e ^ (pi * i) = -1 WHAT!!!

[Dylab]

I'm sorry if this doesn't belong here but I don't know where else to put it. Today I read about the equation and tested it but I'm quite surprised. Can someone give an explanation to why this is true?

[LucyR]

Euler's formula:

e^(i x) = cos(x) + isin(x)

You can use a series expansion to demonstrate this.

In the case of pi:

e^(i pi) = cos(pi) + isin(pi) = -1 + 0 = -1

[daver]

Originally posted by Dylab
I'm sorry if this doesn't belong here but I don't know where else to put it. Today I read about the equation and tested it but I'm quite surprised. Can someone give an explanation to why this is true?

You can use the Taylor series to show that e**ix = cos(x) + i * sin(x). So e**(pi*i) = cos(pi) + i * sin(pi) = cos(pi) = -1.

You can represent a complex number as r * e**(i * theta), where r and theta are real numbers. This is essentialy a polar coordinate representation of a complex number (instead of the rectangular coordinate version x + i*y).

[Brown]

Originally posted by LucyR
Euler's formula:

e^(i x) = cos(x) + isin(x)

Use can use a series expansion to demonstrate this.Check a calculus book for a Maclauren expansion of e^x, cos(x) and sin(x), and you'll see that the equation actually does hold up. (Just remember that you're expanding e^(ix), rather than e^x.)

If you know calculus, try taking the derivative of both sides of the equation four times. Once again, you'll see that the equation holds up.

The expression e^(i x) is one way to express a complex number. This expression is in polar form, with a magnitude and a phase. In particular, a number expressed as y*e^(i x) has a magnitude of y and a phase of x. The number can also be converted to a rectangular or Cartesian form, with a "real part" and an "imaginary part." In the equation above, the "real part" is cos(x) and the "imaginary part" is sin(x).

I've found that it is very difficult to get a grasp on this subject unless there is face-to-face interaction. There are very few books that clearly explain what the equation means or why it is useful.

Basically, this equation is useful to engineers, and in particular, to electrical engineers, who deal with alternating current (which cycles in sums of sines and cosines).

[GreyWanderer]

Originally posted by Brown
Basically, this equation is useful to engineers, and in particular, to electrical engineers, who deal with alternating current (which cycles in sums of sines and cosines).

Yep, and we don't care why it works, only that it works. (;

I'm almost finished width my first year on an engineering degree in space technology, which is a lot like electrical engineering.

[Brown]

Originally posted by GreyWanderer
Yep, and we don't care why it works, only that it works. A few years ago, I was tutoring a room full of confused engineering students who had been exposed to imaginary numbers for the first time. They just didn't understand how you could have a number that, when squared, made a negative number.

I said to them, "I will explain that to you. But first, you have to tell me something. You all know that that multiplying a negative number times a negative number yields a positive number, right?" All the heads nodded. I then asked the group, "Why is that true? I mean, you've been multiplying negative numbers by negative numbers to get positive numbers for over ten years now. Does anyone know why a negative times a negative is a positive?" No one did.

Well, I then briefly explained why a negative times a negative is a positive. In short, the reason is "because it works."

Then I said, "And now we're dealing with imaginary numbers, in which a positive times a positive is a negative. Why do you think that is true?"

"Because it works?" someone asked meekly.

"Exactly!"

I then gave them a quick and dirty explanation of imaginary numbers. The number i (or as engineers call it, j, to distinguish it from current denoted i) is just like any other number. You can add it to other numbers, subtract it from other numbers, multiply it, divide it. You treat it like a variable. But i has one peculiar property. If you multiply it by itself, you get a negative one.

[swellman]

Originally posted by Brown
The number i (or as engineers call it, j, to distinguish it from current denoted i) is just like any other number.

Slight quibble - electrical engineers make the distinction of using j for imaginary numbers. Most other engineering disciplines use i.

[jj]

Originally posted by swellman


Slight quibble - electrical engineers make the distinction of using j for imaginary numbers. Most other engineering disciplines use i.

Well, yes, we EE's use both sometimes, just to confuse you :)

But in fact

e^(i pi) +1 = 0 is perhaps the most classical equation I've seen, it has the 5 most fundamental constants of mathematics all in one neat package.

Additive identity
Multiplicitive identity
Pi
i
and
e

All in one package.

Oh, and EE's have another problem, because 'e' is also 'electromotive force' while 'i' is current.

Heh.

[jj]

Originally posted by Dylab
I'm sorry if this doesn't belong here but I don't know where else to put it. Today I read about the equation and tested it but I'm quite surprised. Can someone give an explanation to why this is true?

Another way to look at it. e^(a+bi) is the same as

e^a * e^bi.

Now the 'i' means that you're rotating around the unit circle, it's a pure phase component, and the 'a' is the magnitude component. Here a=0, so the magnitude is 1.

You have a phase of pi, (b=pi) for which sin (pi)=0, and cos(pi)= -1. That means that the vector expressing that angle is -1 * 0 * i. The magnitude is one, so the value is in fact -1.

[a_unique_person]

Originally posted by Brown
A few years ago, I was tutoring a room full of confused engineering students who had been exposed to imaginary numbers for the first time. They just didn't understand how you could have a number that, when squared, made a negative number.


??? I was taught these in high school. Is that true that you don't teach imaginary numbers till Uni?

[QuarkChild]

Originally posted by jj

e^(i pi) +1 = 0 is perhaps the most classical equation I've seen, it has the 5 most fundamental constants of mathematics all in one neat package.

Yep.
I had a friend in high school who was so impressed by that equation, he wrote it in large letters on the side of his Lit textbook. Now he has a degree in Mathematics.

I once had the misfortune of taking a circuitry class taught by a professor who'd spent too much time around engineers. Writing 'j' instead of 'i'...ugh...

::shudder::

[Iconoclast]

Originally posted by Brown
But i has one peculiar property. If you multiply it by itself, you get a negative one.
That's not a property of i, that's it's Definition.

Originally posted by Brown
Well, I then briefly explained why a negative times a negative is a positive. In short, the reason is "because it works."

