Some remarks on complex numbers

[ClamShell]

They each have modulus \sqrt{2} so are "equivalent" using the equivalence relation "have the same modulus". Since |\sqrt{2}|= \sqrt{2}, those numbers are all "equivalent" to \sqrt{2}. (The "equivalence classes" are circles centered on 0.)

Where do you want me to put the dag-nab bars? @Integral

 

[Integral]

So are we all in agreement that,

$|(1 + i)| = √2 = |(1 - i)|$ ?

You are talking about the magnitude or modulus of the imaginary number so the imaginary numbers need the modulus operator.

Please get a copy of a Complex Analysis text. This a huge rich field there is much to know and learn. It is best done from a good text. I like Complex Variables and Apps by Churchill and Brown

 

[ClamShell]

You are talking about the magnitude or modulus of the imaginary number so the imaginary numbers need the modulus operator.

Please get a copy of a Complex Analysis text. This a huge rich field there is much to know and learn. It is best done from a good text. I like Complex Variables and Apps by Churchill and Brown

I've have "INTRODUCTION TO THE GEOMETRY OF COMPLEX NUMBERS"
by ROLAND DEAUX collecting dust.

I'm beginning to see the error of my ways...Thanks...no sarcasm intended.

 

[LCKurtz]

We know that "1" and "i" are perpendicular and form a right triangle whose
hypotenuse is the sqrt(2), nes pa?

I think you mean "n'est ce pas". It's a French expression which could be translated as "isn't that so".

 

[ClamShell]

I think you mean "n'est ce pas". It's a French expression which could be translated as "isn't that so".

Moocho grassyass, I hope you don't want me to enclose it in modulus bars too. :-)

 

[lendav_rott]

Are you doing this on purpose? :D

 

[ClamShell]

Are you doing this on purpose? :D

No, I spells 'em like I hears 'em.

 

[ClamShell]

While those interested parties wait for the blackout of the "Reality Bit" article
in the Jan, 25-31, 2014 issue of New Scientist...here are some teasers:


http://www.newscientist.com/article...o-u-searching-for-the-quantum-master-bit.html

http://arxiv.org/abs/1210.4535

http://en.wikipedia.org/wiki/U-bit

 

[ClamShell]

BTW, it is said that the Pythagoreans were so horrified to find that SQRT(2) was irrational, they swore themselves to secrecy about this unsettling knowledge. If they could even contemplate imaginary numbers, their heads would probably have exploded.

Rumor has it that Hippasus, a student of Pythagoras, was murdered by the Pythagoreans,
by heaving Hippasus into the Mediterranean with a stone collar, for making the conjecture
that sqrt(2) is irrational. Guess they thought Hippasus couldn't keep his mouth shut.

Closer to the truth is probably that Hippasus got fired and couldn't find a job other than
tending a flock of sheep. And the square root of a flock of sheep is merely a leg of lamb.

 

[ClamShell]

As an EE I use complex numbers all the time. If mathematicians completely drop the subject engineers will keep using (and teaching) them for a long time.

jason

Complex numbers aren't really needed to analyze AC circuits. If you want to
see for yourself, it's only a more "compact" way of viewing the problem.
Actually looks like phase between voltage and current is more easily handled
using complex arithmetic.

See,

http://www.animations.physics.unsw.edu.au//jw/AC.html

if you are interested in the "real" method to analyze AC circuits.

Looks like Dr. Bill Wootten, is trying to do the same with Quantum
Mechanics.

EDIT: And concerning General Relativity...I always silently cringe
when Dr. Hawking refers to "imaginary time" as if "ict", the "imaginary"
space distance, means time can be "imaginary". Time is the real part of
the imaginary part, it's not the imaginary part, nes pa? I mean n'est ce pas?

In short, I am a true believer that "i" is a purely notational construct and
does not have actual physical significance other than to organize equations
into more compact form. And that the absolute operator is an example
of how to recover what the real method already knows. This idea is not
like trying to prove 1 = 0, it's just a total rejection of mystical superstition
that so many fall prey to. Like Pythagoras's fear of irrationals.

 

[lurflurf]

I am a true believer that "17" is a purely notational construct and
does not have actual physical significance other than to organize equations
into more compact form.
What is it with all this talk of physical significance?
Complex numbers are used to make things easier. If someone wants to avoid complex numbers they can. Often it is a bit silly because the "real" method is equivalent as when we introduce a matrix equation like
$$\left[ \begin{array}{ccc}
0 & -1 \\
1 & 0 \end{array} \right]^2=-\left[ \begin{array}{ccc}
1 & 0 \\
0 & 1 \end{array} \right]$$
to avoid using i.

