complex numbers

Sirs,

I am reading tutorial on quadrature signals by Rick Lyons. I did not know
where to clarify some questions, hence i am putting on this forum.

I have read complex numbers before and even this time, i have moments when
I think I have understood and then suddenly so many questions come to
mind.

Anyway, my initial understanding was that imaginary term j is just used to
signify that the corresponding attached number lies in the imaginary axis
(y-axis) but it really confuses me when it is given a value of sqrt (-1)

Why at all does j need a value if it just to represent a point in imaginary
axis. For example, take j3 then we are saying that on the imaginary axis,
the point lies at 3 unit values if we see j as just a character that is
used for notational purposes. 

On the other hand, if j=sqrt(-1) then one can multiply 3 by sqrt(-1) and
then use that as a point on imaginary axis which I believe is incorrect.

Can someone help me please ...

Thanks, manish

>Sirs,
>
>I am reading tutorial on quadrature signals by Rick Lyons. I did not know
>where to clarify some questions, hence i am putting on this forum.
>
>I have read complex numbers before and even this time, i have moments
when
>I think I have understood and then suddenly so many questions come to
>mind.
>
>Anyway, my initial understanding was that imaginary term j is just used
to
>signify that the corresponding attached number lies in the imaginary axis
>(y-axis) but it really confuses me when it is given a value of sqrt (-1)
>
>Why at all does j need a value if it just to represent a point in
imaginary
>axis. For example, take j3 then we are saying that on the imaginary axis,
>the point lies at 3 unit values if we see j as just a character that is
>used for notational purposes. 
>
>On the other hand, if j=sqrt(-1) then one can multiply 3 by sqrt(-1) and
>then use that as a point on imaginary axis which I believe is incorrect.
>
>Can someone help me please ...
>
>Thanks, manish
>

Here is my view:
you can imagine that all real world values are represented as real numbers
only i.e. in units of 1. However in our models(&thinking)an imaginary
element if added can help the model. An imaginary element is how many units
of sqrt(-1) are given to the sample value alongside how many ones (real)
are given. When the calculation then ends up with square(-1) it becomes 1
and is thrown onto real member of sample value.

Kadhiem

On 25.12.2012 15:19, manishp wrote:

> Anyway, my initial understanding was that imaginary term j is just used to
> signify that the corresponding attached number lies in the imaginary axis
> (y-axis) but it really confuses me when it is given a value of sqrt (-1)

"Complex axis" - this implies that there are actually two axis, and 
complex numbers can be seen as pairs of "ordinary" real numbers. In 
fact, the complex numbers are isomorphic to the R^2 vector space.

However, there is something the complex numbers have the R^2 does not, 
namely an additional multiplicative structure. IOW, there is a operation 
* : C x C -> C that does not exist in R^2. The * is defined for a 
complex number (a,b) (here written as a pair of numbers from R^2) as 
(a,b) * (c,d) = (a*c-b*d,a*d+b*c), and i is just a short notation for (0,1).

Now i = (0,1) is (one of the two) square roots of -1 because (0,1) * 
(0,1) = (-1,0) by the above rule. (Thus, i = sqrt(-1) is a bit 
misleading because the same holds true for -i).


> Why at all does j need a value if it just to represent a point in imaginary
> axis.

This is the same thing. A complex value is a point in R^2. It doesn't 
"need" one, it has one by convention. IOW, whether I write a+i*b or 
(a,b) for two real numbers is just the same thing, only different notations.

> For example, take j3 then we are saying that on the imaginary axis,
> the point lies at 3 unit values if we see j as just a character that is
> used for notational purposes.

Yes, which is correct.

> On the other hand, if j=sqrt(-1) then one can multiply 3 by sqrt(-1) and
> then use that as a point on imaginary axis which I believe is incorrect.

No, why? sqrt(-1)*3 defines a (actually two) points on the imaginary 
axis, and indeed, these points are 3i and -3i.

Greetings,
	Thomas

On Tue, 25 Dec 2012 08:19:23 -0600, "manishp" <58525@dsprelated>
wrote:

>Sirs,
>
>I am reading tutorial on quadrature signals by Rick Lyons. I did not know
>where to clarify some questions, hence i am putting on this forum.
>
>I have read complex numbers before and even this time, i have moments when
>I think I have understood and then suddenly so many questions come to
>mind.
>
>Anyway, my initial understanding was that imaginary term j is just used to
>signify that the corresponding attached number lies in the imaginary axis
>(y-axis) but it really confuses me when it is given a value of sqrt (-1)
>
>Why at all does j need a value if it just to represent a point in imaginary
>axis. For example, take j3 then we are saying that on the imaginary axis,
>the point lies at 3 unit values if we see j as just a character that is
>used for notational purposes. 
>
>On the other hand, if j=sqrt(-1) then one can multiply 3 by sqrt(-1) and
>then use that as a point on imaginary axis which I believe is incorrect.
>
>Can someone help me please ...
>
>Thanks, manish

The fact that j = sqrt(-1) is appropriate for making solutions work
out properly for a lot of things.   A simple example is solving
quadratic equations where many solutions involve taking the square
root of a negative number.  

