A complex "number" (don't think of it as of a number! think of it like you would think of a vector, group, ring or any other abstract structure) is just an ordered pair of real numbers that behaves in a certain predefined way when being added to another complex number or multiplied by it. For an introduction, to avoid unnecessary confusion, it is best to write such "numbers" as ordered pairs using the notation: (a,b). The definitions for the operations are the following:
It is useful to have separate names for each part of a complex number, so the a in (a,b) is called the real part, and the b the imaginary part, but for now think about those names as completely devoid of any meaning. Now, observe that under the above definition:
But those are, if you consider only the real parts of the complex numbers, ordinary operations on the real numbers! An example consequence of this is that we can take some equation concerning real numbers like:
2*x + 5 = 21
and write it down in terms of complex "numbers":
(2,0)*x + (5,0) = (21,0)
Since as we have seen pairs of the form (a,0) behave just like real numbers, we have not changed the meaning of the equation, hence we are free to solve it using the rules of complex algebra and if we happen to arrive at another number of the form (a,0), we can take out the real part of it, plug it into the original equation in terms of real numbers and it is certain to be a valid solution.
This is one of the two properties that makes the use of complex "numbers" fruitful. At the other one we arrive if we now look at "numbers" that are NOT of the form (a,0), for example at a curious property of (0,1):
(0,1)*(0,1) = (-1,0)
So, in the domain of complex "numbers", the "number" that corresponds to the real number -1, happens to have the equivalent of what we for real numbers call the "square root". We just talk about the "square root", but it is a different operation when we are talking about complex numbers.
Those two properties combined allowed mathematicians to tackle some problems that previously did not have a solution. One example is the problem of finding a solution to cubic equations. The math here gets more complicated, but basically it turns out that by writing cubic equations in real numbers in complex numbers instead, you can find general formulas in terms of complex numbers for finding all the possible solutions, and as we have discussed if applying such a formula in the end yields a number of the form (a,0), it is guaranteed to be a valid solution for the original real equation. Google for "cubic equations cardano" to see the details.
Now, this going back and forth between complex and real numbers is so useful, that for the purpose of brevity mathematicians sacrificed intelligibility and introduced sort of a shorthand notation of the form: a + bi, so instead of writing (0,1) as we did above, we just write i, instead of (5,0) we just write 5, and instead of (1,2) we write 1 + 2i. This is purely a trick, there is nothing magical about the "i", it is just a "dummy" variable that allows convenient carrying out of the operations with pairs described above in the manner reassembling ordinary high-school algebra we all know and love.
All this is maybe a bit elementary, but I think this is the part most people fail to understand and because of this start treating complex numbers as something mysterious. There is in fact nothing mysterious about them, you have to boil every application you see of them to the above and then you will get a clear understanding of what is happening and why they are useful. Points on the plane happen to be a model for complex numbers with rotation corresponding to multiplication and so forth, this is of course very interesting, but I feel an introduction to the topic should start with what I have just tried to explain.
Understanding a mathematical concept doesn't mean being able to perform computations using it. It means having an intuitive understanding for what it can represent and how to use it, and how to interpret concepts that use it.
Defining complex numbers as a bunch of arbitrary arithmetic operations on tuples lends nearly zero understanding, no matter how good you get at performing that arithmetic. Understanding complex numbers as rotations is much more useful in being able to understand, say, electromagnetism. In fact, I'll go so far as to say it makes the physics easier even if it makes the math more cumbersome.
Teaching mathematics as simple rules for manipulation of symbols is basically like telling only the punchline to a joke. Not even saying the punchline early, but just saying it on its own and providing no context. The fact that math can be (not is) just a bunch of mechanical rules is amazing when those rules can be mapped to complex and abstract phenomenon, and used to provide deeper understanding of them. The mechanical rules, in themselves, are boring. Their boringness is actually what makes them a good punchline, when contrasted with the intricacy of the systems they describe, but without that set-up, they're just boring.
Maybe It's because my mind is different, but I actually found his explanation very nice. I understand that it looks like just "symbol manipulation" but that is just one way to look at it. He shows how to define complex numbers using the bare minimum assumptions. As he mentioned, the reason why complex numbers are useful is precisely because of their properties under addition, subtraction and multiplication.
Like I said, it just might be because our minds are different. But this seems like a good foundation, i.e you first show that this is all there is to it. Now, you can go ahead and explain the nice physical/geometric interpretations.