Then I said, "And now we're dealing with imaginary numbers, in which a positive times a positive is a negative. Why do you think that is true?"

"Because it works?" someone asked meekly.

"Exactly!"

Brown, I think a more correct description than "Because it works" would be "Because it's consistent". It's not obvious to those who've only used simple mathematics that ANY form of maths that you care to invent is perfectly valid if it is found to be always Constitent.

[QuarkChild]

Originally posted by Iconoclast


Brown, I think a more correct description than "Because it works" would be "Because it's consistent". It's not obvious to those who've only used simple mathematics that ANY form of maths that you care to invent is perfectly valid if it is found to be always Constitent.

I thought Gödel proved that any system of algebra is inconsistent.

Or something like that...I read a book about the proof in high school but that was too long ago for me to remember.

Plus I didn't understand most of it.

[QuarkChild]

After some googling, I retract the above statement. Apparently a system can be consistent but not complete.

Reminds me of a quote from my math professor: "The fundamental theorem of algebra cannot be proved using algebra. I remind my algebraist colleauges of this at every possible opportunity."

[Iconoclast]

Originally posted by QuarkChild
After some googling, I retract the above statement. Apparently a system can be consistent but not complete.

Reminds me of a quote from my math professor: "The fundamental theorem of algebra cannot be proved using algebra. I remind my algebraist colleauges of this at every possible opportunity."
Certainly, these are the Axioms, and they are at the very base of mathematical definition so cannot be defined in terms of any simpler constructs, they are self-evident. For example, the 0th Law of Thermodynamics: "if a = b and b = c then a = c" cannot be proved, but we all intuitively know it to be true.

What I was pointing out was that we can invent any system we like in mathematics as long as there is an unfailing coinsistency in the system. One example is Clock Arithmetic, where the set of counting numbers is defined to be finite. In (say) 5-Clock Arithmetic, the only counting numbers that exist are 0,1,2,3, and 4, so this means that 4 + 2 = 1, and that sounds wrong, but only because the number system we usually use is infinite in both directions and doesn't roll back on itself.

Similarly, the invention of negative numbers was opposed by many mathematicians because the concept initially seems absurd. How can I have negative 2 cows?

The same goes for the number 0. Why would I bother counting how many cows I own if I don't own any cows? El Stupido.

[QuarkChild]

Originally posted by Iconoclast

Certainly, these are the Axioms, and they are at the very base of mathematical definition so cannot be defined in terms of any simpler constructs, they are self-evident. For example, the 0th Law of Thermodynamics: "if a = b and b = c then a = c" cannot be proved, but we all intuitively know it to be true

The history of mathematics suggests that one should be careful about what one claims is "self-evident." Axioms of euclidean geometry are "self-evident," but the geometry of our universe is not necessarily Euclidean.

That said, I do understand your point.

[Iconoclast]

Originally posted by QuarkChild
The history of mathematics suggests that one should be careful about what one claims is "self-evident."

Absolutely. Stephen Hawking himself has been known to say that "Common sense tells us that the Sun revolves around the Earth". This was in relation to a question about Relativity and Quantum Mechanics defying common sense.

Originally posted by QuarkChild
Axioms of euclidean geometry are "self-evident," but the geometry of our universe is not necessarily Euclidean.
Certainly, but whether or not the universe is Euclidian in nature has nothing at all to do with the axioms of Euclidian geometry being self-evident.

[Crossbow]

Originally posted by Dylab
I'm sorry if this doesn't belong here but I don't know where else to put it. Today I read about the equation and tested it but I'm quite surprised. Can someone give an explanation to why this is true?

Yep, its true!

I was a bit mystified when it was first discussed in my calculus class, but later we did expansion of it and I used a simple FORTRAN program to verify it numerically.

Go figure!

Or as my one math teacher used to say, "Just relax and enjoy!"

[Hydrogen Cyanide]

Originally posted by Brown
..Basically, this equation is useful to engineers, and in particular, to electrical engineers, who deal with alternating current (which cycles in sums of sines and cosines).

I also used this as a structural dynamics engineer. It is used to analyze a series of equations usually of the form:

mass*dx^2/dt^2 + damping*dx/dt + stiffness*x = 0

By using a Laplace transformation it turns into a quadratic equation where you WANT the solution to include an imaginary number for a stable system.... if the solution is "real" the system is unstable (the first Tacoma Narrows Bridge would be an example of this).

I'm sorry, the details are eluding me... this is what I did in a former life 15 years ago... but I still find the stuff exciting and interesting (especially when the mass, damping and stiffness are variables, not constants). Though I am afraid my "Shock and Vibration Handbook" is presently being used to hold up an unstable document holder (which is essentially a solution to a statics problem).

If you check out basic college text books on vibration, control systems, mechanical systems or anything else with a sinusoidal solution you will see where Euler's stuff is interesting (is explained in the second chapter of this:
http://www.amazon.com/exec/obidos/tg/detail/-/0071370811/qid=1053536787/sr=8-1/ref=sr_8_1/002-2403047-8472007?v=glance&s=books&n=507846 )

[jj]

Originally posted by Hydrogen Cyanide


I also used this as a structural dynamics engineer. It is used to analyze a series of equations usually of the form:

mass*dx^2/dt^2 + damping*dx/dt + stiffness*x = 0

By using a Laplace transformation it turns into a quadratic equation where you WANT the solution to include an imaginary number for a stable system.... if the solution is "real" the system is unstable (the first Tacoma Narrows Bridge would be an example of this).


I'm suspecting that you mean that the real part of the solution has to be negative. Generally, a positive real part in the exponential means that oscillations grow exponentially, and a negative real part that they die out exponentially.

The imaginary part sets the frequency of oscillation.

So, something with two real, negative coefficients would have a beyond critically damped behavior.

However, if you were to point out that's a great lot of damping for some systems, you'd be right. I doubt you can get there in a suspension bridge, I suspect you'll have a (set of?) resonsant frequencie(s).

