complex notation

Sirs,

I have few questions wrt to complex numbers.

1) I would like to know the main purpose of using complex numbers.

2) I would like to know given a number in complex notation, is it possible
to convert this into a real number.

Thanks, Manish

On Thu, 11 Oct 2012 23:27:32 -0500, manishp wrote:

> Sirs,
> 
> I have few questions wrt to complex numbers.
> 
> 1) I would like to know the main purpose of using complex numbers.

Because it makes the math easier (and more systematic) for a wide variety 
of problems.

> 2) I would like to know given a number in complex notation, is it
> possible to convert this into a real number.

Have you ever seen i pennies laying on a table?  No?  Well, how then do 
you convert the imaginary part of a complex number into a real?

-- 
My liberal friends think I'm a conservative kook.
My conservative friends think I'm a liberal kook.
Why am I not happy that they have found common ground?

Tim Wescott, Communications, Control, Circuits & Software
http://www.wescottdesign.com

On Fri, 12 Oct 2012 00:44:50 -0500, Tim Wescott
<tim@seemywebsite.com> wrote:

>> 2) I would like to know given a number in complex notation, is it
>> possible to convert this into a real number.
>
>Have you ever seen i pennies laying on a table?  No?  Well, how then do 
>you convert the imaginary part of a complex number into a real?

I often imagine what it would be like to win the lottery. Does
that count?

>Have you ever seen i pennies laying on a table?  No?  Well, how then do 
>you convert the imaginary part of a complex number into a real?

Sir, the main reason why i asked this is because in literatures I have seen
i raised to 2 = −1. Hence, i should also have a value.

Just a question ...

"manishp" <58525@dsprelated> writes:

>>Have you ever seen i pennies laying on a table?  No?  Well, how then do 
>>you convert the imaginary part of a complex number into a real?
>
> Sir, the main reason why i asked this is because in literatures I have seen
> i raised to 2 = &acirc;&circ;&rsquo;1. Hence, i should also have a value.

It does: sqrt(-1)!
-- 
Randy Yates
Digital Signal Labs
http://www.digitalsignallabs.com

"manishp" <58525@dsprelated> writes:

> Sirs,
>
> I have few questions wrt to complex numbers.
>
> 1) I would like to know the main purpose of using complex numbers.

Unlike others in this group, I disagree that it's merely a matter of
convenience. I believe it is much deeper than that.

The most direct reason that probably appeals to most engineers, without
resorting to more abstract math, is that you cannot find all the roots of
an arbitrary-order polynomial over the reals without using complex
numbers. The simplest example is x^2 + 1 = 0.

More abstractly and deeply, in the field of mathematics referred to as
"abstract algebra" or "modern algebra," complex numbers comprise a
"field" (Google groups, rings, fields), and that field is not the same
as (not "isomorphic" to) the real numbers (which also comprise a field).

> 2) I would like to know given a number in complex notation, is it possible
> to convert this into a real number.

That's like asking if there's a way to convert a real number into an
integer. The answer is either "hell no" or "yes" depending on what is
meaningful for you. 

In the most general sense, the answer is "no." Complex numbers and real
numbers are two different things. 
-- 
Randy Yates
Digital Signal Labs
http://www.digitalsignallabs.com

On Fri, 12 Oct 2012 09:07:11 -0500, manishp wrote:

>>Have you ever seen i pennies laying on a table?  No?  Well, how then do
>>you convert the imaginary part of a complex number into a real?
> 
> Sir, the main reason why i asked this is because in literatures I have
> seen i raised to 2 = −1. Hence, i should also have a value.
> 
> Just a question ...

Wow.  What a question.  I'd like to go into a graduate lounge at a math 
department at some high-falutin' university and ask "does i have a 
value?".  I think the arguing would still be going on when my sons had 
died of old age.

The way of thinking of i that is probably best for your sanity, and will 
provide you the shortest path between using it and getting useful 
results, is to think of it as a handy shortcut for solving mathematical 
problems which has no real meaning in itself, but does have real utility 
in choking real meaning out of certain mathematical constructs.

You will notice, for instance, that any time you grind through some 
problem in its entirety (well, and when you get your math right) that 
when you finally get to the very last step of turning numbers into real 
quantities, that all of the imaginary numbers disappear, and you are left 
with results with all real values.  In fact, not having this happen is an 
excellent indication that you've made an arithmetic error somewhere along 
the way.

This is why a Fourier transform of a real-valued function always has 
Hermetian symmetry, and why the Fourier transform of a Hermetian-
symmetric function always has real symmetry: because -- as it's typically 
defined -- one goes from the time domain (which is all real valued) to 
the frequency domain (Hermetian symmetric), does a bunch of manipulations 
with operators that are Hermetian-symmetric, then transforms the 
Hermetian-symmetric result into the time domain -- and you get all real 
values.

