Complex Numbers
Seairth
Posts: 2,474
(for Heater)
https://greatscottgadgets.com/sdr/
At the URL above, you will find a set of videos related to SDR. Of course, SDR uses DSP. So these tutorials cover some of the topics related to DSP. In particular, there is one related to Complex Numbers, which several people here might find enlightening.
Edit: yes, I should have said "Complex Numbers". Corrected.
https://greatscottgadgets.com/sdr/
At the URL above, you will find a set of videos related to SDR. Of course, SDR uses DSP. So these tutorials cover some of the topics related to DSP. In particular, there is one related to Complex Numbers, which several people here might find enlightening.
Edit: yes, I should have said "Complex Numbers". Corrected.
Comments
Real numbers are graphed on a line, complex numbers are graphed on a plane due to that imaginary component.
It is all video presentation. I guess I am a bit 'old school' in that I'd rather read a text form of presentation. But it certainly does apply to radio.
Where the "imaginary" is component is i * a.
Where the "i" is the square root of -1 and a is real.
Which all sounds very weird and complex. Especially since there is no such thing as the square root of -1
On the other hand many mathematicians don't much like the nomenclature "complex" and "imaginary". Arguably a complex number is a simpler concept than other numbers, like zero or infinity or PI or the square root of two etc.
Complex numbers can be expressed as a simple number pair [a, b] with some simple rules for doing arithmetic with them. Adding, multiplying etc. Which is altogether a lot less intimidating.
For most people the visualization of complex numbers on a plane helps a lot. Eulers formula takes on a meaning when you see it
What exactly is the range of i -- from 1 to -1? Nope i is just -1
The notation is [a +bi]
I'll let Wikipedia do a nicer job of explaining. https://en.wikipedia.org/wiki/Complex_number
"Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part."
1 * 1 = 1
and
-1 * -1 = 1
Unless you consider negative numbers as being some how not real, but that's a whole other debate.
I did not understand the question about "range of i"
What range of numbers does B represent? - any Real number
What range of numbers does i represent? => ???? (a) none, (b) -1, (c) 1 to -1, (d) the negative square root of 1
Real numbers can be negative.
I will give you a hint, i represents the imaginary unit while B represents the imaginary part of a complex number.
"A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying i2 = −1."
There are no imaginary numbers, right? The number set is properly referred to a complex numbers. A so-called imaginary number alone is a worthless item.
There is only one "i". Not a range of values of anything. It is defined such that i * i = -1. Which, traditionally, has no solution. But we call the solution "i" and continue to use it it our reckoning.
Not sure what "B" has got to do with it. Is your "B" a real number, and integer, a whole number, a vector, matrix or quaternion? Or perhaps some value type I have never heard of.
?
Oops, corrected.
Real numbers: minus infinity to plus infinity, including all the integers, and fractions and transcendentals you meet on the way.
Complex numbers: Of the form a + ib where a and b are real and i*i = -1
Imaginary numbers: Of the form ia where a is real and i as above.
As we see imaginary numbers are a perfectly normal set. A subset of the complex numbers.
There is never an "imaginary number on its own", bi = 0 + bi. Which may be a valid solution to some polynomial or some such.
[a, b] + [c, d] = [a + c, b + d]
[a, b] * [c, d] = [ac - bd, bc + cd]
Odd rules perhaps, but that is what it is, that is how you might program a computer to do it, no weird stuff in them after all a computer cannot deal with "imaginary" things. Rather like the rules we have for doing arithmetic with vectors.
Now, you can work with this. It might make things a bit cumbersome to write out. So eventually you give that up and adopt the "i" notation. That is nice because it means you can just use the normal rules for arithmetic when working with it.
Is the "i" thing any more real or imaginary as the "e9" in the floating point number representation "4.9e9".
Potatohead is correct, it can't be on it's own. The complex number requires a pair of values to create the number. But an awful lot of people get lost in missing that all important information. (A+Bi) is 'the complex number, and that is just its notational format.
