Learning to Love Complex Numbers(jeremykun.com)
jeremykun.com
Learning to Love Complex Numbers
http://jeremykun.com/2014/05/26/learning-to-love-complex-numbers/
28 comments
There are interactive plots, along the lines of VCA, here http://puzlet.com/m/b00dd (I am a dev).
+1 for VCA. This book is written with passion and is one of the truly amazing book on Mathematics. It's just too bad it's so priced heavily (hint: International editions are far more cheaper).
I've loved complex numbers since first learning about them, and my first significant programming experience was writing these types of programs for my TI-89 graphing calculator in high school in the late 90s. This is a good introduction.
Another good equation to play around with is x_next = a x (1 - x), for a from 0 to 4. You'll find that x settles to a single value for low a, eventually bifurcating as you increase a, then bifurcating again. At one point, though, it splits into a cycle of three values, then an erratic distribution. Someone once wrote a paper on how "period three implies chaos".
Another good equation to play around with is x_next = a x (1 - x), for a from 0 to 4. You'll find that x settles to a single value for low a, eventually bifurcating as you increase a, then bifurcating again. At one point, though, it splits into a cycle of three values, then an erratic distribution. Someone once wrote a paper on how "period three implies chaos".
aka logistic map
http://en.wikipedia.org/wiki/File:LogisticCobwebChaos.gif
http://en.wikipedia.org/wiki/File:LogisticCobwebChaos.gif
Yes, but the visualizations I programmed started with http://polygeek.com/images/chaos/fig_01_BifurcationGraph.png
Some fantastic visualizations to accompany his essay. I would love an ipython notebook version that I could download and play with.
Great article! I've always intuited the rules with which we operate on imaginary numbers as a hack - we don't know what to do with i other than squaring it, so we avoid doing anything with it and algebra our way around the issue.
Example: (i + 3) + (i + 3) == i^2 + 6i + 9 == 6i + 8 is about as logical to me as (x + 3) + (x + 3) == x^2 + 6x + 9. Sure, if we knew what the heck x was this operation would have been easier, but since we don't, we just use FOIL to work around it. In the former, somehow we know what i^2 is but not i, so we can reduce a little further but not completely.
Of course, I don't know how sound this gutfeel impression is.
Example: (i + 3) + (i + 3) == i^2 + 6i + 9 == 6i + 8 is about as logical to me as (x + 3) + (x + 3) == x^2 + 6x + 9. Sure, if we knew what the heck x was this operation would have been easier, but since we don't, we just use FOIL to work around it. In the former, somehow we know what i^2 is but not i, so we can reduce a little further but not completely.
Of course, I don't know how sound this gutfeel impression is.
It is very sound. You can think of i as being a variable 'x', and therefore, you can see Complex numbers are polynomials over real numbers (denoted as R[x]). But you can simplify the polynomials by using the fact that x2 = 1 for this construction (since x = i on this situation).
This is the exact construction of Complex numbers as an algebraic extension over the field of Real numbers[1]
What is more amazing: there is no such way to do the same on the complex numbers. This is because the Complex numbers form an Algebraically closed field[2]
[1] http://en.wikipedia.org/wiki/Field_extension#Examples
[2] http://en.wikipedia.org/wiki/Algebraically_closed_field
This is the exact construction of Complex numbers as an algebraic extension over the field of Real numbers[1]
What is more amazing: there is no such way to do the same on the complex numbers. This is because the Complex numbers form an Algebraically closed field[2]
[1] http://en.wikipedia.org/wiki/Field_extension#Examples
[2] http://en.wikipedia.org/wiki/Algebraically_closed_field
Very sound.
One formal model of complex numbers is R[i]/(i^2 + 1), which means all real polynomials in the variable i (R[i]) after "simplifying" (/) by taking i^2 + 1 = 0 (equivalently i^2 = -1)... which is exactly the process you are describing.
