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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 orde

by stiff 14y ago

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: (a,b)+(c,d) = (a+c,b+d) (a,b)-(c,d) = (a-c,b-d) (a,b)*(c,d) = (ac-bd,bc+ad) 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: (a,0)+(c,0) = (a+c,0) (a,0)-(c,0) = (a-c,0) (a,0)*(c,0) = (ac,0) 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.

Jun8 14y ago
"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.
- psykotic 14y ago
  > 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.
- michaelochurch 14y ago
  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 stiff 14y ago

Jun8 14y ago

"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.

psykotic 14y ago
> 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.
michaelochurch 14y ago
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.