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Re: So what exactly is a complex number?

 
 
Carsten Haese
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      08-31-2007
On Thu, 2007-08-30 at 20:11 -0500, Lamonte Harris wrote:
> Like in math where you put letters that represent numbers for place
> holders to try to find the answer type complex numbers?


Is English your native language? I'm having a hard time decoding your
question.

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Tim Daneliuk
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      08-31-2007
Carsten Haese wrote:
> On Thu, 2007-08-30 at 20:11 -0500, Lamonte Harris wrote:
>> Like in math where you put letters that represent numbers for place
>> holders to try to find the answer type complex numbers?

>
> Is English your native language? I'm having a hard time decoding your
> question.
>


Here is a simple explanation (and it is not complete by a long shot).

A number by itself is called a "scalar". For example, when I say,
"I have 23 apples", the "23" is a scalar that just represents an
amount in this case.

One of the most common uses for Complex Numbers is in what are
called "vectors". In a vector, you have both an amount and
a *direction*. For example, I can say, "I threw 23 apples in the air
at a 45 degree angle". Complex Numbers let us encode both
the magnitude (23) and the direction (45 degrees) as a "number".

There are actually two ways to represent Complex Numbers.
One is called the "rectangular" form, the other the "polar"
form, but both do the same thing - they encode a vector.

Complex Numbers show up all over the place in engineering and
science problems. Languages like Python that have Complex Numbers
as a first class data type allow you do to *arithmetic* on them
(add, subtract, etc.). This makes Python very useful when solving
problems for engineering, science, navigation, and so forth.


HTH,
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Wildemar Wildenburger
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      08-31-2007
Tim Daneliuk wrote:
> A number by itself is called a "scalar". For example, when I say,
> "I have 23 apples", the "23" is a scalar that just represents an
> amount in this case.
>
> One of the most common uses for Complex Numbers is in what are
> called "vectors". In a vector, you have both an amount and
> a *direction*. For example, I can say, "I threw 23 apples in the air
> at a 45 degree angle". Complex Numbers let us encode both
> the magnitude (23) and the direction (45 degrees) as a "number".
>

1. Thats the most creative use for complex numbers I've ever seen. Or
put differently: That's not what you would normally use complex numbers for.
2. Just to confuse the issue: While complex numbers can be represented
as 2-dimensional vectors, they are usually considered scalars as well
(since they form a field just as real numbers do).


> There are actually two ways to represent Complex Numbers.
> One is called the "rectangular" form, the other the "polar"
> form, but both do the same thing - they encode a vector.
>

Again, that is just one way to interpret them. Complex numbers are not
vectors (at least no moe than real numbers are).


/W
 
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Roy Smith
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      09-01-2007
Wildemar Wildenburger <(E-Mail Removed)> wrote:
> Again, that is just one way to interpret them. Complex numbers are not
> vectors (at least no moe than real numbers are).


OK, let me take a shot at this.

Math folks like to group numbers into sets. One of the most common sets is
the set of integers. I'm not sure what the formal definition of an integer
is, but I expect you know what they are: 0, 1, 2, 3, 4, etc, plus the
negative versions of these: -1, -2, -3, etc.

The set of integers have a few interesting properties. For example, any
integer plus any other integer gives you another integer. Math geeks would
say that as, "The set of integers is closed under addition".

Likewise for subtraction; any integer subtracted from any other integer
gives you another integers. Thus, the set of integers is closed under
subtraction as well. And multiplication. But, division is a bit funky.
Some integers divided by some integers give you integers (i.e. 6 / 2 = 3),
but some done (i.e. 5 / 2 = 2.5).

So, now we need another kind of number, which we call reals (please, no nit
picking about rationals). Reals are cool. Not only are the closed under
addition, subtraction, and multiplication, but division too. Any real
number divided by any other real number gives another real number.

But, it's not closed over *every* possible operation. For example, square
root. If you take the square root of 4.23, you get some real number. But,
if you try to take the square root of a negative number, you can't do it.
There is no real number which, when you square it, gives you (to use the
cannonical example), -1. That's where imaginary numbers come in. The math
geeks invented a wonderful magic number called i (or sometimes j), which
gives you -1 when you square it.