Now, maybe there's some kind of transform... But Laplace is of the form s=a+bi. It's possible to formulate it as either
e^st or e^-st, so the sign can vary, but you want the overall real part to be negative when considering formulation and everything.

[Brown]

Originally posted by Hydrogen Cyanide
I also used this as a structural dynamics engineer. It is used to analyze a series of equations usually of the form:

mass*dx^2/dt^2 + damping*dx/dt + stiffness*x = 0

By using a Laplace transformation it turns into a quadratic equation where you WANT the solution to include an imaginary number for a stable system.... if the solution is "real" the system is unstable (the first Tacoma Narrows Bridge would be an example of this).Hmm, I thought you'd ordinarily want the poles' real part to be negative for stability, i.e., in the left-half plane. The imaginary part of the poles isn't all that critical.

But the point is well-taken. Many of the same mathematics that apply to electrical engineering also apply to other disciplines. You can have an electronic filter that corresponds to pretty much the same equation as a shock absorber that includes a spring and a "dashpot."

[jj]

Originally posted by Brown
Hmm, I thought you'd ordinarily want the poles' real part to be negative for stability, i.e., in the left-half plane. The imaginary part of the poles isn't all that critical.

But the point is well-taken. Many of the same mathematics that apply to electrical engineering also apply to other disciplines. You can have an electronic filter that corresponds to pretty much the same equation as a shock absorber that includes a spring and a "dashpot."

Hey! I just said that!

You do signal processing or EE or something, too?

[Brown]

Originally posted by jj
Hey! I just said that!

You do signal processing or EE or something, too? Yeah, I saw your post when I posted my own! I was glad that someone else had the same take that I did. And yes, I'm a double-E.

[Hydrogen Cyanide]

Originally posted by jj


I'm suspecting that you mean that the real part of the solution has to be negative. Generally, a positive real part in the exponential means that oscillations grow exponentially, and a negative real part that they die out exponentially.

The imaginary part sets the frequency of oscillation.
...

You are right... I am a bit fuzzy. I also remember that there should be an imaginary portion for it to be stable. You do not want a real solution.

Anyway... sorry about that. I guess I meant to say that the title expression is not just for electrical engineering.

[DrMatt]

Originally posted by Brown
Basically, this equation is useful to engineers, and in particular, to electrical engineers, who deal with alternating current (which cycles in sums of sines and cosines).

It's a compact tool for folks who make electronic music, too. Not just alternating currents, but any waveform can be expressed as a Fourier expansion of cosine and sine components, or, in polar coordinates, magnitude and phase pairs.

[jj]

Originally posted by DrMatt


It's a compact tool for folks who make electronic music, too. Not just alternating currents, but any waveform can be expressed as a Fourier expansion of cosine and sine components, or, in polar coordinates, magnitude and phase pairs.

Well, yes, but that's a question of orthonormal expansions :)

[Solitaire]

Originally posted by Dylab
I'm sorry if this doesn't belong here but I don't know where else to put it. Today I read about the equation and tested it but I'm quite surprised. Can someone give an explanation to why this is true?
Oops! I new I forgot about something.
Here's the link. (http://www.well.com/user/davidu/euler.html)

P.S. For fun you might try decomposing pi. Pi Composites (http://members.ispwest.com/r-logan/)

[Underemployed]

Thank you Synchronicity, I'm a maths layman but that link did actually make it almost understandable. Except I still can't see what the 'e' represents. Can someone elucidate?

[rwguinn]

Originally posted by Hydrogen Cyanide


I also used this as a structural dynamics engineer. It is used to analyze a series of equations usually of the form:

mass*dx^2/dt^2 + damping*dx/dt + stiffness*x = 0

By using a Laplace transformation it turns into a quadratic equation where you WANT the solution to include an imaginary number for a stable system.... if the solution is "real" the system is unstable (the first Tacoma Narrows Bridge would be an example of this).



What you are looking for in that case (after taking the LaPlace Transform) is to solve for s, and you end up with a term that has the form 1/((wf^2)-(wn^2)+2*z*wn*wf*i), which, as wf (the forcing frequency) approaches the value of wn (the natural frequency), the term 2*z*wn*wf*i is all that keeps the function from blowing up!
Getting to the above formula requires a transformation to generalized coordinates, and a few other things (an eigen solution, among others), but is very handy for showing why you don't want your automobile sprung at 7-9Hz, for example.
RW

[Solitaire]

Originally posted by Underemployed
Thank you Synchronicity, I'm a maths layman but that link did actually make it almost understandable. Except I still can't see what the 'e' represents. Can someone elucidate?
How about a history page about e? (http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/e.html)

[rwguinn]

Originally posted by jj



However, if you were to point out that's a great lot of damping for some systems, you'd be right. I doubt you can get there in a suspension bridge, I suspect you'll have a (set of?) resonsant frequencie(s).


There is indeed a set of natural frequencies in a suspension bridge, but as "Golloping Gertie" showed, most of the energy is involved in the lowest mode excitable for the forcing function.
See my previous post- the damping (z) is what keeps the function from violating the laws of thermodynamics and venturing into perpetual motion....

[jj]

Originally posted by rwguinn


There is indeed a set of natural frequencies in a suspension bridge, but as "Golloping Gertie" showed, most of the energy is involved in the lowest mode excitable for the forcing function.
See my previous post- the damping (z) is what keeps the function from violating the laws of thermodynamics and venturing into perpetual motion....

Well, yeah, in most of the systems I work with it would be more like F(z)=1/(a+bz^-1+cz^-2) :)

But that's another story.

[Brown]

Here's an obscure mathematical riddle: What do you get when you multiply infinity by i ?

Answer: 8

(The reason is that multiplying by i results in a 90-degree rotation.)

[rwguinn]

Originally posted by jj


Well, yeah, in most of the systems I work with it would be more like F(z)=1/(a+bz^-1+cz^-2) :)

But that's another story.

you lost me there....:)

RW

[Zombified]

Originally posted by rwguinn
you lost me there....:)Those crazy newfangled digital guys.