 

[smize]

The complex numbers are a very important aspect of mathematics. They are utilized often in Analysis (obviously), Mathematical Physics, Algebra, and Number Theory (I am not certain about Geometry/Topology).

There was a problem that was solved in the 19th century: Can one construct a square with the same area as a given circle with a straight edge and compass.

The answer is no. And it deals with the fact that ∏ is transcendental. To prove this, they had to use numbers in the complex plane.

Also, if you know of the beloved i, j, k notation of vectors...they were originally defined as:
i², j². k² = -1. And another notation is gone.

Complex numbers are wonderful. There are cases where I'd rather work with complex number that reals, since it makes things simpler (by making it complex). It has its uses.

 

[ClamShell]

I am a true believer that "17" is a purely notational construct and
does not have actual physical significance other than to organize equations
into more compact form.
What is it with all this talk of physical significance?
Complex numbers are used to make things easier. If someone wants to avoid complex numbers they can. Often it is a bit silly because the "real" method is equivalent as when we introduce a matrix equation like
$$\left[ \begin{array}{ccc}
0 & -1 \\
1 & 0 \end{array} \right]^2=-\left[ \begin{array}{ccc}
1 & 0 \\
0 & 1 \end{array} \right]$$
to avoid using i.

YES, I agree 100%. That's what Dr. Wootten is trying to do with
the standard model, he wants to agree with the standard model.
He wants to reformulate the standard model and at the same time
"avoid" imaginary numbers. But I suspect that he is having a bit
of trouble because so many theorems are formulated in complex
form.

It is my conjecture, that real number reformulations have a much
better chance of revealing the physics that is going on and the
complex number formulations have a better chance of "clouding" the
physics that is going on. I also have no doubt that real number
formulations can be much "uglier".

Complex numbers and their properties ARE wonderful, especially
since some "avoidance" techniques could make proofs millions of
lines long and be too much for even modern computers to handle.

 

[jasonRF]

Complex numbers aren't really needed to analyze AC circuits. If you want to
see for yourself, it's only a more "compact" way of viewing the problem.
Actually looks like phase between voltage and current is more easily handled
using complex arithmetic.

See,

http://www.animations.physics.unsw.edu.au//jw/AC.html

if you are interested in the "real" method to analyze AC circuits.

Yes, for very simple circuits you can do this the hard way and it isn't such a big deal. Now do that style of analysis with dozens of circuit elements. Not so much fun.

For my work it is the complex representation of baseband signals in communications systems that is most useful (I never do circuits). Here we end up estimating correlation matrices (Hermitian), inverting them, and doing all sorts of computation where using complex representation saves work. Sure, you could do this the hard way too ...

But in the end, engineers have been using complex numbers for many decades. If we stop using and teaching it, do we throw away 60+ years of literature and just start from scratch? Do we have our research engineers spend a bunch of years simply translating the complex results into the "new" method instead of doing actual new research? Do we graduate power system engineers that do not even understand the documentation that goes along with our power grid, or communications engineers that do not understand how modern systems internally represent signals?

C
In short, I am a true believer that "i" is a purely notational construct and
does not have actual physical significance other than to organize equations
into more compact form. And that the absolute operator is an example
of how to recover what the real method already knows. This idea is not
like trying to prove 1 = 0, it's just a total rejection of mystical superstition
that so many fall prey to. Like Pythagoras's fear of irrationals.

Agreed. But why tie one hand behind my back just because it is possible to live my life with one hand?

Do I think that this Ubit research is worthwhile - yes! But attempting to purge complex numbers (and by extension complex analysis) from our bag of tools seems silly to me. That is all I am saying.

Thanks for the links, though. Interesting stuff!

jason

 

[ClamShell]

For my work it is the complex representation of baseband signals in communications systems that is most useful (I never do circuits). Here we end up estimating correlation matrices (Hermitian), inverting them, and doing all sorts of computation where using complex representation saves work. Sure, you could do this the hard way too ...

Ahhh, Communications Theory, love that stuff; especially that the entropy(information)
packets come in -Probability *Log(Probability) and it becomes Information Theory.