One formal model of complex numbers is R[i]/(i^2 + 1), which means all real polynomials in the variable i (R[i]) after "simplifying" (/) by taking i^2 + 1 = 0 (equivalently i^2 = -1)... which is exactly the process you are describing.
i's irreducible simply because it's a unit-sized basis vector for how much of a quantity is in the "i direction". It's just something to hang a coefficient on so that it doesn't mix with the real part. We could just as well invent a similar thing for the reals, and call it g. Then your result is 6i + 8g. Or we could use tuples. It doesn't matter, as long as you do proper bookkeeping of the real and imaginary parts.
>One fact we all remember about numbers is that squaring a number gives you something non-negative. 7^2 = 49, (-2)^2 = 4, 0^2 = 0, and so on. But it certainly doesn’t have to be this way. What if we got sick of that stupid fact and decided to invent a new number whose square was negative?
This presentation of complex numbers makes the whole article flawed and useless in my opinion. When I first learned about complex numbers, they were presented in a very similar fashion: let's just invent a number i such that i^2 = -1. This is inane: we have a multiplication operation that we are all familiar with and we know exactly what it does, and then somebody tells us that we can use it on some "imaginary" thing (??) such that it times itself equals -1? How is anybody supposed to make any sense of it? It's like saying: let's invent a "number" j such that j-j=the letter z. What does it even mean? Nothing! it's gibberish, and a similar definition of "i" is also gibberish. We cannot make sense, under our normal understanding of multiplication and under our normal understanding of numbers as including only the real line, of how can something times itself be -1, and neither should we, because there are much better ways to present the whole thing from the beginning.
The correct way to present complex numbers is either in the context of abstract algebra - where there is a very obvious question of whether we can embed the real line as a field inside the real plane, or simply present them geometrically without going into fields. It is simply wrong, in my opinion, to present i as something we "invent" so that i^2=-1 (why would anybody do that??), and then go on and say that after we have invented this, there are ways to imagine this geometrically. No! If you want to talk about geometry, then define i geometrically, then extend the definition of multiplication geometrically to the plane, and then it becomes clear that i^2 = -1, and there is no mystery about anything.
Edit: it should also be noted that historically, complex numbers didn't come into existence because somebody decided on a whim to "invent" a number i such that i^2=-1. Rather, it was a result of the fact that cubic equations such as x^3=15x+4 clearly had a solution (for example x=4), but using the cubic formula to solve them resulted in weird terms such as sqrt(-121). Bombelli, in the 16th century, decided to try and compute with those terms anyway, and through this process eventually succeeded in producing the right results (x=4), so it eventually became clear that the roots of negative numbers aren't just gibberish: they interacted somehow with the reals, and there was some way to "make them work" to produce real results, though the full realization of what was happening probably came much later.
This presentation of complex numbers makes the whole article flawed and useless in my opinion. When I first learned about complex numbers, they were presented in a very similar fashion: let's just invent a number i such that i^2 = -1. This is inane: we have a multiplication operation that we are all familiar with and we know exactly what it does, and then somebody tells us that we can use it on some "imaginary" thing (??) such that it times itself equals -1? How is anybody supposed to make any sense of it? It's like saying: let's invent a "number" j such that j-j=the letter z. What does it even mean? Nothing! it's gibberish, and a similar definition of "i" is also gibberish. We cannot make sense, under our normal understanding of multiplication and under our normal understanding of numbers as including only the real line, of how can something times itself be -1, and neither should we, because there are much better ways to present the whole thing from the beginning.
The correct way to present complex numbers is either in the context of abstract algebra - where there is a very obvious question of whether we can embed the real line as a field inside the real plane, or simply present them geometrically without going into fields. It is simply wrong, in my opinion, to present i as something we "invent" so that i^2=-1 (why would anybody do that??), and then go on and say that after we have invented this, there are ways to imagine this geometrically. No! If you want to talk about geometry, then define i geometrically, then extend the definition of multiplication geometrically to the plane, and then it becomes clear that i^2 = -1, and there is no mystery about anything.
Edit: it should also be noted that historically, complex numbers didn't come into existence because somebody decided on a whim to "invent" a number i such that i^2=-1. Rather, it was a result of the fact that cubic equations such as x^3=15x+4 clearly had a solution (for example x=4), but using the cubic formula to solve them resulted in weird terms such as sqrt(-121). Bombelli, in the 16th century, decided to try and compute with those terms anyway, and through this process eventually succeeded in producing the right results (x=4), so it eventually became clear that the roots of negative numbers aren't just gibberish: they interacted somehow with the reals, and there was some way to "make them work" to produce real results, though the full realization of what was happening probably came much later.
