2 = -1 [/math] and the initial formal arithmetic [math] (x_1 + i y_1)(x_2 + i y_2) = x_1x_2 + x_1 (iy_2) + iy_1(x_2) + (iy_1)(iy_2) = x_1x_2 + i(x_1y_2 + x_2y_1 ) + i^2 y_1 y_2 = x_1 x_2 - y_1 y_2 + i (x_1y_2 + x_2 y_1) [/math]or [math] (x_1, y_1)(x_2, y_2) = (x_1x_2 - y_1 y_2, x_1y_2 + x_2 y_1) [/math]It's almost the multiplication rule in the plane that is invariant on the unit circle. Given two points on the unit circle [math] (x_1, y_1)[/math], [math] (x_2, y_2) [/math]. [math] (x_1x_2 - y_1y_2)^2 + ( x_1y_2 + x_2 y_1)^2 = x_1^2 x_2^2 -2x_1x_2y_1y_2 + y_1^2y_2^2 +x_1^2y_2^2 + 2 x_1 x_2 y_1 y_2 + x_2^2 y_1 ^2 = x_1^2 x_2^2 + x_1^2y_2^2 +y_1^2y_2^2 +y_1^2 x_2^2 = x_1^2 (x_2^2 + y_2 ^2) + y_1^2(x_2^2 + y_2^2) = (x_1^2 + y_1^2)(x_2^2 + y_2 ^2) = 1 \cdot 1 = 1 [/math]since [math] x^2 + y^2 = 1[/math] for all points on the unit circle.
>>16882888>almostshould be also
>>16876881Speaking more on the complex numbers, we cannot neglect the represent which has a the matrix [math] \hat{i} = \begin{pmatrix} 0 & - 1 \\ 1 & 0\end{pmatrix}[/math] and [math] \hat{1} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}} [/math]. Here we have the complex number express as [math] Z = x \hat{1} + y \hat{i} = \begin{pmatrix} x & -y \\ y & x \end{pmatrix} [/math]. The matrices which preserve the circle are the elements of SO(2) which is a very nice. The complex multiplication rule in this representation is just a consequence of normal matrix multiplication. It's quite a notable fact that there is the direct correspondence between SO(2) and the unit circle in the complex plane. We can essential start with SO(2) and get the complex numbers by simply writing them as [math] Z = r \Theta [/math] where [math] \Theta [/math] is an element of SO(2) and [math] r>0 [/math] for ease. For SO(2), we can take the standard parameterization [math] \Theta = \begin{pmatrix} \cos (\theta) & - \sin (\theta) \\ \sin (\theta) & \cos (\theta) \end{pmatrix} [/math] to get the standard polar form of a complex number, or in other words [math] Z = r \exp { \theta \hat{i} } [/math]. Of course SO(2) is the Lie Group which all transformations on the unit circle are invariant.
>>16876796so can i use it to predict the market then?
>>16882933If you want
>>16882933You can use anything to predict the market, people literally just put names of stocks in bowls and buy the ones their pets eat out of.
>>16879428kek
>>16886538My dog is bullish, tells me to buy everything. Including stocks he fished out of the bin
>>16876796>I finally understand imaginary numbers.not based on your post you don't.
>>16879707Agreed. When I teach complex numbers, I like to start like this:0. Review the real number line.1. Explain that we're extending this to a number plane. Draw an imaginary axis and label the pure imaginary numbers. Explain "imaginary" is just a historical name (talk about cubic equations if they're interested); all numbers are imaginary.2. Generalize what they already know abut addition: +3 is 3 steps right of zero, and to add +3 we go 3 steps right on the number line. -2 is 2 steps to the left of zero, and to add -2, we go 2 steps left on the number line. So since i is 1 step up from zero, ...3. Show how every point can be written as the sum of a real number and a pure imaginary number.4. Show how to add complex numbers by combining arrows. Practice it a bit, and show that we get the same result by algebraically combining like terms.5. Subtracting a complex number means we go the same distance in the opposite direction. Verify with a few examples that this is the inverse of addition, and show that you get the same result by algebraically combining like terms.6. Define i times a number as the number rotated 90 degrees counterclockwise. Practice this with pure real / imaginary numbers.7. Show that we can rotate a number by rotating the real and imaginary parts separately. Drawing a rectangle with two sides along the real and imaginary axes and one corner at the number to be rotated makes this easy to see. Practice this, and show that this trick is equivalent to using the distributive property.8. Work out the 0th, 1st, 2nd, 3rd, 4th, and 5th powers of i. Note that the powers of i are cycling around zero.9. Review what fractional exponents mean if needed. Lead them to the conjecture that the (1/2)th power of i lies midway on the unit circle between 1 and i. Have them calculate where this is and verify algebraically that multiplying this number by itself gets you i.10. Explain how multiplication by any number on the unit circle yields a rotation.
>>16876881multiplying them is the easiest thing to understand. i^2=-1 another i makes it -i, another i makes it --1 or just 1, and so it repeats bc you just keep getting 1.The weird part is the distance from i to 0 being 1 or something. But, if you are smart you just grok it when you're learning it. Like it's been a really long time so I forget why, but, it makes perfect sense, just trust me, while everyone else was yelling bullshit I just figured it out like it makes sense. i itself will NEVER make sense the way you want it to if you are a narcissist like narcissists can't into i or .999...=1i is not a placeholder for future apples. That's just negative numbers. Negative numbers aren't even real either like if you set a midpoint you can list something as a negative amount but even 0 is not real like 0 doesn't exist, if you can find nothing you'll be mistaken bc there's always something even in space