Heater seems to have done what I did for many years, and just get caught up with i, which is just a symbolic detail of the concept. It is a trap that many people fall into.
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I think the important thing here that is often overlooked is that complex numbers require a graphic representation as a plane, where as real numbers graphic representation is a line. Accepting that fact begins to may it easier to see what complex number are doing and how they might apply to things like electronic filters or FFT.
So it seem easy for us to concieve of the 'set of real numbers'. It is part of our everyday world and how we keep track of money. It has a nice tidy order of increasing and decreasing values. It has a nice range from infinity to negative infinity.
But the complex number sprawls out onto a plane. Order is not so clear, maybe not important. And the added dimensions of what might be called an axis of positive imaginary infinity to negative imaginary infinity begins to get me dizzy.
Here is the Wikipedia entry on Complex Numbers and Order
"Ordering
Because complex numbers are naturally thought of as existing on a two-dimensional plane, there is no natural linear ordering on the set of complex numbers.
There is no linear ordering on the complex numbers that is compatible with addition and multiplication. Formally, we say that the complex numbers cannot have the structure of an ordered field. This is because any square in an ordered field is at least 0, but i2 = −1."
Understanding that lack of natural linear ordering really may help to get past confusion.
So it all comes down to what are the applications and what does the application need to represent as a plane. And why is the plane representation with the additional detail useful. In other words, what makes that 'Bi' component necessary and useful.
Forget espousing the concept of 'imaginary' for now, and try to grasp the utility of the whole concept of 'complex numbers'.
Here is a link to an explaination of "Common Number Sets" that claims that Complex Numbers are one set and Imaginary Numbers are another set.
My gut feeling is that the so-called Imaginary Numbers are of no use outside the Complex Number set, so I tend to think they really don't exist alone in any useful form.
The questions of about the range of i were just to get your attention to the problem and open your mind to where the real useful maths are.
http://www.mathsisfun.com/sets/number-types.html
BTW, imaginary numbers don't always use i to indicate them, it is some times handy to use j to indicate an imaginary number. (I still prefer to call them the imaginary component of a complex number and avoid the deceptive term of 'imaginary number'.
It shows nested sets of numbers:
Naturals inside Integers inside Rationals inside Reals inside Complex.
So far so good but the imaginaries are in a set of their own outside of the R, Q, Z, N "nest".
Is that to imply that there are not R,Q,Z and N versions of complex numbers?
Like 3.14159i or 2i/3
That cannot be. Perhaps, maybe. On the other hand Real Numbers don't exist outside the Complex Number set either. As the mathisfun page shows, Reals are a subset of Complex
If you are into electromagnetism or quantum mechanics or the humble FFT and so on then Real numbers have no use on their own either!
Or is it that all these things do exist, once we have the concept of them in our minds. They are but concepts after all.
This existential debate is perhaps a waste of time. I mean, do integers exist? That is "exist" in the same way as this laptop I'm typing on or that chair over their.
Once you have run these thoughts around your mind a few times you come to the conclusion that imaginary numbers are no more imaginary than any other numbers. They are all just concepts in human minds.
Which is why many mathematicians think they should not be called "imaginary" numbers. They are no more or less imaginary than any other.
We do have a representation of any real number in natural logs where e is taken to the ix power.
But that same e taken to ix power is equal to (cosx + isinx) which looks like a complex number to me.
I need to thing a lot more about where this all comes together.
Please just consider my opinion was merely a way of getting readers to become more aware of the importance of complex number and that the imaginary unit really is an accessory that rotates values (-1, -i, 1, i) that are squared, cube, and so on in a way that allows mathmatics to represent the square roots of negative real numbers.
https://en.wikipedia.org/wiki/Imaginary_number
What I feel is really important to remember is that imaginary numbers are paired with real number and exist on a plane. That is the big departure from all the other forms of numbers. You have to visualize your answers on a plane, there is no ordering in the everyday sense of real numbers alone.