The reason they invented i was because working directly with sqrt(-121) leads to unexpectedly false proofs (there is a common proof that 1=2 using this method).
In other words, i was invented to give a clear expression to an idea so that people could solve their problems more easily. And then it was discovered to have these nice geometric properties. Whether you find applications first or second doesn't change anything. There has been plenty of mathematics invented for fun or during periods of boredom.
If you want to you can easily invent the number j such that j-j=the letter z. The point is that once you create it you can't control it anymore, and if you made a poor definition then it will turn out to not have existed at all in the first place (due to logical contradictions), or it may turn out to be not interesting, or it may turn out to be wildly fascinating. As a practicing mathematician this is your day to day, and it's only the interesting ideas that reach the outside world.
You make it seem like some math is not "imaginary" while other math is, but this is a fallacy. All mathematics was invented by people, and it was invented mostly for their own pleasure. Applications almost always come later, if ever.
In other words, i was invented to give a clear expression to an idea so that people could solve their problems more easily. And then it was discovered to have these nice geometric properties. Whether you find applications first or second doesn't change anything. There has been plenty of mathematics invented for fun or during periods of boredom.
If you want to you can easily invent the number j such that j-j=the letter z. The point is that once you create it you can't control it anymore, and if you made a poor definition then it will turn out to not have existed at all in the first place (due to logical contradictions), or it may turn out to be not interesting, or it may turn out to be wildly fascinating. As a practicing mathematician this is your day to day, and it's only the interesting ideas that reach the outside world.
You make it seem like some math is not "imaginary" while other math is, but this is a fallacy. All mathematics was invented by people, and it was invented mostly for their own pleasure. Applications almost always come later, if ever.
The only point I made is that there are better ways to present i than saying "let's invent a number i such that i*i = -1", and that the above presentation is pedagogically horrible.
The amount of confusion surrounding complex numbers should make it clear that there is a problem. We should try to alleviate this problem, instead of sweeping it under the rug . "Deal with it, this is how math is" is not a solution, especially when much better ways of presenting the subject do exist.
The amount of confusion surrounding complex numbers should make it clear that there is a problem. We should try to alleviate this problem, instead of sweeping it under the rug . "Deal with it, this is how math is" is not a solution, especially when much better ways of presenting the subject do exist.
And so you would propose to say, "let's invent a number i so that multiplying by i is a rotation!"
A priori, why should anyone have a reason to think of numbers geometrically? The truth is they're both arbitrary. What I'm trying to do is not dictate facts but show the process of exploring an idea. That is correct pedagogy, and this way one can be surprised by the geometric perspective arising from what I think is a more natural question (why can't negative numbers have square roots?). Indeed, I don't think anyone has ever naturally asked "why can't some numbers be thought of as rotations?" (that is, without already knowing about complex numbers, and extrapolating to symmetry groups of various geometric shapes).
Never did I ever say "deal with it, this is how math is." I say, "What if this idea weren't so crazy, then this is what would have to follow. Look at the nice patterns that come up, we must be scratching at the surface of something really interesting!"
Or at least I try to say this. It seems like you didn't get that message.
A priori, why should anyone have a reason to think of numbers geometrically? The truth is they're both arbitrary. What I'm trying to do is not dictate facts but show the process of exploring an idea. That is correct pedagogy, and this way one can be surprised by the geometric perspective arising from what I think is a more natural question (why can't negative numbers have square roots?). Indeed, I don't think anyone has ever naturally asked "why can't some numbers be thought of as rotations?" (that is, without already knowing about complex numbers, and extrapolating to symmetry groups of various geometric shapes).
Never did I ever say "deal with it, this is how math is." I say, "What if this idea weren't so crazy, then this is what would have to follow. Look at the nice patterns that come up, we must be scratching at the surface of something really interesting!"
Or at least I try to say this. It seems like you didn't get that message.