 

[ClamShell]

The complex numbers are a very important aspect of mathematics. They are utilized often in Analysis (obviously), Mathematical Physics, Algebra, and Number Theory (I am not certain about Geometry/Topology).

There was a problem that was solved in the 19th century: Can one construct a square with the same area as a given circle with a straight edge and compass.

The answer is no. And it deals with the fact that ∏ is transcendental. To prove this, they had to use numbers in the complex plane.

Also, if you know of the beloved i, j, k notation of vectors...they were originally defined as:
i², j². k² = -1. And another notation is gone.

Complex numbers are wonderful. There are cases where I'd rather work with complex number that reals, since it makes things simpler (by making it complex). It has its uses.

Agreed

 

[ClamShell]

It is my conjecture, that real number reformulations have a much
better chance of revealing the physics that is going on and the
complex number formulations have a better chance of "clouding" the
physics that is going on. I also have no doubt that real number
formulations can be much "uglier".

Pretty weak conjecture, if I do say myself.

Does no one have an objection to this?

 

[homeomorphic]

In short, I am a true believer that "i" is a purely notational construct and
does not have actual physical significance other than to organize equations
into more compact form.

I am a true believer that "17" is a purely notational construct and
does not have actual physical significance other than to organize equations
into more compact form.
What is it with all this talk of physical significance?

No, no, no, no...

Math is NOT just a bunch of equations. There are IDEAS. And complex numbers are involved in thinking about those ideas. They have geometrical and physical meaning. Does nature use them explicitly? Maybe not. But do we use them to think about what nature is doing? Hell, yes. And not just in equations. In ideas. Physical and geometric.

And yes, 17 has physical significance. If I told you to give me 17 apples, and you gave me 10, that would be wrong. Those are physical actions. It's true that 17 is just a symbol. But it does stand for an abstraction of stuff that is actually out there. We take all collections of 17 objects that we have ever seen and sort of put an equivalence relations on them. And at some point, it gets wishy-washy as to what constitutes an object or something, but we know what I am saying if you tell me to give me 17 apples, so yes, in my book, that's physical significance.

Another point of view is that 17 is the set containing the empty set, set containing the empty set, set containing that...

So, we can define 17 without reference to physical things, but when we talk about 17, I don't think we really have that in mind. We have in mind different things, depending on the context. Sometimes discrete sets of objects like the apples, other times maybe 17 continuous units of something, like time. Sometimes, maybe the set theory definition. It really means something different depending on context. Different, but closely related meanings.

 

[ClamShell]

No, no, no, no...

Math is NOT just a bunch of equations. There are IDEAS. And complex numbers are involved in thinking about those ideas. They have geometrical and physical meaning. Does nature use them explicitly? Maybe not. But do we use them to think about what nature is doing? Hell, yes. And not just in equations. In ideas. Physical and geometric.

And yes, 17 has physical significance. If I told you to give me 17 apples, and you gave me 10, that would be wrong. Those are physical actions. It's true that 17 is just a symbol. But it does stand for an abstraction of stuff that is actually out there. We take all collections of 17 objects that we have ever seen and sort of put an equivalence relations on them. And at some point, it gets wishy-washy as to what constitutes an object or something, but we know what I am saying if you tell me to give me 17 apples, so yes, in my book, that's physical significance.

Another point of view is that 17 is the set containing the empty set, set containing the empty set, set containing that...

So, we can define 17 without reference to physical things, but when we talk about 17, I don't think we really have that in mind. We have in mind different things, depending on the context. Sometimes discrete sets of objects like the apples, other times maybe 17 continuous units of something, like time. Sometimes, maybe the set theory definition. It really means something different depending on context. Different, but closely related meanings.

Thankyou Homeomorphic, I was trying to think of a way to defend the significance of "17",
you put it into a nutshell...thanks again.

 

[ClamShell]

@Homeomorph, now you need to explain the significance of 42.

 

[DrewD]

Pretty weak conjecture, if I do say myself.

Does no one have an objection to this?

Yes. From a mathematical standpoint, it is meaningless to reformulate QM using real matrices since there is an isomorphism between the two representations.

From a computational standpoint, I wouldn't be surprised if Matlab already does complex computations using matrices (I don't know this, but I know that a professor of mine wrote some code that always used the matrix representation because it is easier to deal with on a computer)

Physically, things will be the same. Maybe some problem will be solved by accident while trying to reformulate QM.


