What does e^(pi*i) equal?
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36 members have voted
To address the first three choices, "-1, no ifs ands or buts" means it is a firm -1, and the laws of mathematics can prove it beyond any reasonable doubt. "-1, by convention" means that it is assigned a value of -1, so that all cases of other general formulas work, but the value is more given in the interests of consistency, rather than something that can be proven. "Indeterminate form" means that the answer doesn't exist, like what some would say 0^0 is.
While it is a convention that 0^0 = 1, since an empty-product is not defined, it is a fact that e^(i*pi) = -1, and the proof is given in about the fourth week of any undergraduate course on Complex Variables.Quote: WizardThis is a follow up to the wonderful thread What does 0^0 equal?. This question posed in the title was brought up in that thread here, but I think it is worthy of a thread on its own.
To address the first three choices, "-1, no ifs ands or buts" means it is a firm -1, and the laws of mathematics can prove it beyond any reasonable doubt. "-1, by convention" means that it is assigned a value of -1, so that all cases of general other formulas work, but the value is more given in the interests of consistency, rather than something that can be proven. "Indeterminate form" means that the answer doesn't exist, like what some would say 0^0 is.
Quote: teliot... and the proof is given in about the fourth week of any undergraduate course on Complex Variables.
Is that proof based on Euler's Equation?
Edit : http://scipp.ucsc.edu/~johnson/phys160/ComplexNumbers.pdf looks very familiar to me...
You're not going to ask for a proof of Euler's are you? Hint. Taylor series.Quote: WizardIs that proof based on Euler's Equation?
Quote: thecesspitI see Babs noted that -1 is an imaginary number in a previous thread. It's not. -1 is real and negative. The real and imaginary parts of a complex number (e.g. 3 + 4i) can be used to do all sorts of useful sums, we used for modelling frequency response of a circuit. I forget exactly how, but capacitors and inductors could be modelled in terms of imaginary numbers, resistors as real numbers and the response of a circuit to various inputs could be worked out.
Edit : http://scipp.ucsc.edu/~johnson/phys160/ComplexNumbers.pdf looks very familiar to me...
cess,
Thanks for the correction; I asked for it aand appreciate the opportunity to learn from it.
Quote: teliotYou're not going to ask for a proof of Euler's are you? Hint. Taylor series.
Note: This post is an admission of the weakness of either my math skills or my memory.
Earlier today, when this question was posed in the original thread, I was thinking that a Taylor Series expansion might offer the proof that the Wizard was looking for then. Since I am lazy and like to expand about a=0, I tried to look up the proper spelling of the name for the Maclaurin Series. I was reckless and looked at Wikipedia, where I came across the comment that, "A function may not be equal to its Taylor series, even if its Taylor series converges at every point." They even give an example of a function for which the Taylor Series converges readily and does not represent the function.
If I had heard that when studying the calculus, the notion completely evaporated from my memory, as did so many other things I "studied" in school.
I seem to recall reading somewhere that this is not considered a rigorous proof. Part of the explanation for this might be that the series expansion for sine and cosine is itself a definition.Quote: dwheatleyA Taylor series expansion does prove the result.
Quote: beachbumbabscess,
Thanks for the correction; I asked for it aand appreciate the opportunity to learn from it.
No worries. I spent two years with 'j'. He became a close and intimate friend on many late nights!
e^(pi*i) equals one thing and one thing only: Me going to the Student Services desk and asking to change my schedule for the semester, Daddy has a G.P.A. to protect! At least, I did, when I went to school.
Quote: Mission146Hmmm...
e^(pi*i) equals one thing and one thing only: Me going to the Student Services desk and asking to change my schedule for the semester, Daddy has a G.P.A. to protect! At least, I did, when I went to school.
Yeah, I sorta did that in my first semester of grad school...signed up for 6 classes...dropped the hardest 2. One of those classes was definitely beyond this, mainly solving partial differential equations numerically. The weirdest part was it was taught by the atmospheric sciences dept. Weathermen do some serious modeling math.
And no, there are no ifs, ands, or buts. The ifs, ands, and buts of exponentiation in the complex domain don't enter into it when the base is e.