It isn't an arbitrary assignment just to provide a coefficient for an
orthogonal axis.   It is actually the proper coefficient to make the
trigonometric (and other) math work correctly.


Eric Jacobsen
Anchor Hill Communications
http://www.anchorhill.com

On Tue, 25 Dec 2012 08:19:23 -0600, manishp wrote:

> Anyway, my initial understanding was that imaginary term j is just used to
> signify that the corresponding attached number lies in the imaginary axis
> (y-axis) but it really confuses me when it is given a value of sqrt (-1)
> 
> Why at all does j need a value if it just to represent a point in imaginary
> axis.

So that you can make use of them. A complex number isn't just a 2D vector;
it's a number, on which the usual arithmetic operations can be performed.

E.g.
   (a+bj)(c+dj)
 = (ac + adj + bcj + bdj^2)
 = (ac + adj + bcj - bd)		# j^2 = -1
 = (ac - bd) + (ad + bc)j

   (a+bj)/(c+dj)
 = (a+bj)(c-dj)/(c+dj)(c-dj)
 = (ac - adj + bcj - bdj^2)/(c^2 - d^2j^2)
 = (ac - adj + bcj + bd)/(c^2 + d^2)	# j^2 = -1
 = ((ac + bd) + (bc - ad)j)/(c^2 + d^2)
 = (ac + bd)/(c^2 + d^2) + ((bc - ad)/(c^2 + d^2))j

Similar principles can be used to take practically any mathematical
function which is defined on the reals and extend it to the complex
numbers.

If you consider the variable of a polynomial to be complex, a polynomial
of degree N always has N roots. Some of those roots may have identical
values, and some of them may be complex, but they will always exist.

E.g. the polynomial x^2+1 = 0 has no real roots, but it has complex roots
x=j and x=-j: j^2=-1 => j^2+1 = 0 and (-j)^2+1 = j^2+1 = 0. The polynomial
x^3-1 = 0 has one real root (1), but also two complex roots
((-1+j.sqrt(3))/2 and (-1-j.sqrt(3))/2).

More generally, for any integer N, x^N-1=0 always has N roots (i.e. the
Nth roots of 1), and they always form a regular N-gon with its centroid at
0 (0+0j) and its vertices on the unit circle |z|=1.

Further, by examining the Taylor series for sin(z), cos(z) and e^z where
z is complex, you can obtain Euler's formula:

	e^jx = cos(x) + j.sin(x)

As j^2=-1, j^n is real if n is even and imaginary if j is odd, and the
sign alternates (i.e. j^n = 1,j,-1,-j,1,j,... for n=0,1,2,...). The
result is that the real terms of the Taylor series for e^jx match the
Taylor series for cos(x) while the imaginary terms match that for j.sin(x).

If you study signal theory, you will encounter the above identity
regularly.

Corollary: e^(j.pi)+1=0 (Euler's identity). Probably the five most
significant mathematical constants (0,1,j,e,pi) in one equation.

	http://xkcd.com/179/

On Tue, 25 Dec 2012 08:19:23 -0600, manishp wrote:

> Sirs,
> 
> I am reading tutorial on quadrature signals by Rick Lyons. I did not
> know where to clarify some questions, hence i am putting on this forum.
> 
> I have read complex numbers before and even this time, i have moments
> when I think I have understood and then suddenly so many questions come
> to mind.
> 
> Anyway, my initial understanding was that imaginary term j is just used
> to signify that the corresponding attached number lies in the imaginary
> axis (y-axis) but it really confuses me when it is given a value of sqrt
> (-1)
> 
> Why at all does j need a value if it just to represent a point in
> imaginary axis. For example, take j3 then we are saying that on the
> imaginary axis, the point lies at 3 unit values if we see j as just a
> character that is used for notational purposes.
> 
> On the other hand, if j=sqrt(-1) then one can multiply 3 by sqrt(-1) and
> then use that as a point on imaginary axis which I believe is incorrect.
> 
> Can someone help me please ...

You have your thinking cap on backwards.

The imaginary axis is not defined because the complex plane uses up paper 
neatly.  The imaginary axis was not defined first, with j = sqrt(-1) to 
follow.

