# Re: So what exactly is a complex number?

Discussion in 'Python' started by Carsten Haese, Aug 31, 2007.

1. ### Carsten HaeseGuest

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.

--
Carsten Haese
http://informixdb.sourceforge.net

Carsten Haese, Aug 31, 2007

2. ### Tim DaneliukGuest

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,
--
----------------------------------------------------------------------------
Tim Daneliuk
PGP Key: http://www.tundraware.com/PGP/

Tim Daneliuk, Aug 31, 2007

3. ### Wildemar WildenburgerGuest

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

Wildemar Wildenburger, Aug 31, 2007
4. ### Roy SmithGuest

Wildemar Wildenburger <> 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

Roy Smith, Sep 1, 2007
5. ### Tim DaneliukGuest

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

--
----------------------------------------------------------------------------
Tim Daneliuk
PGP Key: http://www.tundraware.com/PGP/

Tim Daneliuk, Sep 1, 2007
6. ### Tim DaneliukGuest

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" ...

----------------------------------------------------------------------------
Tim Daneliuk
PGP Key: http://www.tundraware.com/PGP/

Tim Daneliuk, Sep 1, 2007
7. ### Lawrence D'OliveiroGuest

In message <46d89ba9\$0\$30380\$-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.

Lawrence D'Oliveiro, Sep 1, 2007
8. ### Roy SmithGuest

In article <fbamkq\$r7q\$>,
Lawrence D'Oliveiro <_zealand> wrote:

> In message <46d89ba9\$0\$30380\$-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.

Roy Smith, Sep 1, 2007
9. ### Dennis Lee BieberGuest

On Sat, 01 Sep 2007 00:06:13 -0400, Roy Smith <> 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

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".

--
Wulfraed Dennis Lee Bieber KD6MOG

HTTP://wlfraed.home.netcom.com/
(Bestiaria Support Staff: )
HTTP://www.bestiaria.com/

Dennis Lee Bieber, Sep 1, 2007
10. ### Lawrence D'OliveiroGuest

In message <>, Roy Smith wrote:

> In article <fbamkq\$r7q\$>,
> Lawrence D'Oliveiro <_zealand> wrote:
>
>> In message <46d89ba9\$0\$30380\$-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?

Lawrence D'Oliveiro, Sep 1, 2007
11. ### Wildemar WildenburgerGuest

Lawrence D'Oliveiro wrote:
> In message <46d89ba9\$0\$30380\$-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.
>

OK, I didn't put that in the right context, I guess. The "magnitude and
direction" thing is fine, I just scratched my head at the "23 apples at
45 degrees" example. Basically because I see no way of adding 2 apples
at 16 degrees to 4 apples at 25 degrees and the result making any sense.
Anyway, that was just humorous nitpicking on my side, don't take it too
seriously .

/W

Wildemar Wildenburger, Sep 1, 2007
12. ### Tim CouperGuest

... FWIW the fundamental difference in using complex number to represent
purely vector information is that the algebra of complex numbers is such
that the product of two of the imaginary components has a result in the
real range (and a product of a real and imaginary components is in the
imaginary range, etc); so what you are modelling should have this
behaviour if you're using complex numbers to represent it; otherwise a
vector representation could be more appropriate. So in areas of physics
where complex number representations are used, rather than vector, it's
because this this algebra *is* appropriate. Now, if you're modelling
entities that has up to 2 numeric components, which dos not have this
"complex number" behaviour, and you're at most going to add and subtract
these, then the representation of this as vector or complex number model
are indeed identical. This could, however, be considered a special, or
at least a simple, case.