I could go on, but instead I recommend that if it still puzzles you, see 
if you can find "i: A History of sqrt(-1)" at the library, check it out, 
and read it.  It gives a good history of the conceptual development of i, 
and kind of opens your eyes to the facile assumptions you make about what 
people know about math (the western world needed to get into the middle 
ages before we really recognized 0 as a number, rather than a separate 
description, and it took centuries after that to accept negative 
numbers.  For that matter, I had a room mate in college who _still_ 
couldn't accept the existence of negative numbers.  Once you know that, 
you can give people slack for not getting their heads wrapped around i).

Once you feel fully comfortable with i, ask about the other six sqrt(-1) 
out there: someone will teach you about quaternions and octonions, and 
your head will catch on fire.

-- 
My liberal friends think I'm a conservative kook.
My conservative friends think I'm a liberal kook.
Why am I not happy that they have found common ground?

Tim Wescott, Communications, Control, Circuits & Software
http://www.wescottdesign.com

On Fri, 12 Oct 2012 10:53:03 -0400, Randy Yates wrote:

> "manishp" <58525@dsprelated> writes:
> 
>> Sirs,
>>
>> I have few questions wrt to complex numbers.
>>
>> 1) I would like to know the main purpose of using complex numbers.
> 
> Unlike others in this group, I disagree that it's merely a matter of
> convenience. I believe it is much deeper than that.
> 
> The most direct reason that probably appeals to most engineers, without
> resorting to more abstract math, is that you cannot find all the roots
> of an arbitrary-order polynomial over the reals without using complex
> numbers. The simplest example is x^2 + 1 = 0.
> 
> More abstractly and deeply, in the field of mathematics referred to as
> "abstract algebra" or "modern algebra," complex numbers comprise a
> "field" (Google groups, rings, fields), and that field is not the same
> as (not "isomorphic" to) the real numbers (which also comprise a field).

I think that complex numbers can have some pretty deep mathematical 
reasons behind them.

But an engineer's _purpose_ in using them is as a convenience, rather 
than as a whole field of mathematics that must be mastered fully before 
proceeding, is going to lead to success sooner rather than later.

I think you could do everything we do using complex numbers without ever 
letting 'i' or 'j' enter into the discussion.  But you'd be doing a lot 
of twisting and turning and application of a list of rules as long as 
your arm, instead of just using 'i * i = -1'.

-- 
My liberal friends think I'm a conservative kook.
My conservative friends think I'm a liberal kook.
Why am I not happy that they have found common ground?

Tim Wescott, Communications, Control, Circuits & Software
http://www.wescottdesign.com

Tim Wescott <tim@seemywebsite.com> wrote:
> On Thu, 11 Oct 2012 23:27:32 -0500, manishp wrote:

 
>> I have few questions wrt to complex numbers.
 
>> 1) I would like to know the main purpose of using complex numbers.

> Because it makes the math easier (and more systematic) for a wide 
> variety of problems.

>> 2) I would like to know given a number in complex notation, is it
>> possible to convert this into a real number.

> Have you ever seen i pennies laying on a table?  No?  Well, how then do 
> you convert the imaginary part of a complex number into a real?

Fortunately, I also haven't been to a store with imaginary pricing.

But, in some sense that is why we have sine and cosine.

Besides the fact that they were found first in geometry, they
are convenient real solutions to what otherwise would be
a complex solution to a differential equation.

The solutions to f''=f are exp(x) and exp(-x), and
to f''=-f they are exp(ix) and exp(-ix). 

The appropriate combinations of exp(ix) and exp(-x) are real.

That is, (exp(ix)+exp(-ix))/2 and (exp(ix)-exp(-ix))/(2i).
More commonly known as cos(x) and sin(x), if x is in radians.

So, you can't convert the imaginary part to a real number,
but with the appropriate complex numbers you can combine
them in such a way that the result is real.

If you had coins worth (1+i)/2 cents and (1-i)/2 cents, you 
could still buy things with real prices. If the person you
were buying from also had some such coins, you could buy
things with imaginary prices.

-- glen

manishp <58525@dsprelated> wrote:
>>Have you ever seen i pennies laying on a table?  No?  Well, how then do 
>>you convert the imaginary part of a complex number into a real?

> Sir, the main reason why i asked this is because in literatures 
> I have seen i raised to 2 = &#4294967295;??1. Hence, i should also have a value.

See the other answers.

Reminds me, though, in high school I managed to learn the
value of i^i before I knew much else about complex exponents.

Many years before the complex exponential operation was added
to Fortran, it was in PL/I.

I wrote a program that would read in two complex values
and print out X**Y. (Now you can do that in some calculators,
but many that claim to do complex math still won't do it.)

Not knowing what else to try, my first try was 1I**1I.

( ** is the 'raise to the power of' operator in Fortran and PL/I.)

I was, then, surprise to see that the result was real!.

(I also tried 2I**1I and 1I**2I.)

Yes, complex math can be fun.

-- glen