Incidentally, this is the way that most books introduce Algebra (well, mathematical algebra, with groups, rings etc, not the algebra from high school)
I'd highly recommend Tristan Needham's Visual Complex Analysis to anyone interested in the subject, for exactly this reason . . . and, subsequently, Henri Cartan's Elementary Theory of Analytic Functions of One or Several Complex Variables as proof that, nevertheless, "algebraic" need not imply "boring" or "computational". As a silly example, I'll never forget Cartan's definition of "2π" as the unique positive real number such that the kernel of the homomorphism "t -> e^it" from the additive group of reals to the multiplicative group of unit-length complexes is the set of integer multiples of 2π.
I saw, and experienced, this approach in school and see no value in it for improving your understanding complex numbers. The point of the approach is to teach students about abstraction and formalism, and complex numbers happen to be a convenient place to do it. But the formalism is a barrier for building up a more convenient mental model about what is really going on.
Before disputing this, in any calculation that you've ever done by hand with complex numbers, do you naturally write it as (a, b) or a + bi? I always do the latter, and it saves me both time and conceptual effort.
And a random note. If you go on past advanced Calculus, you'll encounter two subjects that take Calculus and go back to the basic foundations and build them up. The first is real analysis, for which you have to learn all of the ways that things fail to work out like you would want them to. The other is complex analysis, where you wind up learning all of the ways that everything has to work out amazingly perfectly.
The difference between the two subjects is that "differentiable" in the 2-dimensional structure of complex numbers is a far, far stronger condition than "differentiable" is for the real numbers. Indeed there actually exist functions that you can construct which are infinitely differentiable everywhere in the real numbers, but for which on no interval can you extend them to a function that is differentiable in the complex plane.
> But the formalism is a barrier for building up a more convenient mental model about what is really going on.
Actually, I'd say that complex numbers are a wonderful opportunity for illustrating that math is nothing but a mental model, not necessarily with anything "really going on", AND at the same time incredibly useful.
IMO the best way to think about complex numbers is to start with the observation that sqrt(-1) cannot exist in the realm of real numbers, quite easily provable. Then you make the bold-assed assumption that it exists anyway and simply call it i. Then play around with it a little, and find out that adding the axiom "there is something called i, where i * i == -1" does not lead to any contradictions; instead you can work with this i just fine, and numbers of the form a+bi are closed under all your everyday operations, just by applying your everyday arithmetic rules. Neat!
Then you notice that those numbers can be interpreted as 2D vectors and some operations on them as geometrical transformations thereof. Super Neat!
Then someone notices that those numbers and the operations on them can actually be used to model certain aspects of the physical world. Holy Crap!
First, I find that having more different ways to look at a problem, the better I am able to deal with it. I can look at an equation algebraically, or as a graph, e.g. I can use rectangular coordinates or polar coordinates. I can look at complex numbers as abstract entities or as points in a plane.
Second, if you look at the history of complex numbers, mathematicians were just not sure what to make of them, and had no way to have confidence that what they were doing was even consistent. Being able to interpret them as point in a plane with intuitive geometric operations gave them a huge boost.
Third, thinking of them this way led to the search for generalizations. Gauss and Hamilton tried to find a way to do arithmetic in three dimensions, or prove that it couldn't be done. Hamilton eventually found the four-dimensional quaternions. And the (ac-bd,bc+ad) definition was generalized to the Cayley-Dickson construction.
And not long afterwards, Clifford generalized real numbers, complex numbers, and quaternions into what are now known as "Clifford algebras". Handy stuff. Certain algebras allow you to express geometric shapes such as points and lines using very simple equations. Other algebras show how quaternions (for example) arise naturally as the even subalgebra of Cl0,3(R). The "spacetime algebra" appears as CL1,3(R), which makes it easy to express special relativity.
"Young man, in mathematics you don't understand things. You just get used to them."
-- John von Neumann
For what it's worth, I have sometimes manipulated complex numbers as (a, b), usually when using them as 2-D vectors. But I make no claims to being a mathematician.
That's not unique to complex analysis; there are smooth real functions that are nowhere real analytic. It's far easier to come up with (and understand!) examples of functions that are smooth when considered as real maps from R^2 to itself, but nowhere homomorphic when considered as complex functions of a single complex variable, though. Complex conjugation is an obvious example: as the real map (x,y) -> (x,-y), it's linear, so smooth, but only holomorphic at points where 1 = -1, so nowhere. Complex conjugation restricted to the real line does have an obvious holomorphic extension, of course!
That's not unique to complex analysis; there are smooth real functions that are nowhere real analytic.