Dr Tim Couper
CTO, SciVisum Ltd

www.scivisum.com

Wildemar Wildenburger wrote:
> Lawrence D'Oliveiro wrote:
>
>> In message <46d89ba9\$0\$30380\$-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.
>>
> >

> OK, I didn't put that in the right context, I guess. The "magnitude and
> direction" thing is fine, I just scratched my head at the "23 apples at
> 45 degrees" example. Basically because I see no way of adding 2 apples
> at 16 degrees to 4 apples at 25 degrees and the result making any sense.
> Anyway, that was just humorous nitpicking on my side, don't take it too
> seriously .
>
> /W
>

Tim Couper, Sep 1, 2007
13. ### =?UTF-8?B?R3J6ZWdvcnogU8WCb2Rrb3dpY3o=?=Guest

> 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,
>

You're mixing definition with application. You didn't say a word about
what complex numbers are, not a word about the imaginary unit, where
does it come from, why is it 'imaginary' etc. Since we're being arses
here I'd hazard a guess you were educated in the USA where doing without
understanding has been mastered by teachers and students alike. You're
explanation of what vectors are is equally bogus but this has already
been pointed out. I'd also like to see a three-dimensional vector
represented by a complex number.

Besides, I find Wikipedia extremely unhelpful for learning maths, partly
beacuse of it's non-linear nature (while reading an article you come
and over the next 4 hours you find out lots of interesting things which
are sadly at best tangential to the initial subject) and because it's
written by people who already know a lot about the subject and take many
things for granted. This seems to be a decent introduction:
http://www.ping.be/~ping1339/complget.htm

Regards,
Greg

=?UTF-8?B?R3J6ZWdvcnogU8WCb2Rrb3dpY3o=?=, Sep 1, 2007
14. ### Tim CouperGuest

"... I'd hazard a guess you were educated in the USA where doing without understanding has been mastered by teachers and students alike. You're explanation ... """

Grzegorz

I think that this is unnecessarily offensive both to the poster and to the many teachers and students of quality in the USA. Maybe it wouldn't be so offensive in Poland, but in the world of first-language English speakers, it is.

Tim "European" Couper

Dr Tim Couper
CTO, SciVisum Ltd

www.scivisum.com

Grzegorz S?odkowicz wrote:
>> 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,
>>
>>

> You're mixing definition with application. You didn't say a word about
> what complex numbers are, not a word about the imaginary unit, where
> does it come from, why is it 'imaginary' etc. Since we're being arses
> here I'd hazard a guess you were educated in the USA where doing without
> understanding has been mastered by teachers and students alike. You're
> explanation of what vectors are is equally bogus but this has already
> been pointed out. I'd also like to see a three-dimensional vector
> represented by a complex number.
>
> Besides, I find Wikipedia extremely unhelpful for learning maths, partly
> beacuse of it's non-linear nature (while reading an article you come
> and over the next 4 hours you find out lots of interesting things which
> are sadly at best tangential to the initial subject) and because it's
> written by people who already know a lot about the subject and take many
> things for granted. This seems to be a decent introduction:
> http://www.ping.be/~ping1339/complget.htm
>
> Regards,
> Greg
>

Tim Couper, Sep 1, 2007
15. ### Tim DaneliukGuest

Grzegorz SÅ‚odkowicz wrote:
>
>> 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,
>>

> You're mixing definition with application. You didn't say a word about
> what complex numbers are, not a word about the imaginary unit, where

I was trying to motivate the idea by means of analogy. This is a
legitimate thing to do. It helps lead people to a conceptual understanding
long before they understand the minutae. I am well aware of the
imaginary unit and from whence complex analysis springs. I just didn't
think that was the best place to start explicating the *concept*.
I find concrete examples that then can lead to theoretical underpinnings
a better way to go than the reverse.

It is so hard to grasp that learning happens in layers, and that each
layer need not be complete or even precise? Evidently you've either never
taught (or were very bad at it). You have to motivate concept and interest
before you can get to the precise detail. For instance, you start with
Newtonian physics, not quantum physics. The entry level physics classes
ignore things like the the non-linear behavior of springs, or the effects
when you don't actually do things in a vacuum. By your definition these
lectures would be "wrong" .. but they're not. They are attempting to introduce
a topic painlessly. And that's what I was doing.

> does it come from, why is it 'imaginary' etc. Since we're being arses
> here I'd hazard a guess you were educated in the USA where doing without
> understanding has been mastered by teachers and students alike. You're

I was initially educated in Europe where being rude was sometimes encouraged
to mask insecurity with a false sense of self-importance. I was later educated
in both Canada and the US wherein I learned both they "why" and the how".

