# complex notation

Started by October 12, 2012
```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 = &minus;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 = &minus;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

--
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

--
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
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.

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

```