T(s) = w^2/(s^2 + w/q s + w^2), but it's just a hobby for me, so I get to be old-fashioned and eccentric. :) (And that's nearly the limit of my self-taught understanding, as well...)

[synaesthesia]

Originally posted by Brown
Here's an obscure mathematical riddle: What do you get when you multiply infinity by i ?

Answer: 8

(The reason is that multiplying by i results in a 90-degree rotation.)

Fascinating.
How does the eight represent a 90 degree rotation?

[Jethro]

Originally posted by synaesthesia


Fascinating.
How does the eight represent a 90 degree rotation? Turn your head 90 degrees and you'll see.

[Underemployed]

Unfortunately that link to the history of 'e' did not work.

[Jethro]

Originally posted by Underemployed
Unfortunately that link to the history of 'e' did not work. Worked fine for me.

[rwguinn]

Originally posted by Jethro
Turn your head 90 degrees and you'll see.

Oh.

:rolleyes:

well, DUH! Nice one, Jethro.:p

[Underemployed]

Got the link now, I see that e is....2.71 plus a bunch of other numbers.

I wish I felt the same glow of understanding and wonder at the nature of the universe that you guys do when reading this equation.

[Iconoclast]

Originally posted by Underemployed
Got the link now, I see that e is....2.71 plus a bunch of other numbers.
No, the other digits are interesting because e is one of those numbers that sucks you in the first time you see it. See, it starts off as 2.718281828 and you think "hey, there's a recurring pattern here", but then it turns out the gods were only teasing you, the 1828 1828 was just a glitch in the matrix and then the digits go back to being seemingly random.

[Ike]

e^(pi*i) = -1

I seem to remember from one of my old calculus textbooxs a rather humorous aside to this famous equation. I cannot for the life of me remember who, maybe Pascal, apparently used this equation as proof that god exists, by saying that since such a simple equation that contains every important number in mathematics exists at all must be proof that god exists. Again, my memory is a little fuzzy on this, but since then, I have jokingly referred to this equation as 'The God Equation'.

Ike

[Brown]

Originally posted by Ike
e^(pi*i) = -1

I seem to remember from one of my old calculus textbooxs a rather humorous aside to this famous equation. I cannot for the life of me remember who, maybe Pascal, apparently used this equation as proof that god exists, by saying that since such a simple equation that contains every important number in mathematics exists at all must be proof that god exists. I suspect you are referring to an incident in which Euler supposedly "proved" the existence of God by reciting a nonsense equation (not e^(pi*i) = -1) and loudly demanding that the "proof" be refuted.

According to the story, Euler did this to silence a blowhard, who claimed to know a great deal about the existence or nonexistence of God, but knew nothing about mathematics.

This story is recounted briefly in Carl Sagan's book, "Broca's Brain."

[Soapy Sam]

A question:-
I was clearly told in school
Multiplying two negative numbers always yields a positive product.

It seems to me that if this is true, then i cannot be a number.

If i is a number, then multiplying negatives CAN give a negative product.

I asked this question in high school and never received a straight answer. It's a question about definitions, in simple English. So, with no hedging please. No "complex / imaginary" voodoo. No Argand diagrams or Laws of Form.

Which , if either, of these statements, is true? Is i a number or a process?

[Sundog]

Originally posted by Soapy Sam
A question:-
I was clearly told in school
Multiplying two negative numbers always yields a positive product.

It seems to me that if this is true, then i cannot be a number.

If i is a number, then multiplying negatives CAN give a negative product.

I asked this question in high school and never received a straight answer. It's a question about definitions, in simple English. So, with no hedging please. No "complex / imaginary" voodoo. No Argand diagrams or Laws of Form.

Which , if either, of these statements, is true? Is i a number or a process?

It's completely simple. Unfortunately, you have already rejected the answer as "voodoo".

Why do you accept some things you're told, like multiplying two negatives equalling a positive, and reject others?

\Math is not English, and you can't learn it if you fight the knowledge. An accepting mind is essential, or you'll never get it.

[rwguinn]

Originally posted by Soapy Sam
A question:-
I was clearly told in school
Multiplying two negative numbers always yields a positive product.

It seems to me that if this is true, then i cannot be a number.

If i is a number, then multiplying negatives CAN give a negative product.

I asked this question in high school and never received a straight answer. It's a question about definitions, in simple English. So, with no hedging please. No "complex / imaginary" voodoo. No Argand diagrams or Laws of Form.

Which , if either, of these statements, is true? Is i a number or a process?

We will do that for you, right after you explain why water flows down hill-but you cannot use words such as gravity, potential, density, hydrogen, oxygen, compound, slope, mass or kinetic--or God, Jesus, Zeus, Thor, or any other deity.

[Hydrogen Cyanide]

Originally posted by Soapy Sam
..
It seems to me that if this is true, then i cannot be a number.

..

It is an "imaginary" number.

It is just a definition. But a very useful definition.

See: http://mathforum.org/library/drmath/view/58251.html
and http://mathworld.wolfram.com/ImaginaryUnit.html

Oh, and to confuse things a bit more and to make them even more metaphysical: pi and e are also transcendental numbers(http://mathworld.wolfram.com/TranscendentalNumber.html)

You seem to be asking the wrong high school teacher, or rejecting the standard math vocabulary. Many of your math questions can be answered at http://mathforum.org/library/drmath/drmath.high.html and http://mathworld.wolfram.com/about.html (this is really cool, it has the terms hypertexted so that you can get a more detailed explanation).

[Sundog]

Originally posted by Soapy Sam

I asked this question in high school and never received a straight answer. It's a question about definitions, in simple English. So, with no hedging please. No "complex / imaginary" voodoo. No Argand diagrams or Laws of Form.

Which , if either, of these statements, is true? Is i a number or a process?

Soapy:

Think about this for a second. You accept the arithmetic rules regarding negative numbers. Did you know that when negative numbers were introduced, many mathematicians thought the idea was as ridiculous as you think imaginary numbers are?

Also think about this. Mathematicians are about as far from being woo-woo as you can get. It behooves us to try to understand them, not to tell them they are foolish for believing voodoo. That voodoo happens to describe the Universe we live in quite well.