No, I don't propose we say that. I don't propose we start with "let's invent... now let's 'explore' if this makes sense". This is bad pedagogy: It is confusing, it gives no motivation for what is being invented, and ultimately it produces bad results and demotivated students.
What I do propose is that we start with an interpretation of multiplication as a geometric transformation of of the real line by scaling. Then think about equations such as 1xx=-1, and think what solving them means in terms of our new interpretation: is there a geometric transformation x that we can apply to 1 such that doing it twice results in -1? Yes there is: rotation by pi/2. Now we call this transformation i, and work out the rest of the details.
There is no "let's invent", no "believe me, it works", and no leaps of faith required. A simple problem is presented, and its solution naturally leads to to the definition of i. There is no mystery, there is no need to somehow force ourselves to accept this new symbol into our number systems without knowing why we are doing it or what it is, and most importantly: there are far less confused students.
Many non-geometric presentations are also good, but it should never start with "let's invent". It should start with a problem that people can relate to (just as it started historically), and the definition should arise from our attempt to solve this problem. More generally, it should come out of something we understand, not dropped on our heads by the teacher who "knows" math. Geometry is often slightly better because our intuition works better there (this is why VCA is so successful), but it is not necessary.
Exploring ideas is good. Presenting some hocus pocus definition out of the blue and expecting young students to then "explore" it because somebody said it's interesting - this is bad, and it's definitely not how math is done.
What I do propose is that we start with an interpretation of multiplication as a geometric transformation of of the real line by scaling. Then think about equations such as 1xx=-1, and think what solving them means in terms of our new interpretation: is there a geometric transformation x that we can apply to 1 such that doing it twice results in -1? Yes there is: rotation by pi/2. Now we call this transformation i, and work out the rest of the details.
There is no "let's invent", no "believe me, it works", and no leaps of faith required. A simple problem is presented, and its solution naturally leads to to the definition of i. There is no mystery, there is no need to somehow force ourselves to accept this new symbol into our number systems without knowing why we are doing it or what it is, and most importantly: there are far less confused students.
Many non-geometric presentations are also good, but it should never start with "let's invent". It should start with a problem that people can relate to (just as it started historically), and the definition should arise from our attempt to solve this problem. More generally, it should come out of something we understand, not dropped on our heads by the teacher who "knows" math. Geometry is often slightly better because our intuition works better there (this is why VCA is so successful), but it is not necessary.
Exploring ideas is good. Presenting some hocus pocus definition out of the blue and expecting young students to then "explore" it because somebody said it's interesting - this is bad, and it's definitely not how math is done.
You "invent" when you extend the interpretation of "geometric transformation by scaling" to "geometric transformation in general." You're saying "believe me it works" when you do that. It's just as arbitrary, but you just don't see it that way because somehow geometry on a line and geometry in the plane are more difficult to distinguish than square roots of positive and negative numbers.
Why can't people relate to the question of negative square roots? Students ask this question all the time, along with why we can't divide by zero and what happens when you do 0^0. Rather than address their questions, you're telling then, "Now you have to think about multiplication as a geometric scaling (why? just do it, it's 'natural'), and now we get to add rotations because... it's interesting, believe me!" There is more hocus pocus in that than you're willing to admit.
The problem here is: can we have square roots of negative numbers, and the natural definition arising from an attempt to answer it in the positive is i. It's way more obvious to just declare something to be the square root of a negative number than to try to rack your brain for clever geometric transformations. We do this in mathematics all the time: if we don't know what the roots of a polynomial are, we just denote them by a letter and see what we can deduce about them. If you don't know whether something exists, you suppose it does and either arrive at a contradiction or don't.
Plenty of math is invented without motivation. I see it happen all the time. In fact, people will often give a problem or definition motivation after the fact to try to sell it better! I also see this happen regularly, and it all starts with "let's invent." Your gripe seems to be that you do always want the teacher to have the answer, and that every interesting pattern must follow from a non-arbitrary motivation. This is just not how mathematics works.
