What are imaginary numbers? • Hacker News Board

Re: What are imaginary numbers?

#131

Post by jasomill » Sat, Sep 22, 2012, 11:36 AM UTC

btilly wrote: post #65

stiff wrote: post #49

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)…

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 y…

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!

Re: What are imaginary numbers?

#132

Post by yogrish » Sat, Sep 22, 2012, 12:09 PM UTC

btilly wrote: post #76

diminish wrote: post #63

Earlier quoted context omitted.

or 0.008983291.. Are there countably infinite solutions or otherwise? See infinity discussion few days ago at HN http://news.ycombinator.com/item?id=4526049 PS: Just curious, i am not a mathematician.

It is countably infinite. Let's try to find them all. A relatively simple way to understand this is that i^i = e^(i * log(i)) for every possible log of i. So all we need to do is understand what values log(i) could have (there are actually many), and then we can work it out. But log(z) just undoes e^z, so we need to understand e^z. Now let's work backwards. If z = x + y i with x and y real, then x tells us the absolu…

simple explanation of PI: http://en.wikipedia.org/wiki/File:Pi-unrolled-720.gif

Re: What are imaginary numbers?

#133

Post by michaelochurch » Sat, Sep 22, 2012, 1:41 PM UTC

Jun8 wrote: post #52

stiff wrote: post #49

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)…

"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.

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.

Re: What are imaginary numbers?

#134

Post by btilly » Sat, Sep 22, 2012, 3:19 PM UTC

jasomill wrote: post #131

btilly wrote: post #65

Earlier quoted context omitted.

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 y…

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…

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.

Re: What are imaginary numbers?

#135

Post by gizmo686 » Sat, Sep 22, 2012, 4:36 PM UTC

ColinWright wrote: post #130

gizmo686 wrote: post #125

Earlier quoted context omitted.

There do exist number systems where .999.../=1, however, they are not a strict superset of the reals. If it were, then any operation involving only real numbers would behave identicly to the real number system. Also, this property is not a mere convention, but rather a nessasary result of what we want the number line to be. For example, assume that X<Y. Consider Z=(X+Y)/2. Z=X/2+Y/2. X<Z<Y. I have just shown, using b…

> There do exist number systems where .999.../=1, > however, they are not a strict superset of the reals. The hyper-reals of non-standard analysis are, in fact, a strict superset of the reals. > If it were, then any operation involving only real numbers > would behave identically to the real number system. Why is that a problem? Seems to me that that's desirable.

> The hyper-reals of non-standard analysis are, in fact, a strict superset of the reals.

And .9999==1 in the hyper-reals

>>If it were, then any operation involving only real numbers >>would behave identically to the real number system. >Why is that a problem? Seems to me that that's desirable.

Consider the expression ".999... - 1" in the real number system. As has been established earlier, this simplifies to 0.

Consider the same expression ".999... - 1" in the hyper-real number system. As all the numbers are real, this expression simplifies just as it would in the real number system. Meaning that ".999... - 1 =0" in the hyper-real system. Adding 1 to both sides ".999... = 1" in the hyper-real system.

One could also argue that because the statement ".999... = 1" is true in the real number system, and the hyper-reals are a superset of the reals. Then ".999... = 1" is true in the hyper-reals.

Re: What are imaginary numbers?

#136

Post by pm90 » Sat, Sep 22, 2012, 4:40 PM UTC

lmkg wrote: post #83

stiff wrote: post #49

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)…

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…

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)

Re: What are imaginary numbers?

#137

Post by Someone » Sat, Sep 22, 2012, 5:07 PM UTC

Evbn wrote: post #109

alok-g wrote: post #77

Earlier quoted context omitted.

Do you know of something that I can read further on this? I wonder about the impact this may have on mathematics. I did not know of this property of real numbers of nonzero infinitesimals. (Possibly there is where my confusion is starting from.)

Look up nonstandard analysis. It is a formalism that can define calculus, but doesn't have any practical difference on physical computations on real world entities.

For a more playful and, some would say, practical view, read "Winning Ways for Your Mathematical Plays" (http://en.wikipedia.org/wiki/Winning_Ways_for_your_Mathemati...)

Re: What are imaginary numbers?

#138

Post by ColinWright » Sat, Sep 22, 2012, 5:40 PM UTC

gizmo686 wrote: post #135

ColinWright wrote: post #130

Earlier quoted context omitted.

> There do exist number systems where .999.../=1, > however, they are not a strict superset of the reals. The hyper-reals of non-standard analysis are, in fact, a strict superset of the reals. > If it were, then any operation involving only real numbers > would behave identically to the real number system. Why is that a problem? Seems to me that that's desirable.

> The hyper-reals of non-standard analysis are, in fact, a strict superset of the reals. And .9999==1 in the hyper-reals >>If it were, then any operation involving only real numbers >>would behave identically to the real number system. >Why is that a problem? Seems to me that that's desirable. Consider the expression ".999... - 1" in the real number system. As has been established earlier, this simplifies to 0. Consi…

Ah, I see your point. I was trying to address the more general "intuition" that people have about this, that 0.9999... falls short of 1.0, and that there's some sort of infinitesimal between them. You can use that to talk about the hyper-reals, but, as you say, it still doesn't "solve" the "problem" that people perceive.

So you're right, and I was answering a different (although related) question.

Re: What are imaginary numbers?

#139

Post by crntaylor » Sat, Sep 22, 2012, 6:44 PM UTC

btilly wrote: post #110

psykotic wrote: post #107

Earlier quoted context omitted.

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…

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,…

The naturals form a monoid, not a group. Or am I completely misinterpreting you?

Re: What are imaginary numbers?

#140

Post by btilly » Sat, Sep 22, 2012, 10:33 PM UTC

crntaylor wrote: post #139

btilly wrote: post #110

Earlier quoted context omitted.

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,…

The naturals form a monoid, not a group. Or am I completely misinterpreting you?

Oops, I meant integers form a group. Minor mistake.