Rather, the concept that you can take the square root of a negative 
number, and that this square root will be an entity called an "imaginary 
number", and that all imaginary numbers can be denoted as a real times 
sqrt(-1), comes first.  (And, in the history of mathematics, rather 
grudgingly and over strong protest over the course of several hundred 
years.  So if you're worried about it taking you months to get your brain 
wrapped around it, just remember that you're centuries ahead of some of 
the greatest mathematicians in the 16th century).

Once you accept the concept of j = sqrt(-1), then you find that the 
arithmetic behavior of complex numbers makes sense if you give it a 
geometrical interpretation by mapping complex numbers to the complex 
plane using a Cartesian graph, with the imaginary numbers on a line at 
right angles to the real numbers.

-- 
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com

On Tuesday, December 25, 2012 11:35:44 AM UTC-7, Eric Jacobsen wrote:
> On Tue, 25 Dec 2012 08:19:23 -0600, "manishp" <58525@dsprelated> wrote: >Sirs, > >I am reading tutorial on quadrature signals by Rick Lyons. I did not know >where to clarify some questions, hence i am putting on this forum. > >I have read complex numbers before and even this time, i have moments when >I think I have understood and then suddenly so many questions come to >mind. > >Anyway, my initial understanding was that imaginary term j is just used to >signify that the corresponding attached number lies in the imaginary axis >(y-axis) but it really confuses me when it is given a value of sqrt (-1) > >Why at all does j need a value if it just to represent a point in imaginary >axis. For example, take j3 then we are saying that on the imaginary axis, >the point lies at 3 unit values if we see j as just a character that is >used for notational purposes. > >On the other hand, if j=sqrt(-1) then one can multiply 3 by sqrt(-1) and >then use that as a point on imaginary axis which I believe is incorrect. > >Can someone help me please ... > >Thanks, manish The fact that j = sqrt(-1) is appropriate for making solutions work out properly for a lot of things. A simple example is solving quadratic equations where many solutions involve taking the square root of a negative number. It isn't an arbitrary assignment just to provide a coefficient for an orthogonal axis. It is actually the proper coefficient to make the trigonometric (and other) math work correctly. Eric Jacobsen Anchor Hill Communications http://www.anchorhill.com

This is the example I use as well to convince people 'j' is not used arbitrarily or for convenience. It finally made sense to me when learning about polynomials over Galois fields in abstract algebra for coding theory. A polynomial over GF(2) can have roots from other fields (GF(2^m) actually), and learning it from this perspective makes it easy to see that polynomials over R can have roots from C.

The book Visual Complex Analysis is another source I use. It starts with a great history of 'j', and stories of early dissention about the validity of 'j' (which is why we're unfortunately left with rhetoric such as "imaginary").

Bryan

On Tuesday, December 25, 2012 9:19:23 AM UTC-5, manishp wrote:
> Sirs,
> 
> 
> 
> I am reading tutorial on quadrature signals by Rick Lyons. I did not know
> 
> where to clarify some questions, hence i am putting on this forum.
> 
> 
> 
> I have read complex numbers before and even this time, i have moments when
> 
> I think I have understood and then suddenly so many questions come to
> 
> mind.
> 
> 
> 
> Anyway, my initial understanding was that imaginary term j is just used to
> 
> signify that the corresponding attached number lies in the imaginary axis
> 
> (y-axis) but it really confuses me when it is given a value of sqrt (-1)
> 
> 
> 
> Why at all does j need a value if it just to represent a point in imaginary
> 
> axis. For example, take j3 then we are saying that on the imaginary axis,
> 
> the point lies at 3 unit values if we see j as just a character that is
> 
> used for notational purposes. 
> 
> 
> 
> On the other hand, if j=sqrt(-1) then one can multiply 3 by sqrt(-1) and
> 
> then use that as a point on imaginary axis which I believe is incorrect.
> 
> 
> 
> Can someone help me please ...
> 
> 
> 
> Thanks, manish

Based upon this and some of your past questions, I think you would highly benefit from the book:

"An Imaginary Tale : The Story of sqrt(-1)" by Paul J. Nahin


It is a great read, and I think he really gives a compelling story to the necessity of sqrt(-1). He says it was really needed to solve cubic equations.

He has written some other great books which I would highly recommend to anyone studying engineering or in particular DSP.