Also, 17 is a grouping of pixels that we use to communicate the idea of a particular quantity. If I rewrite 17 as !&, math will not change. Similarly, if I rewrite $i\equiv\left[\begin{array}{cc} 0&1\\-1&0\end{array}\right]$, math will not change. The numbers we write exist only for equations (and theorems etc.). The algebraic relationship between elements of the set $\mathbb{C}$ exist with or without the notation, but $a+bi$ is only one of infinitely many ways to write those relationships, so why should it be especially meaningful? I side with lurflurf. At least it is a legitimate philosophical position even if you don't personally agree with it.

 

[ClamShell]

@DrewD. 17 balls in a hat; 10 white and 7 black. Looks like
probability of an observation has something to do with
"significance" too. We will never pick a sqrt(-1) ball from the
hat, even if someone says some of the balls are positive
sqrt(-1) balls. That's all I've got, maybe math was invented
and not discovered. I have often considered that math is entirely
notational. If you've got 17 pigs and I have 17 sheep, then maybe
both of us would be better off with 17 pigs and sheep each.

 

[homeomorphic]

@Homeomorph, now you need to explain the significance of 42.

Well, you can extrapolate to any number by counting. We have a system for generating numbers. If I say, give me 43 millions apples, we know what that means. You just count it out until you get to 43 million. That's all that is really needed. We have a way of referring to something physical that is out there. The only condition is that you have to be able to interpret what I am saying as more than just a string of letters or sounds.

The numbers we write exist only for equations (and theorems etc.)

It gets a little better if you add theorems, etc, but obviously there are real world uses for numbers, like money, counting things, etc. Not restricted to math-land.

If I rewrite 17 as !&, math will not change.

I don't think you want to just talk about the symbols, though. There are concepts behind the symbols. The notation isn't important, but it isn't a bunch of meaningless symbols. Meaningless symbols are for computers, not human beings. Human beings think in pictures or concepts, and we can manipulate symbols as well, but that's not the whole picture. For me, i is simply the point (0,1) on the complex plane and it acts on the plane by rotating it 90 degrees. This is an equivalent way to think about it to the algebra and is not in terms of equations. You can use this in any way you like. You can use it to think about actual planes and rotations, you can use it to think about wave functions, or you can use it as a visual representation of things that you could calculate with the arithmetic of complex numbers, using the formal rules. These points of view are all different sides of the same thing.

It doesn't matter if we use i or if we use the matrix, but that has nothing to do with whether they have physical significance. They both have the same physical significance. So, yes, it is superfluous to chose one or the other, but that is because they share the same physical significance, not because they have no physical significance. Complex numbers correspond to points on a plane. So do matrices for rotation and dilation. When you interpret them this way, that gives them physical significance. They can be used to symbolize something out there. If I give you a point on a plane, if we set up the right coordinate system, it could be used to symbolize an actual point out there. Like a point on the chalkboard, just as natural numbers can refer to a physical collection of apples. We could change the symbols around, but that misses the point.

Denying any physical or geometric significance would just be tying your hands behind your back, psychologically speaking. Even great mathematicians like Euler made retarded mistakes with complex numbers because they didn't have a geometric representation to help them to spot the errors. Once people started thinking about them as points on a plane, magically everything started making sense, and a lot of nonsense was finally put to rest. Let's not go back to the 1700s level of understanding of complex numbers. Once the complex plane was revealed, it was really sort of an embarrassment to the previous couple hundreds years of mathematicians how simple it actually was. I mean, what a cheap, stupid, dead obvious little trick. Points on a plane. But it made all the difference and suddenly there was huge progress in understanding complex numbers.

 

[homeomorphic]

@DrewD. 17 balls in a hat; 10 white and 7 black. Looks like
probability of an observation has something to do with
"significance" too. We will never pick a sqrt(-1) ball from the
hat, even if someone says some of the balls are positive
sqrt(-1) balls. That's all I've got, maybe math was invented
and not discovered. I have often considered that math is entirely
notational. If you've got 17 pigs and I have 17 sheep, then maybe
both of us would be better off with 17 pigs and sheep each.

Well, the problem here is not that i doesn't have a physical interpretation. It's just that it doesn't have one that extends the representation given to natural numbers by counting, unless perhaps i means 1 unit up, 2i means 2 units up, etc. So, it's sort of a 2-dimensional quantity, whereas the usual counting we 2 is only applicable to 1-dimensional quantities.