So, the next step is to take an imaginary number and add it to a real
number. Now you've got a complex number. There's all kinds of wonderful
things you can do with complex numbers, but this posting is long enough
already
 
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Tim Daneliuk
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      09-01-2007
Wildemar Wildenburger wrote:
> Tim Daneliuk wrote:
>> A number by itself is called a "scalar". For example, when I say,
>> "I have 23 apples", the "23" is a scalar that just represents an
>> amount in this case.
>>
>> One of the most common uses for Complex Numbers is in what are
>> called "vectors". In a vector, you have both an amount and
>> a *direction*. For example, I can say, "I threw 23 apples in the air
>> at a 45 degree angle". Complex Numbers let us encode both
>> the magnitude (23) and the direction (45 degrees) as a "number".
>>

> 1. Thats the most creative use for complex numbers I've ever seen. Or
> put differently: That's not what you would normally use complex numbers
> for.
> 2. Just to confuse the issue: While complex numbers can be represented
> as 2-dimensional vectors, they are usually considered scalars as well
> (since they form a field just as real numbers do).
>
>
>> There are actually two ways to represent Complex Numbers.
>> One is called the "rectangular" form, the other the "polar"
>> form, but both do the same thing - they encode a vector.
>>

> Again, that is just one way to interpret them. Complex numbers are not
> vectors (at least no moe than real numbers are).
>
>
> /W


Yeah, I know it's a simplification - perhaps even a vast simplification -
but one eats the elephant a bite at a time. FWIW, the aforementioned
was my first entre' into complex arithmetic, long before I waded through
complex analysis and all the more esoteric stuff later in school. I
wonder why you think it is "creative", though. Most every engineer I've
ever know (myself included) was first exposed to complex numbers in much
this way. Then again, I was never smart enough to be a pure mathematician

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Tim Daneliuk
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      09-01-2007
Wildemar Wildenburger wrote:
> Tim Daneliuk wrote:
>> A number by itself is called a "scalar". For example, when I say,
>> "I have 23 apples", the "23" is a scalar that just represents an
>> amount in this case.
>>
>> One of the most common uses for Complex Numbers is in what are
>> called "vectors". In a vector, you have both an amount and
>> a *direction*. For example, I can say, "I threw 23 apples in the air
>> at a 45 degree angle". Complex Numbers let us encode both
>> the magnitude (23) and the direction (45 degrees) as a "number".
>>

> 1. Thats the most creative use for complex numbers I've ever seen. Or
> put differently: That's not what you would normally use complex numbers
> for.



Oh, one other thing I neglected to mention. My use of "vector" here
is certainly incorrect in the mathematician's sense. But I first
ran into complex arithmetic when learning to fly an airplane.
The airplane in flight has a speed (magnitude) and a bearing (direction).
The winds aloft also have speed and bearing. These are called
the aircraft "vector" and the wind "vector" respectively. They must
be added to compute the actual (effective) speed/direction the aircraft
is flying. In the Olden Days (tm), we did this graphically on a
plastic flight computer and a grease pencil. With the advent of
calculators like the HP 45 that could do polar <-> rectangular
conversion, this sort of problem became a snap to do. It is from
this experience that I used the (non-mathematical) sense of the
word "vector" ...

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Lawrence D'Oliveiro
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      09-01-2007
In message <46d89ba9$0$30380$(E-Mail Removed)-online.net>, Wildemar
Wildenburger wrote:

> Tim Daneliuk wrote:
>>
>> One of the most common uses for Complex Numbers is in what are
>> called "vectors". In a vector, you have both an amount and
>> a *direction*. For example, I can say, "I threw 23 apples in the air
>> at a 45 degree angle". Complex Numbers let us encode both
>> the magnitude (23) and the direction (45 degrees) as a "number".
>>

> 1. Thats the most creative use for complex numbers I've ever seen. Or
> put differently: That's not what you would normally use complex numbers
> for.