[LaserCool]

Originally posted by swellman
Slight quibble - electrical engineers make the distinction of using j for imaginary numbers. Most other engineering disciplines use i.

How amusing. Physicists use i since j is already taken for current density.

The term i also appears frequently as a component of complex waves (e.g. Schrodinger's equation, decomposition of EM waves in E-fields, B-fields and polarization), and in eigenstates of a given wavefunction.

[Walter Wayne]

EE's use J for current density and j for iota. So we distinguish with capatilization.

Walt

[Brown]

Where I went to school, EEs also had to have a good background in mechanical concepts, particularly statics and dynamics. In those fields, i and j (and k) were orthogonal unit vectors.

[Hydrogen Cyanide]

Originally posted by Brown
Where I went to school, EEs also had to have a good background in mechanical concepts, particularly statics and dynamics. In those fields, i and j (and k) were orthogonal unit vectors.

And those of us who use the terminology can usually keep track of them within context... including the use of the required list of definitions in front of the memo/report.

Unfortunately I was never able to convince the finance guy to include definitions in his memos. They were alphabetic gobbly-gook, and I could not convince him that the "PSD" he required my boss to submit was to ME either a "Power Spectral Density" or "Pressure Spectral Density". He just thought it was just a bunch of noise.

I like the idea of your school teaching EE's some statics and dynamics. My EE hubby managed to skip that part... including barely passing the required Differential Equations. Oh, well. He is a computer guy, he really only has to count in 0's and 1's (okay, that is not quite true since he has been working in database management lately).

[jj]

Originally posted by Soapy Sam
A question:-
I was clearly told in school
Multiplying two negative numbers always yields a positive product.

It seems to me that if this is true, then i cannot be a number.

If i is a number, then multiplying negatives CAN give a negative product.

I asked this question in high school and never received a straight answer. It's a question about definitions, in simple English. So, with no hedging please. No "complex / imaginary" voodoo. No Argand diagrams or Laws of Form.

Which , if either, of these statements, is true? Is i a number or a process?

Error of excluded middle. i is neither negative or positive, it is imaginary.

That is the fact. You may reject the facts as voodoo, but then you're simply rejecting the answer by resorting to religion.

[69dodge]

Originally posted by Soapy Sam
A question:-
I was clearly told in school
Multiplying two negative numbers always yields a positive product.

It seems to me that if this is true, then i cannot be a number.

If i is a number, then multiplying negatives CAN give a negative product.

I asked this question in high school and never received a straight answer. It's a question about definitions, in simple English. So, with no hedging please. No "complex / imaginary" voodoo. No Argand diagrams or Laws of Form.

Which , if either, of these statements, is true? Is i a number or a process? There are different kinds of number, each with its own definition. What is voodoo-like about defining a certain concept and giving it the name "complex number"?

2 is a real number; multiplication by 2 can be thought of as the process of doubling in size. i is a complex number; multiplication by i can be thought of as the process of rotation by 90 degrees about the origin.

i is a number. Which two negative numbers do you think yield a negative product when multiplied? Keep in mind that the real numbers can be partitioned into negative numbers, zero, and positive numbers; but the complex numbers cannot. A complex number that isn't real is neither negative nor positive.

Did I help at all?

[Soapy Sam]

Hoo! I stepped on some toes this time.
OK cards on the table. I'm piss poor at math, always was. I have always been frustrated by the fact. Pardon my pencil envy. I think many others are like me- not wholly stupid, but taught in a way they could not understand. Could be bad pupils, could be bad teaching. A few responses to some of the points lobbed at me may make my position clearer. Keep yelling. I'm listening. I suspect some folk here may be better teachers than I had in school.
I apologise for the length of this, but it's important to me and possibly to others.

Sundog said:-
'It's completely simple. Unfortunately, you have already rejected the answer as "voodoo".'

I don't think it is completely simple. If it was, a lot more people would understand. If it truly IS simple, then there must really be a lot of poor math teachers out there.
+++
'Why do you accept some things you're told, like multiplying two negatives equalling a positive, and reject others?'

Fair question. (Remember we are talking 32 years ago). The honest answer would have to be that I was told one thing first, by a teacher, in a subject where, so far as I could tell, the rules were arbitrarily made up by teachers.
When subsequently told something contradictory, I was stumped. If it was not contradictory, then the reason why was not explained.
+++

'\Math is not English, and you can't learn it if you fight the knowledge. An accepting mind is essential, or you'll never get it.'

Swap the word "Religion" for "Math" in that sentence. Does it make as much sense? It does to me, and I never heard a sensible religion yet.

Sundog, I'm serious here. I did not (and, sadly still do not) speak "mathematics". I speak English. When you explain a complex idea , whether in math, physics, or morality, you may REQUIRE a specialised language; but first you must teach that language. If the only common language a teacher and pupil have is (eg) English, then the concept must be taught in English, not in "Mathematics", whatever that is.
If two statements in English are contradictory, then one, or both must be false.
Statement 1. Negative numbers multiply to a positive product. Always.
Statement 2. i is the square root of a negative number.

That's linguistically unacceptable. It was to me then. It still is.
If there exists a third possibility it needs to be stated. Better, it should have been clarified way back when negative numbers were introduced. Tell kids only two possibilities exist, then slip in a third. Is this clarity in education?
+++


Rgwuinn said 'We will do that for you, right after you explain why water flows down hill-but you cannot use words such as gravity, potential, density, hydrogen, oxygen, compound, slope, mass or kinetic--or God, Jesus, Zeus, Thor, or any other deity.'

You illustrate my point well. A common vocabulary is required to explain any complex subject.You start simple and get complex. I'm not clear in what way your comment is intended to move this discussion along though.
+++
Sundog again..

'Think about this for a second. You accept the arithmetic rules regarding negative numbers. Did you know that when negative numbers were introduced, many mathematicians thought the idea was as ridiculous as you think imaginary numbers are?'