--
----------------------------------------------------------------------------
Tim Daneliuk
PGP Key: http://www.tundraware.com/PGP/

Tim Daneliuk, Sep 2, 2007
16. ### Tim DaneliukGuest

Wildemar Wildenburger wrote:
> Lawrence D'Oliveiro wrote:
>> In message <46d89ba9\$0\$30380\$-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.
> >

> OK, I didn't put that in the right context, I guess. The "magnitude and
> direction" thing is fine, I just scratched my head at the "23 apples at
> 45 degrees" example. Basically because I see no way of adding 2 apples

It was badly stated, I'll agree. What I should have said is something
like "An apple is launched at 45 degrees." thereby sticking to the
magnitude and direction thing.

> at 16 degrees to 4 apples at 25 degrees and the result making any sense.

No, but go to my other example of an aircraft in flight and winds
aloft. It is exactly the case that complex numbers provide a convenient
way to add these two "vectors" (don't wince, please) to provide the
effective speed and direction of the aircraft. Numerous such examples
abound in physics, circuit analysis, the analysis of rotating machinery,
etc.

> Anyway, that was just humorous nitpicking on my side, don't take it too
> seriously .

I didn't and I concede I could have provided a better and crisper example.

> /W

--
----------------------------------------------------------------------------
Tim Daneliuk
PGP Key: http://www.tundraware.com/PGP/

Tim Daneliuk, Sep 2, 2007
17. ### Bryan OlsonGuest

Tim Daneliuk wrote:
> Grzegorz SÅ‚odkowicz wrote:

[...]
>> You're mixing definition with application. You didn't say a word about
>> what complex numbers are, not a word about the imaginary unit, where

>
> I was trying to motivate the idea by means of analogy. This is a
> legitimate thing to do. It helps lead people to a conceptual understanding
> long before they understand the minutae. I am well aware of the
> imaginary unit and from whence complex analysis springs. I just didn't
> think that was the best place to start explicating the *concept*.

Gotta side with Grzegorz on this. Simplifying an explanation
of complex numbers to the point of omitting the imaginary unit
helps lead people to a conceptual *misunderstanding*. I don't
like feeling confused, but where I really screw up is where I
think I understand what I do not.

--
--Bryan

Bryan Olson, Sep 2, 2007
18. ### Lawrence D'OliveiroGuest

In message <>, Tim Daneliuk wrote:

> No, but go to my other example of an aircraft in flight and winds
> aloft. It is exactly the case that complex numbers provide a convenient
> way to add these two "vectors" (don't wince, please) to provide the
> effective speed and direction of the aircraft. Numerous such examples
> abound in physics, circuit analysis, the analysis of rotating machinery,
> etc.

Not really. The thing with complex numbers is that they're
numbers--mathematically, they comprise a "number system" with operations
called "addition", "subtraction", "multiplication" and "division" having
certain well-defined properties (e.g. associativity of multiplication, all
numbers except possibly one not having a multiplicative inverse).

An aircraft in flight amidst winds needs only vector addition (and possibly
scalar multiplication) among the basic operations to compute its path--you
don't need to work with complex numbers as such for that purpose.

For my AC circuit theory example, however, you do.

Lawrence D'Oliveiro, Sep 2, 2007
19. ### =?UTF-8?B?R3J6ZWdvcnogU8WCb2Rrb3dpY3o=?=Guest

> I was trying to motivate the idea by means of analogy. This is a
> legitimate thing to do. It helps lead people to a conceptual
> understanding long before they understand the minutae.

You're mixing terms again. Analogy is saying 'something is like
something else.' What you are saying is 'Something is used to represent
something.' You also make a scalar - vector distinction which is neither
here nor there and implies that complex numbers are vectors, as opposed
to real numbers.

Are you seriously saying that the imaginary unit is a small detail when
explaining complex numbers? If you don't even mention the form 'a + bi'
how can anyone make heads or tails of it? You omit many logical steps in
what you think to be a simple explanation.