Lagrange's work on Taylor series gives us that Maclaurin series converge to the result so long as it's infinitely differentiable at every point from x to the origin. (I know there are stronger techniques, but I think they might be assuming the consequent in this case, and I'm not going to dig through the proofs to check.) Since d(e^x)/dx = e^x, e^x is trivially infinitely differentiable wherever it exists.
Quote: tringlomaneYeah, I sorta did that in my first semester of grad school...signed up for 6 classes...dropped the hardest 2. One of those classes was definitely beyond this, mainly solving partial differential equations numerically. The weirdest part was it was taught by the atmospheric sciences dept. Weathermen do some serious modeling math.
I didn't do that, exactly, I'd always just replace my hardest class or two with a cupcake class. The main goal was to try to figure out the easiest possible way to obtain your degree, with Honors, in terms of what classes you were going to take, so my method would give you the opportunity to weed everything out. For example, my Major was Economics, but my toughest Math class was a 300-level Intermediate Statistical Analysis. Certainly, many of my higher level classes were Math-related, but in terms of what was just a straight Math class, that was my toughest.
It could have been worse, but you have to take the specific required classes, and then you have to earn so many credits in each individual category, but you can plug that up with absolute crap to GPA-protect. I NEVER took a Science beyond 100-Level science classes, because I'm awful at Science. The electives? Absolute garbage. Bowling? You're going to give me a credit hour to bowl?
All of my Science classes have sucked, though. Art, also, hate it, but that's Middle School you're talking, there.
D's in Art, D's & C's in Science, and then A's & B's everywhere else, mostly A's. Except the second semester of my senior year of High School, had so many 5.0 Honors classes that my 3.5+ GPA was well-enough protected that I messed around and failed second semester Business Management just because I thought it'd be funny, given that would be my major...before deciding on Economics.
"A minor from the school of business or any University discipline," Sweet, freakin' Philosophy, playa! Easy A's.
Hmmm....That's interesting, though, my Major was such that it was important for me to work with numbers, but I must question whether or not numbers actually mean anything. What a mess!
"Imaginary" has a specific meaning in math, and a general meaning in everyday language. As such, I agree that "-1" is somehow imaginary, in that you don't "see" -1. But things you don't see may still exist (the mind, the boson, kindness, etc.); you only see their consequences. They are not imaginary, in the sense of "not existing".
As a (very weak) comparison, let us say that "-1" is like a proper name: some convention to be able to speak easily of a concept, and use it.
The number i, by contrast, could be like the noun "unicorn" or "God". There are no unicorns in the real world, but the concept is clearly defined, and usable in the appropriate contexts. Many phrases or elaborate concepts are usefully referring to "God" even though that does not exist.
And some people even believe it exists. I have no problem with that. Be it God or i.
Quote: thecesspitI forget exactly how, but capacitors and inductors could be modelled in terms of imaginary numbers, resistors as real numbers and the response of a circuit to various inputs could be worked out.
In another thread a while back I argued that infinity is a philosophical/mathematical concept, but nothing in nature is actually infinite. This, mind you, is just my opinion. Some bright people I respect disagree with it.
However, I'd like to float out the question for i. I don't doubt that imaginary numbers are used in electrical engineering. However, are they just "modelling" properties of capacitors, or is there something truly ... imaginary ... about them? Likewise, my father has a PhD in physics, and I know a lot of his advanced texts have i all over the place. However, are they there to help solve real problems, or do they really exist?
The word "imaginary" has nothing to do with what these numbers are, any more than the word "spot" has anything to do with the particular dog it names. It is just a historical accident that this word was chosen. Unfortunately, the name gives the appearance that there is some dispute. There is none. This number exists as much as any other number exists. "i" is the name of a number, it has no connection with the nature of the object itself, any more than "two" has anything to do with the object it refers to.Quote: WizardIs there something truly ... imaginary ... about them?
The algebraic closure of the real numbers is the complex numbers. This is the smallest field that contains roots of all polynomial equations with real coefficients. You learned this in 8-th grade, it's called the Fundamental Theorem of Algebra, Gauss proved it at age 19, I believe. Without this theorem, the subject known as "algebra" would not be possible, that is, it is always possible to solve for x.
Galois later proved that there is no closed formula for solving polynomials of degree 5 or larger. This is usually the Theorem that is proven at the culmination of an undergraduate course in Abstract Algebra (groups, rings and fields).