The electronics application is usually in AC and the first part of a complex number represents Amplitude (Voltage), while the so-called 'imaginary number' part represents the phase shift (Current).
Complex numbers are also important in filters.
It seems that the cyclical nature of AC, or radio frequency is where complex numbers really help resolve things.
In all honesty, I am reading and learning more than I understood before. I just started out as a devil's advocate for the idea that there are no imaginary numbers, but math called the Bi exactly that in the complex number form A +Bi. A is the real number component.
Heater annoys me, so I annoy him back. I think his cleverness sometimes muddles up being concise about important ideas.
In rotation mathematics, imaginary numbers just might be used alone in sets of three when using quaternion. So I guess my assertion that there is no such thing is a lost cause. (I previously thought quaternions used complex numbers. But now I see that it not be so. Quaternons 'extend' the use complex numbers by using triplets of imaginary numbers.)
https://en.wikipedia.org/wiki/Quaternion
What is both cool and curious here is that
A. Real numbers are represented on a line
B. Complex numbers are represented on a plane
C. Quaternion numbers are represented on a sphere.
I consider these infinite subsets. We use a number to represent the set to facilitate computation.
Yes! However, the dilemma is we cannot point to the real world entity the "imaginary" numbers represent easily. That's really the "imaginary" part, not so much the number. Those are just representation entities like all of the math entities we employ everyday.
And frankly, all of that boils down to axioms. Long ago, when I learned about these, I had a thought about how those impact a lot of things! The axioms we build on will have artifacts higher up in the computations. Pick a different set, or choose to represent them differently, and you get different artifacts.
And by artifact, I mean gaps in the representations. Things we have trouble with, given a particular representation structure and it's implications on computation.
It's a lot like base numbers and how that plays out for fractions.
The real problem here is that some people use 'imaginary numbers' interchangably with 'complex numbers' and other use it to mean the component of a complex number that has the imaginary unit. If you read carefully, the Wikipedia entry for Imaginary Numbers, that author seems to slip in to making them equal to Complex Numbers.
This is basically a semantic mess. And I feel that in teaching, it puts too much focus on the i and distracts from how the pair create planes to visualize in useful ways.
Take a look at a Mandelbrot set. It is an image of that Complex Number plane. But be careful, not all fractal use the complex plane for their creation.
http://www.fractal-explorer.com/mandelbrotset.html
A dilemma: how can we just scratch a surface, if we don't know, what a surface is?
There are idealized surface representations, and that's geometry.
The real world is just atoms and forces. A scratch is a change to the arrangement.
We have an expression (A + Bi) where A is the Real component that is well-known to all and Bi is the Imaginary component that learners tend to struggle with. All the philosophical discussion is a distraction toward what I am hoping we can achieve in the way of understanding. The human mind is always attempting to balance the real and the unreal. We even have grammar for both it English.
Real "If you are a man, I can do what you want."
Unreal "If you were a frog, you would have to eat flies everday."
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Consider this important difference. The 'imaginary unit' cycles though four different values. i,-1, -i, and 1. for a clockwise rotation (and can indicate both clock-wise or counter-clockwise motion if useful by reversing the sequence).
So at times it remains imaginary, and at other times it converts to a real component that can either retain the positive or negative value of B, and at other times change the positive or negative value of B.
Bi is useful because the imaginary unit creates a cyclical indication. It actually determines which of the four Cartesian quadrants the Complex Number should be in the Complex Plane.
So we see a tremendous utility of Complex Number notation in analysis of cyclical data, where a Real Number alone doesn't really achieve any clarity.
Take the Complex Number notation (cos x + isin x). The information is a vector that is a sum of the Real number component on the horizontal axis (that is what the cos indicates) and the Imaginary number component (that is what the sin indicates).
Take note of the confusion in notation....