Yes, I had read that. It seems to illustrate my point, that people have difficulty fitting new concepts into their mindset. This is worsened if the concept seems to contradict their experience. (What does "negative two" cows weigh, for instance. This is a valid question. It can't be shrugged off as meaningless. It must be explained). I recall when we were shown "negative" numbers for the first time I did feel similar confusion. I got over that.
I don't know why I could live with that and not with the concept of complex number, which came years later; I suppose because there is no actual contradiction in negative number-it's just "counting downward". (Oddly, I have no memory of being given the concept of a "number line". I encountered it years later in a book about the psychology of teaching mathematics in schools.)
+++
Sundog once again-
'Also think about this. Mathematicians are about as far from being woo-woo as you can get. It behooves us to try to understand them, not to tell them they are foolish for believing voodoo. That voodoo happens to describe the Universe we live in quite well.'

Here I disagree big time. Isaac Newton was as big a woo-woo as anyone: Several mathematicians of my acquaintance hold some very strange views. John Napier was widely held to be in league with the devil. I know of no evidence that being a mathematician makes someone less prone to wierd beliefs than anyone else. Also, I can't agree that it is my job to understand mathematicians; It is the job of any academic or scientist to understand his subject. It may be incumbent on him to teach it as well and as widely as possible, though this is a matter of opinion. Why must we understand mathematicians, as opposed to biologists, bakers, ballerinas etc? Understand mathematics yes. I agree. That's what I'm trying to do. You may break the logjam. Keep trying.
+++

jj-( woops, the heavies are wading in)
'Error of excluded middle. i is neither negative or positive, it is imaginary.That is the fact. You may reject the facts as voodoo, but then you're simply rejecting the answer by resorting to religion.'

THIS is interesting. I'll come back to why when looking at what 69Dodge has to say, but first, I beg you to read that sentence with the mind of a not very mathematical 16 year old:-

"i is neither positive nor negative, it is imaginary" I ask you in all seriousness, jj, what meaning has that sentence IN ENGLISH.
In English, it means. "i does not exist." It's imaginary. Like Dr.Who and Santa Claus.
I'm not resorting to religion here, I'm resorting to simple English.

Now jj, I appreciate the label "imaginary" was not your choice. But in the entire English lexicon, could we have chosen a stupider label to use?

About now you are likely throwing up your hands in disgust and excoriating me as a pedantic, MP3 liking loonie. But I return to my earlier theme- If English is the only common language you and your pupil have, then you must teach in English. I'm not arguing about complex numbers. I'm arguing about teaching. If there are only two "signs " of number, (+ and -), there IS no middle to exclude. What does "imaginary" MEAN, in English?
+++
69Dodge said 'There are different kinds of number, each with its own definition. What is voodoo-like about defining a certain concept and giving it the name "complex number"?'

Nothing; but equally, what would be wrong with defining "the number of the beast" and assigning it properties? We would both agree it's nonsense (I hope),we may even agree it's an "imaginary" number, but there is nothing in your statement which would enable us to do so; we would use our experience of "the real world", which is precisely what high school students of mathematics lack. I knew in primary school that there was no such thing as "minus two" cows. It's an accounting mechanism (as imaginaries are in circuit design- a useful tool.) Negative numbers can't be concrete quantities. Imaginary numbers may have any number of properties, but these can't be defined in contradictory terms if you expect kids to get the idea. They must make sense in terms of what they know already.

'2 is a real number; multiplication by 2 can be thought of as the process of doubling in size. i is a complex number; multiplication by i can be thought of as the process of rotation by 90 degrees about the origin.'

Thaks for that, it's a succinct description. Back to English though. How do you rotate a number about an origin? Explain the operation, if you will, in terms of the properties of numbers already known to most 16 year olds. (ie addition/ subtraction/ roots etc and the operational properties such as commutation. Please. In words I can grasp. How do you rotate a number about an origin?
+++
'i is a number. Which two negative numbers do you think yield a negative product when multiplied? Keep in mind that the real numbers can be partitioned into negative numbers, zero, and positive numbers; but the complex numbers cannot. A complex number that isn't real is neither negative nor positive.'

I'm not saying ANY -ve numbers give a -ve product. But is i not being defined as the root of -1?
If a complex number is neither negative nor positive, then how is its sign (if any) defined? This seems to be the same thing jj is trying to get across. Between the two of you, a chink may be opening here. What is the third option?
(Please don't say "imaginary", that's what I mean by "voodoo". Try to explain it , as to a slow, but striving teenager. What is this third option?)

'Did I help at all?'
Yes. If nothing else I'm getting my primal rage at failure all those years ago into the open. I truly appreciate everyone's attempts. I genuinely am baffled, and annoyed that I'm baffled. If there really is a simple way to grasp this , then I want to know it , so I can write it down in simple English, wrap it around a brick and throw it through the window of several high school textbook publishers. Keep trying.I'm listening. (Probably some other good folk are too.)

[Zombified]

The most general "number" is a complex number, which takes the form

a + bi

where a and b are real numbers and i is sqrt(-1). If you've got two such numbers, you can add them:

(a+bi) + (c+di) = (a+c) + (b+d)i

(Rearrange terms and factor out i).

You probably seen lines of numbers laid out like a ruler. Instead of a line, think of a plane, where the real part (the part not multiplied by i) is along the x axis, and the imaginary part (multiplied by i) is along the y axis. A complex number is a point on this plane. If the number is a+bi, the coordinates are (a,b).

If you have just a real number with no imaginary part, it lies on the x axis. If you multiply it by i, you get a point on the y axis.

If you take an arbitrary complex number and multiply it by i, you get:

(a+bi)i = ai + b (i^2) = -b + ai

So by multiplying by i, you've switched the components, and changed a sign. This is, in fact, a rotation by 90 degrees (try a few examples on graph paper if you don't believe me).

Imaginary numbers are just a subset of complex numbers where the real part is zero. Real numbers are just a subset of complex numbers where the imaginary part is zero.

(Compare: integers are just fractions where the denominator is 1. Similar situation.)