Did you accidentally get my intended point backwards? My point is that real analysis admits all sorts of fine gradations of counter-examples, and complex analysis does not. In this case I was indeed thinking of a smooth real function that is nowhere analytic. (For non-mathematicians, that would mean a function that can be differentiated any number of times at any point, but which cannot be written as a power series.)
Contrast with complex analysis where continuous at a point and differentiable in a neighborhood of that point implies analytic. Even something as ill-behaved as the point at 0 of the absolute value function is not possible in complex analysis.
I was going to post something like this but you did a much better job of it.
In some sense, TLDR: "Imaginary number" is a misnomer that confuses people. 1+2i is also just a convenient but potentially confusing notation to those who don't know what it is a short-hand for.
Complex numbers are basically 2D vectors with a funny multiply operation ((a,b)(c,d) = (ac-bd,bc+ad)) and it just so happens that defining it this way leads to some interesting properties and convenient formalism when dealing with many problems.
EDIT: there is also a nice graphical way of interpreting what the operator does. It adds the angles of the original vectors (wrt x-axis) and multiplies their lengths. Better explained has a nice article on this: http://betterexplained.com/articles/intuitive-arithmetic-wit...
Complex numbers are not vectors. If you have (a + b) in an expression, you can do algebra with it. If you then say b=ci, (a + b) would be a complex number, but the work you just did as if it were 2 real numbers is still valid.
"A complex "number" (don't think of it as of a number! ..."
Excellent point! Another example of misleading mathematical terminology is "random variable", which are not random or variable but instead are well-defined mappings.
> Another example of misleading mathematical terminology is "random variable", which are not random or variable but instead are well-defined mappings.
Indeed. Kolmogorov's random variables are a special case of observables in mathematical physics.
In classical physics, the state space for a free particle is S = R^3 x R^3. As an observable we might take the particle's velocity in some particular direction, which defines a mapping f : S -> R. We want such mappings to respect the relevant structure of the state space. A classical state space usually has the structure of a smooth manifold, so the mappings should be smooth as well.
In classical stochastic physics, the states now have the structure of a probability space. We want the mappings to be measurable so we can take the preimage of a measurable set of observable values to calculate its probability. This is exactly the Kolmogorov definition of a random variable.
Quantum theory doesn't quite fit into the above scheme, but there are several ways these three cases can be unified. For example, from the viewpoint of C* algebras of observables, classical systems have commutative algebras and quantum systems have noncommutative algebras.
By the way, there is a connection to monads that I find enlightening. With monads, there is both an internal semantics and an external semantics. From the internal point of view, random variables are indeed random and variable. From the external point of view, they are neither. (This can be given a Bayesian spin by thinking of the internal and external observers as someone with respectively imperfect and perfect knowledge.) This is analogous to someone's perspective on the state monad from the internal and external points of view. From the viewpoint of someone who lives in the state monad, the same expression can return different values depending on context. But from an outsider's explicit state-passing point of view, everything is referentially transparent.
The way I think of "random variable" is that it's an attempt to move toward a stateless world (a la functional programming) so we can reason about it.
What we actually observe in real life is a sample from the corresponding distribution. The "random variable" concept (a function over a sample space with a probability measure) is the stateless analogue that is more usable in abstract reasoning. In probability, we don't care about the specific values the variable actually ends up producing, but all the values it can produce and how likely each is.
In the observed world, it's unknown (pre-measurement) and appears to be random. In the mathematical world, it's a function over a sample space.
It's a mechanism to reason about randomness by (though the magic of probability) taking the randomness and state out.
By the same token, you shouldn't think of a real "number" as a number, but as an infinite sequence of rational numbers. Yes, all of the reals are infinite sequences of rationals.
Also, a rational isn't a number but an ordered pair of integers. (Technically, the set of the rationals is the set of all ordered pairs of integers (excluding the ones with zero in the second position) with an equivalence class defined on them.)
The integers are also not numbers, but the naturals with a sign adjoined to them.
Finally, the naturals aren't numbers but sets of sets, with zero the empty set and all further integers built inductively from that.
So... what does it really gain you to look 'inside' the numbers you use?
One minor correction. The integers are more naturally defined as an equivalence class of pairs of natural numbers, with (a, b) equivalent to (c, d) if and only if a+d = b+c. (Your approach does not eliminate the equivalence class issue, because you have to specify that +0 and -0 are the same.)