Why can't people relate to the question of negative square roots? Students ask this question all the time, along with why we can't divide by zero and what happens when you do 0^0. Rather than address their questions, you're telling then, "Now you have to think about multiplication as a geometric scaling (why? just do it, it's 'natural'), and now we get to add rotations because... it's interesting, believe me!" There is more hocus pocus in that than you're willing to admit.
The problem here is: can we have square roots of negative numbers, and the natural definition arising from an attempt to answer it in the positive is i. It's way more obvious to just declare something to be the square root of a negative number than to try to rack your brain for clever geometric transformations. We do this in mathematics all the time: if we don't know what the roots of a polynomial are, we just denote them by a letter and see what we can deduce about them. If you don't know whether something exists, you suppose it does and either arrive at a contradiction or don't.
Plenty of math is invented without motivation. I see it happen all the time. In fact, people will often give a problem or definition motivation after the fact to try to sell it better! I also see this happen regularly, and it all starts with "let's invent." Your gripe seems to be that you do always want the teacher to have the answer, and that every interesting pattern must follow from a non-arbitrary motivation. This is just not how mathematics works.
I "invent" it in the same kind of way that I "invent" that a certain cloud has the shape of a rabbit. Your "invention", on the other hand, is like inventing a new animal just so that it will fit the cloud. But it's not only the invention that matters - it is where it starts and what are its foundations. Your "invention" starts at the end and its foundations are hidden. You keep drawing those general analogies between things as if they mean anything, but they are so general that the only thing they do is hide the difference, instead of exposing the similarities.
By the same kind of reasoning we can justify an extremely large amount of bad expository articles on mathematics. There are clear conceptual differences between making people "see" multiplication as scaling, then asking a question about the possibility of extending it, out of which i arises naturally and it's clear to everybody how it fits in and in what sense i^2=-1 despite the seeming contradiction with everything we know about numbers and multiplication up to that point, and between your way of inventing this "number" because "if you think about it long enough, there are ways to make it work without contradictions". People don't understand it just as they don't understand what is the square root of an apple: how can you even take the square root of a fruit without extending the definition of what it means to take a square root? We are not dealing with mathematicians who have experience in abstract algebra where multiplication has this abstract meaning and we can just add new stuff to our set and see if it works. We are talking about people who's experience with multiplication and numbers comes from calculating the acceleration of a ball and counting coins. It shouldn't come as a surprise that one approach to teaching something works better than the other: Tim Gowers presents normal subgroups and quotient groups in the much better way [1] than the traditional approach, even though the same very broad analogies can be drawn between the two. VCA presents certain ideas about complex analysis in a much more intuitive and clear way than the traditional approach, even though all it does is place a much higher emphasis on geometry.
Why don't you grab a bunch of students with little prior knowledge of complex numbers, present them the complex numbers using those two ways, then ask them what seems more obvious and natural.
I don't care about what seems natural to you. I care about what produces better results for the majority of people. Your way of explaining complex numbers produces bad results. You can either accept this, or you can keep ignoring it because you know better. There is no shortage of smart mathematicians with the wrong ideas about how to teach mathematics, and they all have plenty of excuses.
[1] - http://gowers.wordpress.com/2011/11/20/normal-subgroups-and-...
By the same kind of reasoning we can justify an extremely large amount of bad expository articles on mathematics. There are clear conceptual differences between making people "see" multiplication as scaling, then asking a question about the possibility of extending it, out of which i arises naturally and it's clear to everybody how it fits in and in what sense i^2=-1 despite the seeming contradiction with everything we know about numbers and multiplication up to that point, and between your way of inventing this "number" because "if you think about it long enough, there are ways to make it work without contradictions". People don't understand it just as they don't understand what is the square root of an apple: how can you even take the square root of a fruit without extending the definition of what it means to take a square root? We are not dealing with mathematicians who have experience in abstract algebra where multiplication has this abstract meaning and we can just add new stuff to our set and see if it works. We are talking about people who's experience with multiplication and numbers comes from calculating the acceleration of a ball and counting coins. It shouldn't come as a surprise that one approach to teaching something works better than the other: Tim Gowers presents normal subgroups and quotient groups in the much better way [1] than the traditional approach, even though the same very broad analogies can be drawn between the two. VCA presents certain ideas about complex analysis in a much more intuitive and clear way than the traditional approach, even though all it does is place a much higher emphasis on geometry.