Now, why bother? Well, one example, not all polynomials can be factored if you don't have complex numbers. For example,

(x^2 - 1) = (x+1)(x-1)

but

(x^2 + 1) = (x+i)(x-i)

(Multiply and see.)

More importantly, complex numbers are very useful. There are a lot of problems in physics and many engineering fields that are easier to solve (and take less algebra) when you can deal with complex numbers.

Mathematics is just logic. There is nothing arbitrary or circular about it. Once you start with a set of definitions, there are conclusions that inevitably follow. A lot of these things have been worked out and written down, and a lot of them are very useful.

There are a lot of abstractions in mathematics, not all of them correspond to things that people are immediately familiar with. There's nothing to be done about that except to learn what the definitions mean and what things must follow from those definitions.

Hope that helps.

[Martin]

Right, you lot. F*cking no one mentions quaternions.

[rwguinn]

Originally posted by Soapy Sam
Hoo! I stepped on some toes this time.




Rgwuinn said 'We will do that for you, right after you explain why water flows down hill-but you cannot use words such as gravity, potential, density, hydrogen, oxygen, compound, slope, mass or kinetic--or God, Jesus, Zeus, Thor, or any other deity.'

You illustrate my point well. A common vocabulary is required to explain any complex subject.You start simple and get complex. I'm not clear in what way your comment is intended to move this discussion along though.
+++


Sorry if I sounded petulent. The real point here is that the "imaginary" number "i" requires a mathematical language to prove its nature. Its actual value can be stated in English and arithmetic as "the square root of negative 1", that is, i^2=(-1)
Start with that as a hypothosis. Now arrange experiments to prove it. That would be very difficult, semanticaly. Mathematically, it fits and works.
In a similar vein, the existance of the planet Pluto was hypothosized long before it was found, simply by its effect on the other planets. Its nature was known-even to where it should be, but it was not visible.
"i' can be judged to exist, because certain physical phenomina demand that it exist in order that they may be expressed mathematically. Physical vibration and electrical current/voltage relationships are among them.
So when I state that it cannot be described with words, other than the definition, that is what I am getting at.It it not a matter of "faith", but a provable phenominon.

[jj]

Originally posted by Soapy Sam
"i is neither positive nor negative, it is imaginary" I ask you in all seriousness, jj, what meaning has that sentence IN ENGLISH.
In English, it means. "i does not exist." It's imaginary. Like Dr.Who and Santa Claus.
I'm not resorting to religion here, I'm resorting to simple English.

Now jj, I appreciate the label "imaginary" was not your choice. But in the entire English lexicon, could we have chosen a stupider label to use?


Well, I must agree that the choice of the term "imaginary" was somewhat, well, interesting.

But it's a number, defined by the idea that it times itself is equal to -1.

It is neither positive or negative. Since they chose to use the term "imaginary", that's what it is. On a line of real numbers (btw, "real numbers" came first, so that is part of why "imaginary" came into play) it's at zero, because it has no "real" part.

The contradiction you see in - times - is + and sqrt(-) is i, i*i is -, is not a contradiction. 'i' is not negative or positive. It is something else, and the term chosen, since it was not on the real axis, was "imaginary".

Btw, I'm hardly the only "heavy" here...

And you might be surprised to find that you have better math skills than you think. You can at least describe what you're thinking, which passes most of the population in turn 1.

[Soapy Sam]

"i' can be judged to exist, because certain physical phenomina demand that it exist in order that they may be expressed mathematically.
+++
This looks worryingly like circular argument to me, RG.- "x must exist, because if it exists it helps explain y." But there might be other ways to explain y. Would they all be true?
I think we might also haggle over the meaning of "exist". If a symbolic concept exists in someone's mind, does this imply the existence of the object symbolised? I hope not, or we have to accept psi, esp and angels dancing on pinheads as real phenomena.

I'm playing with words here though. I think to move me on, I need to know about the alternative to "plus" and " minus", which seems to be what is meant by "imaginary" in the mathematical sense. I have no iea what this means and hope someone can give me a way into understanding it.

I find understanding a concept may be triggered by a number of approaches. The best is either a graspable example or a sound analogy. The problem in abstract math is finding them. I wonder how many engineers understand imaginary numbers because they have an understanding of circuit behaviour to exemplify it, rather than the process operating the other way around?

One thing about this forum- you may not get smarter, but you get to be ignorant with greater certainty. :)

[jj]

Originally posted by Martinm
Right, you lot. F*cking no one mentions quaternions.

(whack!) Splat! Nerf bat to Martinm! SPLAT! (whack!)

:D :D :D :D

I'll square your quaternions and raise you an infintesimal!

:p

[rwguinn]

Originally posted by Soapy Sam

This looks worryingly like circular argument to me, RG.- "x must exist, because if it exists it helps explain y." But there might be other ways to explain y. Would they all be true?


That's where the circular argument breaks down.:) Nothing else can explain the phenomina! While something else might work for particular instances, only a value where i^2=(-1) fits ALL cases (lets stay with Newtonian physics, where x' (velocity) is much less than C, OK? I haven't checked it out in relativistic physics, and can't speak for that)!
Rather than go into a bunch of equations where I try to prove how smart I am and actually prove the reverse :D , let it suffice to say that when the equation of motion is converted from the time domain to the frequency domain, the only thing which allows the equations to reflect reality involves the square root of -1.

[Walter Wayne]

The argument of existence seems wierd to me. 'i' is a purely mathematical construct which just so happens to make solutions to many equations simpler.

For instance, in electrical circuit if you add the right imaginary source to your circuit the solution is easier to find. The solution found has a real and imaginary part; the real part being the solution to the original circuit, and the imaginary part of the solution being a result of the imaginary source.

No engineer believes the imaginary source ever existed in the circuit. However, defining one simplifies many problems.

Do imaginary numbers exist in the world? No. Do they exist in math? Yes, because we define it and derive its properties based on the definition.

Walt

[69dodge]

Originally posted by Soapy Sam

"i is neither positive nor negative, it is imaginary" I ask you in all seriousness, jj, what meaning has that sentence IN ENGLISH.
In English, it means. "i does not exist." It's imaginary. Like Dr.Who and Santa Claus.