Clay

"manishp" <58525@dsprelated> writes:

> Sirs,
>
> I am reading tutorial on quadrature signals by Rick Lyons. I did not know
> where to clarify some questions, hence i am putting on this forum.
>
> I have read complex numbers before and even this time, i have moments when
> I think I have understood and then suddenly so many questions come to
> mind.
>
> Anyway, my initial understanding was that imaginary term j is just used to
> signify that the corresponding attached number lies in the imaginary axis
> (y-axis) but it really confuses me when it is given a value of sqrt (-1)
>
> Why at all does j need a value if it just to represent a point in imaginary
> axis. For example, take j3 then we are saying that on the imaginary axis,
> the point lies at 3 unit values if we see j as just a character that is
> used for notational purposes. 
>
> On the other hand, if j=sqrt(-1) then one can multiply 3 by sqrt(-1) and
> then use that as a point on imaginary axis which I believe is incorrect.
>
> Can someone help me please ...
>
> Thanks, manish

Hi manish,

You've received some good answers here already. I'd like to present
another perspective.

If a three-year-old asked, "How does an automatic transmission work?",
would you attempt to explain it to him in two or three sentences?

The point is, there is a complexity (pun not intended!) to some things
that just can't be explained with a few minutes of study or a few
sentences on a usenet newsgroup. They require in-depth study, with a
textbook and pencil and paper, over months or even years.

I think complex numbers is one of those things. I'll give you a target
though: If you can study abstract algebra (AKA "modern algebra") and get
to the complex field, so that you can see that the complex field is a
splitting field for any polynomial over the rationals, you will have
come very far in understanding the complexity of the subject. E.g.,
see section 5.3 in [herstein].

--Randy

@book{herstein,
  title = "Topics in Algebra",
  author = "I.N. Herstein",
  publisher = "Wiley",
  edition = "second",
  year = "1975"}

-- 
Randy Yates
Digital Signal Labs
http://www.digitalsignallabs.com

On Tue, 25 Dec 2012 08:19:23 -0600, "manishp" <58525@dsprelated>
wrote:

>Sirs,
>
>I am reading tutorial on quadrature signals by Rick Lyons. I did not know
>where to clarify some questions, hence i am putting on this forum.
>
>I have read complex numbers before and even this time, i have moments when
>I think I have understood and then suddenly so many questions come to
>mind.
>
>Anyway, my initial understanding was that imaginary term j is just used to
>signify that the corresponding attached number lies in the imaginary axis
>(y-axis) but it really confuses me when it is given a value of sqrt (-1)
>
>Why at all does j need a value if it just to represent a point in imaginary
>axis. For example, take j3 then we are saying that on the imaginary axis,
>the point lies at 3 unit values if we see j as just a character that is
>used for notational purposes. 
>
>On the other hand, if j=sqrt(-1) then one can multiply 3 by sqrt(-1) and
>then use that as a point on imaginary axis which I believe is incorrect.
>
>Can someone help me please ...
>
>Thanks, manish

Hello manish,
   I first encountered 'j' in engineering college.
It made no sense to me then, and nothing a professor 
could say helped me understand the weird idea 
of j = sqrt(-1).  (Yes, I saw how it originated from 
finding the roots of certain polynomials, but I wanted 
to understand what 'j' meant in case I encountered it 
in my upcoming engineering career.)  I worked as an 
engineer for 20 years and never encountered the 
concept of 'j' in my particular engineering work.)

Then when I started to learn about DSP, I wanted to understand 
how some digital radios operated.  That's when I had to 
try and figure out what is the meaning of 'j'.

At this point in my life I never think of 'j' as being a 
number.  I now think of 'j' as an operation that can be 
performed on a number.  For example, "negation" is an operation 
you can perform on a number.  Negate the number 7 and the result 
is -7 (minus seven).

In quadrature signal processing, digital radios for example, 
we use signals that have two parts (just like when we 
use latitude & longitude to describe a single physical location). 
We usually call those signals "complex", or "quadrature" signals 
and describe them using complex numbers.

When thinking about complex numbers, if someone said, "Rick, 
think about the number 7" I think of a complex number whose 
East-West value is 'plus seven' and its North-South value is zero.
If someone then said, "Now apply the j-operator to that number 
7", I next think of a new number whose East-West value is zero 
and its North-South value is 'plus seven' (0 + j7).  And I don't 
worry about what the East-West or North-South values actually 
mean in the physical world at this point.

Now having a graphical picture of complex numbers helps us 
understand their nature.  That's where the complex plane comes 
in. Viewing a complex number 'C' on the complex plane is merely 
a point somewhere on the plane.  And applying the j-operator 
to 'C' gives us the number 'jC' that shows up on the complex 
plane as a point rotated by +90 degrees from where the 
original 'C" point was.  

So Again, I don't think of 'j' as a number, I think of 'j' as 
an operation that can be applied to (performed on) a number.

[-Rick-]



 
And a