The (cos x + isin x) is merely Trig notation on a Unit circle and it is NOT the general Complex Number polar notation. It just represented cycles where radius is always 1 ( known as the Unit Circle, right?) If you desired to have Polar notation, you would have add a radius - so it would look like this r(cos x + isinx). And there are at least three other Polar notation methods. That's another reason why students balk at trying to learn Complex Numbers. They get lost in all the ways they are presented (and there is a Matrix notation and a Natural log notation too.)
So we find Complex Numbers extremely useful in representing where in a Sine wave (or any other cyclical phenomina) we want to calculate. It is a very clean model for Impedances in AC electronics sources; where resistance only really works well with DC or an average voltage of AC (The RMS voltage and not the peak-to-peak voltage). It also works with discharge of capacitors, and charge of inductors to create a precise curve. (So it applies to just about everything electronic, and is not just useful for FFT calculations - which are more advanced, and learners may not be ready for.)
So I hope we can get back on track with discussion of the Applied Mathematics of a Complex Number or an Imaginary Number. There are lots of topics that it can apply to - once you see how that pesky unit i contributes to the model.
Again, in Quaternions, the Imaginary Unit is used for rotation through quadrants. But the model is simply inclusive of the x-axis, the y-axis, and the z-axis. To any Quaternion with represent an vector in 3-D, in contrast the Complex Number represention a rotational movement in 2-D. Thus we use this quarternion math which gyros and accelerometers to get a stable position in those little quadcopters we all like. It also helps to do a lot of fun computer animation.
When you understand how that pesky 'imaginary unit' i applies to cyclical phenomina, motion on circles, and motion on spheres; it just may get easier to see what the data is trying to tell you.
The jump from line to plane finally to sphere is incorrect. It should be 1-D to 2-D to 3-D.
I found a clearer explaination of quaternion notation in a few reference books I have. Rotation is a sub-set of that notation on a unit sphere; similar to cycles on a unit circe.
So the general version of a Quaternion is actually a Complex number with One real number components and three imaginary number components.
In other words a Quaternion is equal to ( a + bi + cj + dk).
So the dimensional relations go from point to circle to sphere for rotation, where a point seems trival.
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I really hope this is helpful to someone else. For me personally, it has brought a lot of different math topics into focus and shown why I was personally confused about Complex Numbers, but no longer am.
I have absolutely no idea what I personally can apply Quaternions to besides rotation in 3-dimensional space. But I am finding it easier to go back and reread the applications for Complex Numbers with the ability to understand the different uses.
There is a lot more interesting stuff that I am not sharing just because it is Mathematics History about both the Complex Number and Quaternions. Each has opened up new horizons in match, like Newton's Calculus.
It is nice to confirm that a guy going on 67 (birthday this month) can continue to learn new stuff. And maybe share it with others that are blocked.
What an excellent presentation of complex numbers in that greatscottgadgets video. I just now had time to watch it. Did anyone else?
He starts from the basics, even the existential debate we have been having here "the existence of negative numbers has not always been obvious people, mathematicians argued over the legitimacy of negative numbers for centuries." Sorry that annoyed you so much Loopy.
Interestingly he introduces the concept of "rotation" even with just addition and multiplication on the real number line. The idea that negatives introduce a rotation of half a turn is a little take on things I have never seen before.
I'm pleased to see he dislikes the nomenclature "imaginary" and "complex".
I love that he like to use tau.
But BAM! He flies into space, far away from explanation, when he gets to e and Euler's identity.
I'm totally with his suggestion: When ever you see a sinusoid think of it as a helix in three dimensional space. It always has a real and imaginary part. They are both a real as each other.
I'm amazed to find that in the notes on that web page he links to a presentation about all of this which does not use the imaginary number, i, at all. Or all that "e" stuff. Only the idea of using a number pair to represent these "complex" quantities and the special rules for doing arithmetic with them as I suggested above. By Professor Norman Wildberger http://web.maths.unsw.edu.au/~norman/Rational1.htm. Perhaps that is where I got the idea in the first place and is what inspires me to want to develop an explanation of the FFT without "i", "e" etc.