Read: http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra
I've got to stop reading threads about mathematics here. They are just nonsense. It's a lot easier to "think" things than to take the time to *learn* things.
Quote: WizardIn another thread a while back I argued that infinity is a philosophical/mathematical concept, but nothing in nature is actually infinite. This, mind you, is just my opinion. Some bright people I respect disagree with it.
However, I'd like to float out the question for i. I don't doubt that imaginary numbers are used in electrical engineering. However, are they just "modelling" properties of capacitors, or is there something truly ... imaginary ... about them? Likewise, my father has a PhD in physics, and I know a lot of his advanced texts have i all over the place. However, are they there to help solve real problems, or do they really exist?
I think there is a definite correlation between i, -1, and the bible or other religious texts. They are all creations of the mind to explain the (for the present) unknowable certain values. As there is a correlation between this thread and the "Life after Death" thread. One is symbolic and the other prosaic, but both describe the same effort to understand that which is beyond the threshhold of current methodologies. I believe the concept of imaginary numbers is a convention, even a placeholder that allows explanation and manipulation of results where no other proof exists. And from that same well of "leaps of faith" comes nearly all creativity, invention, and progress, so I would not give up believing for all the world.
Right ... go for it. What does it matter if people actually know what these numbers are?Quote: beachbumbabsI think there is a definite correlation between i, -1, and the bible or other religious texts.
Quote: WizardIs that proof based on Euler's Equation?
It was in my education. I do recall the Taylor Series proof now, but I think e^ix = i sinx + cos x is more often used to prove e^i*pi = -1.
Quote: teliotRight ... go for it. What does it matter if people actually know what these numbers are?
I'm reading what you're writing, Eliot, and trying to learn from it, but I think there is a basis for recognizing that there is a concept involved, not just a firm value, or else "i" would not exist; it would have an assigned numerical value rather than a derived one. Like your previous example of "two" as a quantifying symbol, not the thing itself, except that "two" is firmly defined across all languages and cultures as a specific quantity of objects, and you can't point to an "i" amount of things anywhere, except as used in a formula to help define another concept or value. Maybe it's a philosophical argument requiring rejection by mathematical processes to simply accept the definition of "i" as it exists in order to explain other things.
Quote: WizardHowever, are they just "modelling" properties of capacitors, or is there something truly ... imaginary ... about them?
There is no "true" imaginary about capacitors in physics (or anything in physics).
In mathematics, i and -i are very different. Yet in physics, you can replace every i by -i (and vice versa), and all laws will still hold. This proves that the imaginary property of capacitors (or any other object) is purely a model, yet handy but not to be confused with reality.
Quote: beachbumbabsI'm reading what you're writing, Eliot, and trying to learn from it, but I think there is a basis for recognizing that there is a concept involved, not just a firm value, or else "i" would not exist; it would have an assigned numerical value rather than a derived one. Like your previous example of "two" as a quantifying symbol, not the thing itself, except that "two" is firmly defined across all languages and cultures as a specific quantity of objects, and you can't point to an "i" amount of things anywhere, except as used in a formula to help define another concept or value. Maybe it's a philosophical argument requiring rejection by mathematical processes to simply accept the definition of "i" as it exists in order to explain other things.
2 is defined to be the cardinality of the set {{},{{}}}. The fact that a set can be said to have cardinality 2 is simply a statement that there is a bijection between that set and the reference above. I am not kidding.
In a mathematics major, calculus, differential equations, that sort of stuff is meaningless when it comes to having any real understanding of mathematics. Usually about the junior year the student takes courses like "Set Theory," "Abstract Algebra," "Real Variables," etc. At that point, the subject starts from scratch and you get to do actual mathematics. Without those foundations, this conversation really is meaningless. If you want to have a real conversation about foundational questions, get some books on those topics and start reading.
Quote: teliotThe word "imaginary" has nothing to do with what these numbers are,
I'd be interested to know what they are called in other languages.
Quote:This number exists as much as any other number exists. "i" is the name of a number, it has no connection with the nature of the object itself, any more than "two" has anything to do with the object it refers to.
Well, I know we could argue that no number really exists, and they are all just concepts. However, the number two describes quantities of things commonly found in nature. Like the number of boobs on Jeri Ryan's chest.