Why don't you grab a bunch of students with little prior knowledge of complex numbers, present them the complex numbers using those two ways, then ask them what seems more obvious and natural.
I don't care about what seems natural to you. I care about what produces better results for the majority of people. Your way of explaining complex numbers produces bad results. You can either accept this, or you can keep ignoring it because you know better. There is no shortage of smart mathematicians with the wrong ideas about how to teach mathematics, and they all have plenty of excuses.
[1] - http://gowers.wordpress.com/2011/11/20/normal-subgroups-and-...
>This is inane: we have a multiplication operation that we are all familiar with and we know exactly what it does, and then somebody tells us that we can use it on some "imaginary" thing (??) such that it times itself equals -1?
How is this more inane than defining 0, or negative numbers? Those didn't always exist either, somebody invented them for various purposes. When a person learns arithmetic, negative numbers are introduced in a similar way. By the time a person learns about complex numbers, they already know how to deal with variables in equations, so i*i=-1 shouldn't be too hard to comprehend.
How is this more inane than defining 0, or negative numbers? Those didn't always exist either, somebody invented them for various purposes. When a person learns arithmetic, negative numbers are introduced in a similar way. By the time a person learns about complex numbers, they already know how to deal with variables in equations, so i*i=-1 shouldn't be too hard to comprehend.
The fact remains that complex numbers are confusing to an extremely large number of people, and negative numbers and 0 aren't. This is probably because 0 can be given direct meaning in terms of everyday applications: I don't have money = I have 0 dollars. I have 0 dollars and I owe you 5 dollars = I have -5 dollars. But what is i dollars? The switch from real to complex is much more complicated than the switch from the naturals to integers or from the integers to the reals. Algebraically, it is also a completely different kind of extension.
i^2 = -1 should be hard to comprehend, and people should be suspicious when things are presented this way. If your experience with equations make it seem easy, I would argue that you don't fully comprehend the significance of such a definition (for example - lets invent a number j such that 1/j = 0. Does this also seem easy?).
i^2 = -1 should be hard to comprehend, and people should be suspicious when things are presented this way. If your experience with equations make it seem easy, I would argue that you don't fully comprehend the significance of such a definition (for example - lets invent a number j such that 1/j = 0. Does this also seem easy?).
I would argue that very few people understand the reals any better than they understand the complex numbers, though they think they do.
The lack of knowledge people have regarding the reals is different in kind from the confusion surrounding complex numbers, and has nothing to do with what is being discussed.
I just hate them.. Why don't we just use vectors and make math easier to learn?
And basically, negative numbers can be avoided as well using the concept of vectors
Complex numbers are the key to complex analysis which is different than just real multivariate calculus (it has certain constraints).
> Why don't we just use vectors and make math easier to learn?
We do just use vectors; over complex numbers. You really want both, where the vectors themselves are made up of[1] complex numbers. Using a single complex number to represent a vector has gone out of style in some sense because it's restricted to two dimensions. For example historically it was quite common to represent the velocity of a fluid at each point with a complex number. While you can do this, and there are many of advantages, most don't generalize to higher dimensions.
[1] technically the field is complex numbers https://en.wikipedia.org/wiki/Field_%28mathematics%29
We do just use vectors; over complex numbers. You really want both, where the vectors themselves are made up of[1] complex numbers. Using a single complex number to represent a vector has gone out of style in some sense because it's restricted to two dimensions. For example historically it was quite common to represent the velocity of a fluid at each point with a complex number. While you can do this, and there are many of advantages, most don't generalize to higher dimensions.
[1] technically the field is complex numbers https://en.wikipedia.org/wiki/Field_%28mathematics%29
Complicated math concepts (like differential equations) are much easier to learn with a solid background in complex numbers. Vectors are important, but so are complex numbers.
How would you compute exp(v) for a vector v? (for example)
Kalid Azad's oft-posted article: http://betterexplained.com/articles/a-visual-intuitive-guide...
Tristn Needham's book Visual Complex Analysis http://usf.usfca.edu/vca//