Thanks, Seairth, for the heads up on this video.
Oh my, that takes be off into the whole Golden Section world of spirals and icosohedrons, dodecahedrons, and so one. It ties into having a circle divided evenly by fifths or 72 degrees -- a whole different world from Cartesian x, y, and z axis. The best one can do is have six axises representing a sphere constructed of tetrahedrons.
I guess I'll just have to watch it and see what he says............
A Sinusoidals as a helix evokes thoughts of electromagnetism.
In any event, mathematics is the basis of all engineering, not just electronics... a a good deal of science. Once one gets that hang of advanced maths, one can jump around to many interesting fields of engineering and science to discover new applications.
But one does have to be careful with presentations from living mathematicans as they just might be trying to get across a 'pet theory or hypothesis' that is not widely understood or accepted. You could be chasing a dead-end.
I think the problem here is that I read to learn, but the new way to do so is supposedly to watch videos. These videos can take hours to watch and I go away feeling I have to watch it again and again. The media is slower to me.
And in this case, it is a whole course that covers the basics of Software Defined Radio usually taught over two days of classes with homework. That is really a lot of information to take in... much more than Complex and Imaginary numbers.
If you have difficulty with complex numbers, ask me and I will try to help. If you have trouble with the video, I will have to first watch it before I can help you with that.
I guess you comment about "careful with presentations from living mathematicians" is a reference to my mention of the Norman Wildberger video.
Norman is viewed by many in the maths community as being a bit a crank as far as I can tell. The reason for that is that he refuses to deal with those numbers on the number line that give us headache. Infinity is out for starters, if you can't get there and can't write it down, lets not even think about it. Which of course eliminates all the transcendental numbers, PI, e, square root of two and an infinity of others. And that necessarily removes all consideration of trig functions, sin, cos, etc.
Infinite sets are out. Instead he does everything with rational numbers. Those that can be expressed as p/q for finite integer values of p and q. Basically he is not happy with the foundations that infinite sets rest on. Or at least claims they are resting on concepts and proofs far in advance of what beginners to maths can see to be true for themselves.
He famously points out that in school we are presented with many things in maths and told "trust me it's correct, as you will learn as post graduate student". Then, when the post grad student asks about them he gets "It's obvious that was covered in high school"
Is what he presents wrong? Not as far as I have seen and obviously mathematicians the world over will have been eagerly trying to spot holes in his arguments, just for fun, like they do. It's a tough and competitive world in maths.
Are his suggested methods easier than hitting young minds with infinities? Sometimes as I listen I think he has a point.
As a simple example, why say that "1 / 0 = infinity" and drag in all the conceptual complication that brings with it? Why not just have "1 / 0" as a rational number value like 1/2 or 2/3 as we are used to? Just point out that it cannot be reduced further. In the same way that the complex a + ib cannot be reduced further. It's just [a, b] in Norman's world.
Hey, wouldn't it be great if processors worked in rational numbers, as Norman suggests, instead of using floats. Then that pesky 0.1 + 0.3 problem and many others would be gone, results would always be correct! (barring register overflow of course).
Aside: The idea of "rational number" tickles me. It implies that any other numbers you may think of are "irrational", not rational, that is to say insane!
@Erna,
Yes, often we can do so much "on autopilot", as it were, we forget that a beginner with a clean slate in his head has no idea what we are talking about.
Norman Wildberger is making an even deeper point, there is a ton of stuff in the maths we learn in school and accept without any watertight proof. Later, we have known this stuff for so long it seems "obvious", we are not even aware that we can't prove it!
I value the history of math, sciences, and invention as much as the tools that we have to work with today. When one is blocked in understanding, often a review of the history around the topic clarifies how others also refused to understand the idea at first.
Enough -- I really have to just watch the videos before I have more to say here.