The question at hand is can you point to anything in nature which has an imaginary or complex quantity? If not, then I would submit for your consideration that real numbers are more ... real ... than imaginary ones.
Quote:The algebraic closure of the real numbers is the complex numbers. This is the smallest field that contains roots of all polynomial equations with real coefficients. You learned this in 8-th grade, it's called the Fundamental Theorem of Algebra, Gauss proved it at age 19, I believe. Without this theorem, the subject known as "algebra" would not be possible, that is, it is always possible to solve for x.
I have no problem that they are useful to the study of mathematics. However, I'm looking for something I can sink my teeth into here.
Quote:I've got to stop reading threads about mathematics here. They are just nonsense. It's a lot easier to "think" things than to take the time to *learn* things.
Come on Eliot, we need you. Think of it as a good deed to the those who need your enlightenment. If you'd prefer, I can post some difficult functions to integrate, but I don't think anybody would bother.
I would feel needed if there was some recognition that you have a Ph.D. in mathematics here, who was a professor of mathematics for 15 years, who has 20+ journal publications, and has somewhat of an expertise in foundational questions. That way I wouldn't feel like I was banging my head.Quote: WizardCome on Eliot, we need you. Think of it as a good deed to the those who need your enlightenment. If you'd prefer, I can post some difficult functions to integrate, but I don't think anybody would bother.
Mathematics is not a sequence of ever more difficult puzzles.
Look up topics like Goedel's Incompleteness Theorem or the Banach-Tarski Theorem and get into some interesting conversations where there really is something to talk about. Or, when questions of infinity come up, don't talk about if "it exists" but rather discuss the cardinality of the rationals vs. the cardinality of the real numbers, Cantor's diagonal argument, and once that's understood, have a go at the Continuum Hypothesis.
Asking if a dog is a "dog" may be a Zen Koan, but it is not math.
Quote: MangoJThe exponential function is a true representation of such beauty, that people ignorant of its very properties should be slapped in the face with a good math book.
Hey, I represent that remark, and your book hertz!
Quote: teliotI would feel needed if there was some recognition that you have a Ph.D. in mathematics here, who was a professor of mathematics for 15 years, who has 20+ journal publications, and has somewhat of an expertise in foundational questions. That way I wouldn't feel like I was banging my head.
If you take no pleasure in helping others climb up the ladder of mathematical understanding, then don't bother. I'm sure you'll find peers at your level elsewhere.
Quote: teliotI would feel needed if there was some recognition that you have a Ph.D. in mathematics here, who was a professor of mathematics for 15 years, who has 20+ journal publications, and has somewhat of an expertise in foundational questions. That way I wouldn't feel like I was banging my head.
Mathematics is not a sequence of ever more difficult puzzles.
Look up topics like Goedel's Incompleteness Theorem or the Banach-Tarski Theorem and get into some interesting conversations where there really is something to talk about. Or, when questions of infinity come up, don't talk about if "it exists" but rather discuss the cardinality of the rationals vs. the cardinality of the real numbers, Cantor's diagonal argument, and once that's understood, have a go at the Continuum Hypothesis.
Asking if a dog is a "dog" may be a Zen Koan, but it is not math.
This has become somewhat of a two person discussion, but I would like to interject that you, Dr. Jacobson, and several others, are exactly the reason I find this forum so compelling and worthwhile. I absolutely respect and acknowledge your talents and education.
I spent an hour listening to you and two others two weeks ago (the day before you invited me to address you as "Eliot" rather than "Dr. Jacobson") on your G2E panel, and you were interesting, colloquial, and armed with several useful and illustrative anecdotes that made you approachable and affable despite your formidable credentials. This was entirely to your credit, as was your attendance at the WoV dinner the next evening.
These same attributes made your book very readable and enjoyable, and as your reader I was quite comfortable in trusting the information you provided, in part because you wrote with a perfect mixture of expertise and humanity. If my game is a success, it will be partially due to your efforts there.
I respectfully request that you continue to participate and teach those of us who are interested in what you have to say, bringing those same laudable qualities here that you demonstrated elsewhere. Your contributions are appreciated.
Quote: beachbumbabs
I respectfully request that you continue to participate and teach those of us who are interested in what you have to say, bringing those same laudable qualities here that you demonstrated elsewhere. Your contributions are appreciated.