In fact, a proper vector in physics has 4 features: point of
application, magnitude, direction and sense. In case of a vector in two
dimensions (a special case, which you also fail to stress not to mention
that you were talking about space) the magnitude and sense can be
described by one number and the direction as another. Since a complex
number is composed of two numbers it can be used to describe those
features of a vector.
Numbers are commonly used to represent breast sizes. This doesn't mean
numbers are breasts, alas. If someone asks you what numbers are and you
say only that they are used to denote breasts sizes, you might create
the wrong impression.

> I just didn't think that was the best place to start explicating the
> *concept*. I find concrete examples that then can lead to theoretical
> underpinnings a better way to go than the reverse.

1. Actual answer to the question can hardly be regarded as 'theoretical
underpinnings'
2. You don't mention that answer at all.
3. Your 'concrete example' is equally sloppy and full of omissions.

> It is so hard to grasp that learning happens in layers, and that each
> layer need not be complete or even precise?

Is it so hard to grasp that you completely omit certain important layers?

> Evidently you've either never taught (or were very bad at it). You
> have to motivate concept and interest before you can get to the
> precise detail.

If someone asks a question their interest is evident, I think. I also
reckon they expect a simple answer which in this case is either a link
or something along the lines:

'As you know, square roots of negative numbers aren't real numbers. We
are taught at school that they "don't exist". However, if we denote
square root of -1 as i we can represent any such root as a product of a
real number and i.
Eg.
sqrt(-9) = sqrt(9 * -1) = 3 * i
sqrt(-0.25) = sqrt(0.25 * -1) = 0.5i etc.

Since sqrt(-1) has little representation in the surrounding world
(saying "I have 2i apples" makes no sense) it's called the imaginary
unit. Now, complex numbers are numbers of the form
a + bi
where a and b are real numbers. They have been invented because they
have many applications in mathematics, physics, engineering etc. They
have many useful properties, perhaps most important of which being that
i2 = -1. <Explanation of complex plane as an example of application>

Note that real numbers are a subset of complex numbers. They are complex
numbers with imaginary part equal to 0.
Also, since i also means current in physics and electrical engineering,
the imaginary unit is denoted as j in those contexts. This is also the
case in Python (presumably because i is a common name for an integer
variable). Speaking of which, they are a built-in type in Py. Try
performing operations on them in the interactive mode:
> >> c = 4 + 1j d = 1 + 2j c + d

(5+5j)
> >> c = 1j # 1 is necessary because 'j' is a legit name for a
> >> variable c * c

(-1+0j)'

And no, I've never taught professionally. Have you?

> The entry level physics classes ignore things like the the non-linear
> behavior of springs, or the effects when you don't actually do things
> in a vacuum. By your definition these lectures would be "wrong" ..
> but they're not. They are attempting to introduce a topic painlessly.
> And that's what I was doing.

That's not what you were doing. You were doing the equivalent of 'Hello,
today we shall discuss gravity. I have 23 apples here, and I will throw
them at 45 degrees - let's denote this by a vector. I throw them. Oh
look, they fell! Well, that's gravity. Thank you for your attention.'

> > does it come from, why is it 'imaginary' etc. Since we're being
> > arses here I'd hazard a guess you were educated in the USA where
> > doing without understanding has been mastered by teachers and
> > students alike. You're

>
> I was initially educated in Europe where being rude was sometimes
> encouraged to mask insecurity with a false sense of self-importance.
> I was later educated in both Canada and the US wherein I learned both
> they "why" and the how"

I'm sorry I made a generalisation from a tendency that I encountered. I
shouldn't have done that. But the fact remains that your explanation,
further statements and general mental sloppiness are excellent examples
of what I said. Whether you learned how and why is debatable but you
certainly don't know how to pass knowledge on.

GS.

=?UTF-8?B?R3J6ZWdvcnogU8WCb2Rrb3dpY3o=?=, Sep 2, 2007
20. ### Evil BertGuest

Grzegorz SÅ‚odkowicz wrote:
> You're mixing terms again. Analogy is saying 'something is like
> something else.'

Actually, that's a simile.

Evil Bert, Sep 3, 2007