In this formulation, you define (a, b) + (c, d) to be (a+c, b+d). And you define (a, b) * (c, d) to be (a * c + b * d, a * d + b * c). After a little work you can prove that both operations send equivalence classes to equivalence classes.
Then you can embed as follows. The natural n gets mapped to the integer (n, 0). The integer m gets mapped to the rational (m, 1). The rational r gets mapped to the real (r, r, r, r, ....). And the real x gets mapped onto the complex number x + 0i.
For those who find the above so-called Grothendieck construction somewhat puzzling, here's a little motivation and background. It is the group completion of a monoid. The simplest case is the group completion of the free monoid on one generator. The monoid has as elements all the finite strings (including the empty string) built from the symbol x. Its operation is concatenation. Any such string is uniquely defined by the number of x symbols that occur, so this monoid is isomorphic to the natural numbers.
Its group completion must add an inverse for all elements. We will denote the formal inverse of x by the symbol x' with the relations xx' = x'x = 1 where 1 denotes the empty string. This cancellative concatenation is commutative because xx' and x'x both equal 1. Thus a string can be identified (though not uniquely) by a pair of natural numbers (m,n) that count the number of x and x' occurrences.
To see the connection with the sign representation of integers, add an orientation to the relations xx' = x'x = 1 to get the length-shortening reduction rules xx' -> 1, x'x -> 1. It's then easy to see that every string has a unique normal form that is either empty or consists entirely of x symbols or entirely of x' symbols, corresponding to the cases 0, +n and -n, respectively. For if a string is not of this form it must have at least one x and one x' element. But then there must be at least one adjacent pair of x and x' elements. Hence the string admits a reduction and cannot be a normal form.
It should be noted for non-mathematical readers that a group has a single operation called multiplication, and the multiplicative identity is called 1.
The natural numbers form a group with "multiplication" being addition, and "1" being 0. In which case the Grothendieck construction on the natural numbers gives the integers with addition.
If calling the basic operation multiplication instead of addition is confusing, remember that one of the inspirations for group theory are permutations of a set of things, which can be represented by matrices using matrix multiplication to perform the permutations.
>For those who find the above so-called Grothendieck construction somewhat puzzling, here's a little motivation and background. It is the group completion of a monoid. The simplest case is the group completion of the free monoid on one generator
A number is whatever we want it to be, whatever it is useful to treat as a number.
That's it. That's the whole gag. Saying something isn't a number is pointless: It is if it can be, and it isn't if it's more interesting to treat it as something else.
This leads to something more fundamental:
Math is all about modeling. Math is a language for making models that are logically consistent.
Confusing a model with the thing being modeled is wrong. Saying complex numbers aren't numbers because they're rotations is going at it backwards: Complex numbers can be used to model rotations. That doesn't tie them to that one model. They can be used in other ways, too.
This is one of the two properties that makes the use of complex "numbers" fruitful. At the other one we arrive if we now look at "numbers" that are NOT of the form (a,0), for example at a curious property of (0,1):
So, in the domain of complex "numbers", the "number" that corresponds to the real number -1, happens to have the equivalent of what we for real numbers call the "square root". We just talk about the "square root", but it is a different operation when we are talking about complex numbers.Those two properties combined allowed mathematicians to tackle some problems that previously did not have a solution. One example is the problem of finding a solution to cubic equations. The math here gets more complicated, but basically it turns out that by writing cubic equations in real numbers in complex numbers instead, you can find general formulas in terms of complex numbers for finding all the possible solutions, and as we have discussed if applying such a formula in the end yields a number of the form (a,0), it is guaranteed to be a valid solution for the original real equation. Google for "cubic equations cardano" to see the details.
Now, this going back and forth between complex and real numbers is so useful, that for the purpose of brevity mathematicians sacrificed intelligibility and introduced sort of a shorthand notation of the form: a + bi, so instead of writing (0,1) as we did above, we just write i, instead of (5,0) we just write 5, and instead of (1,2) we write 1 + 2i. This is purely a trick, there is nothing magical about the "i", it is just a "dummy" variable that allows convenient carrying out of the operations with pairs described above in the manner reassembling ordinary high-school algebra we all know and love.
All this is maybe a bit elementary, but I think this is the part most people fail to understand and because of this start treating complex numbers as something mysterious. There is in fact nothing mysterious about them, you have to boil every application you see of them to the above and then you will get a clear understanding of what is happening and why they are useful. Points on the plane happen to be a model for complex numbers with rotation corresponding to multiplication and so forth, this is of course very interesting, but I feel an introduction to the topic should start with what I have just tried to explain.