Yeah, when Eliot speaks, I do listen. Unfortunately, I don't think he's as personally into the topic nearly as much as the Wiz and others here are. It's like how I generally stay out of craps threads, they just aren't worth it to me. But I am being amused by the thread because it reminds me of the days of circuit theory. I also found this nice website that gives illustrative applications on how imaginary numbers are particularly useful (note: it needs Java for the demos).
Quote: tringlomaneYeah, when Eliot speaks, I do listen.
So do I. That is why I kept bothering him to join the math threads. Maybe my last post was too harsh, but if he doesn't enjoy teaching others, then I will stop pressing him on it.
Quote: WizardSo do I. That is why I kept bothering him to join the math threads. Maybe my last post was too harsh, but if he doesn't enjoy teaching others, then I will stop pressing him on it.
Fair enough. It is, of course, his choice.
No one has asked me to teach. No on has asked me to tell me what something is. Rather, what happens is that people express their opinion about what something is, and for the most part it is nonsense. Whether it is 0^0, e^(i*pi), 0.9999... or whatever. Mike asks me to chime in and I say what is true about the thing.Quote: beachbumbabsFair enough. It is, of course, his choice.
On the Internet, all opinions are equal. It doesn't matter that I spent almost 20 years learning mathematics because of my love of the subject, nor demonstrated my expertise by a Ph.D. and a professional career in the subject. It just doesn't matter.
Babs, all I ask is that you pick up a book. You are as smart as anyone on this site. Just read.
Quote: teliotOn the Internet, all opinions are equal. It doesn't matter that I spent almost 20 years learning mathematics because of my love of the subject, nor demonstrated my expertise by a Ph.D. and a professional career in the subject. It just doesn't matter.
It does matter, at least to the people who should matter to you. Not all opinions are equal, but anyone can post one. To that extent, many opinions get the same amount of airtime. Unlike academia, there is no peer review on a newsgroup or forum. But don't let that discourage you.
Quote: MathExtremistIt does matter, at least to the people who should matter to you. Not all opinions are equal, but anyone can post one. To that extent, many opinions get the same amount of airtime. Unlike academia, there is no peer review on a newsgroup or forum. But don't let that discourage you.
And I'm pretty sure opinions from you, Eliot, the Wizard, etc. generally get greater weight than someone that hasn't previously demonstrated their knowledge on the subject. However, I do understand were Eliot is coming from here. This particular question has been answered by others elsewhere, so whatever he says would just mostly be a rehash of what's already been done. I'm personally good for doing that myself, especially on basic video poker questions.
Yes, I know all of this. It my failing for getting so frustrated. Thanks for the reminder.Quote: MathExtremistIt does matter, at least to the people who should matter to you. Not all opinions are equal, but anyone can post one. To that extent, many opinions get the same amount of airtime. Unlike academia, there is no peer review on a newsgroup or forum. But don't let that discourage you.
I had breakfast a few weeks back with Mike and he invited me to join-in on the math threads. I explained to him then how frustrated I get when I post in threads such as this, and said I would not be posting. So, I explained some ideas about 0^0 to him and he generously posted on my behalf. It is my own fault for going against my own good advice for myself.
It was my error for not recognizing that this was the Math forum; I thought the group was having a metaphysical conversation and stretching boundaries. I was not aware that 300 level math courses and above threw everything out and started over - I got distracted during Calc 3 sophomore year and had to withdraw from the course, and my life went a different way from there. The most valuable thing (to me, anyway) you said in this thread was that post, especially the references to specific topics of higher maths that I can search for and learn about; I had already started. Thanks for your counsel.
Babs
What you are seeing come up for me in this thread is the left-over angst from my years as a professor.Quote: beachbumbabst was my error for not recognizing that this was the Math forum; I thought the group was having a metaphysical conversation and stretcIhing boundaries. I was not aware that 300 level math courses and above threw everything out and started over - I got distracted during Calc 3 sophomore year and had to withdraw from the course, and my life went a different way from there. The most valuable thing (to me, anyway) you said in this thread was that post, especially the references to specific topics of higher maths that I can search for and learn about; I had already started. Thanks for your counsel.