But that's how they're used in AC circuit theory, as a common example.
 
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Roy Smith
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      09-01-2007
In article <fbamkq$r7q$(E-Mail Removed)>,
Lawrence D'Oliveiro <(E-Mail Removed)_zealand> wrote:

> In message <46d89ba9$0$30380$(E-Mail Removed)-online.net>, Wildemar
> Wildenburger wrote:
>
> > Tim Daneliuk wrote:
> >>
> >> One of the most common uses for Complex Numbers is in what are
> >> called "vectors". In a vector, you have both an amount and
> >> a *direction*. For example, I can say, "I threw 23 apples in the air
> >> at a 45 degree angle". Complex Numbers let us encode both
> >> the magnitude (23) and the direction (45 degrees) as a "number".
> >>

> > 1. Thats the most creative use for complex numbers I've ever seen. Or
> > put differently: That's not what you would normally use complex numbers
> > for.

>
> But that's how they're used in AC circuit theory, as a common example.


Well, not really. They're often talked about as vectors, when people are
being sloppy, but they really aren't.

In the physical world, let's say I take out a compass, mark off a bearing
of 045 (north-east), and walk in that direction at a speed of 5 MPH.
That's a vector. The "north" and "east" components of the vector are both
measuring fundamentally identical quantities, along perpendicular axes. I
could pick any arbitrary direction to call 0 (magnetic north, true north,
grid north, or for those into air navigation, the 000 VOR radial) and all
that happens is I have to rotate my map.

But, if I talk about complex impedance in an AC circuit, I'm measuring two
fundamentally different things; resistance and reactance. One of these
consumes power, the other doesn't. There is a real, physical, difference
between these two things. When I talk about having a pole in the left-hand
plane, it's critical that I'm talking about negative values for the real
component. I can't just pick a different set of axis for my complex plane
and expect things to still make sense.
 
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Dennis Lee Bieber
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      09-01-2007
On Sat, 01 Sep 2007 00:06:13 -0400, Roy Smith <(E-Mail Removed)> declaimed
the following in comp.lang.python:

> In the physical world, let's say I take out a compass, mark off a bearing
> of 045 (north-east), and walk in that direction at a speed of 5 MPH.
> That's a vector. The "north" and "east" components of the vector are both


<Devil to the stand please>

As I learned it, 5mph at 45deg is a "velocity" (speed +
direction)...

About the only place I encounter "vector"s these days is satellite
work -- and they are commonly the infamouse "unit vector" in which the
magnitude of the distance is normalized to "1.0".



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Lawrence D'Oliveiro
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      09-01-2007
In message <(E-Mail Removed)>, Roy Smith wrote:

> In article <fbamkq$r7q$(E-Mail Removed)>,
> Lawrence D'Oliveiro <(E-Mail Removed)_zealand> wrote:
>
>> In message <46d89ba9$0$30380$(E-Mail Removed)-online.net>,
>> Wildemar Wildenburger wrote:
>>
>> > Tim Daneliuk wrote:
>> >>
>> >> One of the most common uses for Complex Numbers is in what are
>> >> called "vectors". In a vector, you have both an amount and
>> >> a *direction*. For example, I can say, "I threw 23 apples in the air
>> >> at a 45 degree angle". Complex Numbers let us encode both
>> >> the magnitude (23) and the direction (45 degrees) as a "number".
>> >>
>> > 1. Thats the most creative use for complex numbers I've ever seen. Or
>> > put differently: That's not what you would normally use complex numbers
>> > for.

>>
>> But that's how they're used in AC circuit theory, as a common example.

>
> But, if I talk about complex impedance in an AC circuit, I'm measuring two
> fundamentally different things; resistance and reactance. One of these
> consumes power, the other doesn't. There is a real, physical, difference
> between these two things. When I talk about having a pole in the
> left-hand plane, it's critical that I'm talking about negative values for
> the real component. I can't just pick a different set of axis for my
> complex plane and expect things to still make sense.


In other words, there is a preferred coordinate system for the vectors. Why
does that make them any the less vectors?

 
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