Maybe you can think of it this way. You can GIVE it a physical interpretation. It's not a God-given one. You can just use it to refer to stuff. It's conventional now to identify complex numbers with points on a plane. So, that's a standard interpretation that any mathematician will know.

 

[oneamp]

The number i is a gadget that represents a counterclockwise quarter turn of the plane.

Why don't we have gadgets to represent turns of the plane in different dimensions? If i moves it counterclockwise as the viewer sees it, where's the 'imaginary' number to tilt it anterior?

 

[AlephZero]

Why don't we have gadgets to represent turns of the plane in different dimensions?

'And Hamilton said, "Let there be quaternions". And Hamilton saw the quaternions, and behold, they were very good...'

Sorry, I got carried away by homeomorphic's "not-god-given" interpretations.

 

[ClamShell]

Notations seem to possesses qualities of "protocol".
Real numbers seem to possesses some "material", "down-home" quality.

 

[homeomorphic]

Why don't we have gadgets to represent turns of the plane in different dimensions? If i moves it counterclockwise as the viewer sees it, where's the 'imaginary' number to tilt it anterior?

We do. They are called matrices (specifically, orthogonal matrices, those that preserve distances, if you just want rotations with no distortion). As we have mentioned, you can do everything i does by using a matrix.

If you wanted something a little bit more like complex numbers, that gets a little trickier. In general, what you would get in higher dimensions are called Clifford algebras, which also include complex numbers and quaternions as special cases. These are combinations of different square roots of -1 (in the most basic version--there are more general formulations). However, Clifford algebras have a lot of quirks that make them not quite like complex numbers. For one thing, they are not division algebras, so things don't always have inverses anymore. So you can't divide by guys in the Clifford algebra, like you can with complex numbers.

Also, the geometric interpretation of how they operate isn't so straight forward in general. You can still use them to describe rotations in higher dimensions, though. The square roots of -1 in the Clifford algebra act by reflection across a coordinate hyper-plane--a funny "spinorial" reflection, in which you have to reflect 4 times to get back to where you started because although the square is -1, which acts on the space by doing nothing, the Clifford algebra secretly knows that something is different until you square it once more and get back to 1. It's a bit of a long story.

Another thing about Clifford algebras is that, while you could think of them as a higher dimensional space because they are just vector spaces with some kind of extra multiplication, it's not as natural to think of them that way as it is with complex numbers and the 2-dimensional plane. You have to throw in all the products of the square roots of -1 which boost the dimension up way higher than the space they act on (2^n for n-dimensional space), and algebraically, it's a lot more messy. Very different from the complex plane which acts on itself.

 

[ClamShell]

Notations seem to possesses qualities of "protocol".
Real numbers seem to possesses some "material", "down-home" quality.

Yikes, this is probably close to what Pythagoras thought about rationals.
Par day m'wah.

 

[ClamShell]

OK, let me collect my thoughts again..."Where did I leave them, in the refrigerator again?"

This relates back to my first post, "sqrt(-1) enters the picture when we attempt to factor a sum."

It would seem to me that trying to get rid of "i", after it appears, such as by "substituting" a
two by two matrix for "i", is absolutely TOO trivial a thing to do to purge "i" from the analysis.
The two by two matrix is just a "wolf in sheep's clothing"; an isomorphism as revealed
above.

We've got to stop its introduction BEFORE it appears, so let's start thinking of removing the
factoring of sums(polynomials) from the analysis. I know...then how will we find out the
zero crossings and those precious eigenvalues. I'm thinkin' that Dr. Wootters isn't merely
performing an isomorphism on QM, but it is more like a reformulation avoiding "i" by not
ever seeing it in the first place. This could be done by avoiding factoring polynomials, but
rightfully so...I could not really understand Dr. Wooters' lecture.

So how about avoiding the factoring of polynomials as a way to avoid "i"?
Maybe subtractions factoring into a conjugate real pair would be OK,
but real conjugate pairs might need to be avoided too, for consistency.

Could analysis even be performed without this factoring?

Could reality not even know how to factor, and a model that does factor, is
expecting too much from Mother Nature or some other deity?

Is the real meaning of "i", simply that "No FISHING IS ALLOWED"...
I mean "no factoring is allowed"; dag-nab keyboard.