I still have ratings on ratemyprofessor,
http://www.ratemyprofessors.com/ShowRatings.jsp?tid=105743
These show that I was either loved or hated, and that I was a very tough teacher -- tests and assignments were very hard. That's how it went for many years. I had little patience for stupid questions, while the best students had plenty to do. That same stuff comes up here. Arrogance, pure and simple.
As far as advanced math starting over, it can do so many times if that's the direction you want to take. Such was the case with my Ph.D. area, "Category Theory", in which the basic ingredients are called classes, objects and functors.
Quote: teliot
I still have ratings on ratemyprofessor,
http://www.ratemyprofessors.com/ShowRatings.jsp?tid=105743
First, no hotness grade? what's that about??
Second, many of these ratings include helpful hints like "don't cheat...ever." That must be much more prevalent than I'd thought. Can't say I ever considered that in my own studies. Does it really happen that often, and what did you DO to the people you caught?!?
Quote: Wizard
Well, I know we could argue that no number really exists, and they are all just concepts. However, the number two describes quantities of things commonly found in nature. Like the number of boobs on Jeri Ryan's chest.
+2
Best. Example. Ever.
On a Star Trek Voyager kick?
Quote: teliotI had breakfast a few weeks back with Mike and he invited me to join-in on the math threads. I explained to him then how frustrated I get when I post in threads such as this, and said I would not be posting.
You're right, you did explain to me why you didn't want to participate in the 0^0 thread at the breakfast. I should have remembered your feelings about it. Instead I tried to pull you into a new thread you didn't want to me in with my "Eliot, we need you" remark. For that, I apologize.
By the way, before you remarked to Babs to "open a book," I did. I went through my old beat-up Calculus 3 series book from UCSB to brush up on the Taylor Series, which I haven't looked at in 30 years. I've been meaning to post a solution to epi*i using Taylor, but between a lack of time and that I don't know how to make mathematical symbols in any kind of electronic document, I haven't. Yet. I plan to just write out a solution by hand and scan it.
Quote: teliotI still have ratings on ratemyprofessor,
http://www.ratemyprofessors.com/ShowRatings.jsp?tid=105743
I had a good laugh at your reviews! Thanks for sharing them.
Quote:"Jacobson is the worst teacher I have ever heard of. He made me change my major from comp. sci to business."
No, you changed your major because, like me, you're an idiot.
How can people possibly blame the professor for all of this?
Oh, he's too hard.
OK, so he's hard. Do what winners do, buckle down, grit your teeth, dig in, walk to Student Services, and switch classes!
Sorry, Teliot, I like you a ton, but if your classes are tough, I'm not having it!!!
DROP before you FAIL!!!
Quote: WizardI've been meaning to post a solution to epi*i using Taylor, but between a lack of time and that I don't know how to make mathematical symbols in any kind of electronic document, I haven't. Yet. I plan to just write out a solution by hand and scan it.
Wiz,
I don't know if you have Word, but if you do, once you start a new document, go to the "insert" tab and look at the top right. It has all common mathematical symbols and many common formulae pre-formatted for insertion into your documents. I don't know if that formatting will translate to this forum using a cut-and-paste, but at least you can easily type your solution out before scanning.
Quote: beachbumbabsI don't know if you have Word, but if you do, once you start a new document, go to the "insert" tab and look at the top right.
Thanks. I never figured how to do that with Word 97, but I see now that Word 07 seems to make it easier. Let me try...
Quote: WizardIs that proof based on Euler's Equation?
Euler published the general equation exp(ix)=cos(x)+i*sin(x) in 1748, and his proof was based on the infinite series of both sides being equal. That is the way that I learned it in school. The geometric definition historically came later.
Euler's identity which is the special case of the more general formula: exp(i*PI)+1=0 was not actually mentioned by Euler. There is some evidence to suggest that the identity was known to August de Moivre who did not see the more general formula. But I don't know the proof that he used.
The identity carries almost a mystical aura since it contains all five significant constants ( 0, 1, e, i, and PI) in one simple equation,
Quote: pacomartinEuler published the general equation exp(ix)=cos(x)+i*sin(x) in 1748,
It is too easy to prove with that. For full credit, I expect the user to get the solution based on Taylor's Formula.
By the way, I was expecting you to illustrate my Jeri Ryan example with a carefully chosen photo.