What does "imaginary" MEAN, in English?If you want to know what a "big house" is, it suffices to look up the words "big" and "house" in a dictionary: a "big house" is simply a house that's big. I don't think the same approach is useful in trying to understand what an "imaginary number" is, however. There are no definitions of "imaginary" and "number" I could give you that would enable you to say, "Ok, I get it now. An 'imaginary number' is simply a number that's imaginary."

Treat the phrases "imaginary number" and "complex number" as indivisible units. They're just technical terms whose meanings need not bear any relation to the familiar meanings of the words "imaginary", "complex", or "number".
I knew in primary school that there was no such thing as "minus two" cows. It's an accounting mechanism (as imaginaries are in circuit design- a useful tool.) Negative numbers can't be concrete quantities.Negative numbers aren't cows, but neither are positive numbers. All numbers are just mathematical constructs. They're just ideas. They can be usefully associated with properties of real-world objects, but they aren't identical to them.

Is the temperature in winter any less concrete than the temperature in summer, simply because it's represented by a negative number? Numbers can be used to count cows, but they have many other uses too. Don't confuse a particular application of numbers with the abstract concept of number.
'2 is a real number; multiplication by 2 can be thought of as the process of doubling in size. i is a complex number; multiplication by i can be thought of as the process of rotation by 90 degrees about the origin.'

Thaks for that, it's a succinct description. Back to English though. How do you rotate a number about an origin? Explain the operation, if you will, in terms of the properties of numbers already known to most 16 year olds. (ie addition/ subtraction/ roots etc and the operational properties such as commutation. Please. In words I can grasp. How do you rotate a number about an origin?I guess I was a bit too succinct. :)

You don't rotate complex numbers. You multiply them. You rotate points in a plane. However, you can set up a correspondence between complex numbers and points in a plane. (The real and imaginary components of a complex number are the x- and y-coordinates, respectively, of the corresponding point.) Then, the (complex number) result of multiplying a complex number by i corresponds to the (point) result of rotating the corresponding point.If a complex number is neither negative nor positive, then how is its sign (if any) defined?It's not defined. Complex numbers don't have signs. (This troubles astrologers greatly, but otherwise it's not really a problem.)

I am going to tell you exactly what complex numbers are. What I say will probably not help you understand why they are so useful. For that, you need to learn about all the ways that they are used. But it should at least convince you that they have nothing to do with voodoo.

A complex number is a pair of real numbers.

That's it. Really.

Here's a complex number: (3.5, -17).
Here's another complex number: (-17, 3.5)
(These two are not the same. The order matters.)

Well, ok, that's not quite it. There are some rules for adding and multiplying complex numbers. There are good reasons, of course, why people picked the rules they did. I won't discuss those reasons right now. But, in principle, the rules are arbitrary. If some mathematician makes up a crazy rule and calls it "addition", well, then, that's what addition is. Definitions can't be wrong. They're definitions.

The rule for addition is easy. You just add the first numbers together, and add the second numbers together. For example,<blockquote>(1, -2) + (10, 20) = (11, 18)</blockquote>Multiplication is a bit stranger. Here's its rule:<blockquote>(a, b) x (c, d) = (ac - bd, ad + bc)</blockquote>For example,<blockquote>(1, 2) x (3, 4) = (3 - 8, 4 + 6) = (-5, 10)</blockquote>Now, here comes the interesting part of the story. Let's look at only those complex numbers whose second component is zero, for example, (-1, 0) or (5, 0), and try adding and multiplying them using the rules for complex arithmetic given above.<blockquote>(-1, 0) + (5, 0) = (-1 + 5, 0 + 0) = (4, 0)
(-1, 0) x (5, 0) = (-5 - 0, 0 + 0) = (-5, 0)</blockquote>How about that? They behave just as real numbers do using the normal rules of arithmetic, except for that second component of zero tagging along. If we were being somewhat sloppy in our speech, we might even say that the complex number (-1, 0) is the real number -1, for example.

And now for the clincher. Try multiplying the complex number (0, 1) by itself. What do you get?

Now you know what i is. And there's no voodoo in sight.

[xouper]

69dodge: Complex numbers don't have signs. (This troubles astrologers greatly, ...<blockquote>http://www.xoup.net/img/roflmao.gif</blockquote>Who says mathematics has to be dry and humorless.

[Soapy Sam]

WW- As you imply, this is one of those areas where math shades into philosophy. (Or do I mean epistemology? Gods, all that beer, all those lost neurons!)

I still am uneasy with the business of defining something to suit the result we want to get, then taking it to be in some sense actual. (I'm avoiding the word "real"). The argument that "it works" held for lots of other ideas which turned out to be approximations to the truth, or just plain wrong.

If, as RWguinn says, 'the equation of time can only be solved using root-1', why did that lead mathematicians to assume that root-1 existed, rather than concluding the equation of time was in error somehow? To be dangerously simplistic, if I was balancing my bankbook and found the only way I could do so was by invoking root-1, I would assume that I had made a serious (if possibly brilliantly original) error in arithmetic, not that I had discovered a useful technique, no matter how good the accounts looked!

I suppose this is a line of divergence between math and (observational) science. Lacking that intuitive ability to mentally use the concepts, I and people like me, tend to fall back on the assumption that we just CAN'T understand mathematics. I know I will never use such techniques with comfort, but I keep hoping I can understand in my terms. For that I need a non mathematical explanation. Aye, there's the rub.

However, several of the comments here have got me thinking. Like Sundog, I think it's time to haul down some dusty books and try again. They may not be math books though.

Finally, let me apologise for arguing from a position of ignorance. I realise it's annoying to those who understand, but ignorance is all I've got.:D

[Soapy Sam]

69Dodge. Your post appeared while I was composing mine.
Thanks for that. I'm off to lie in a cold bath and think.
The comment that definitions can't be wrong is something I spent several years trying to get a math teacher to admit. It worried me deeply as a kid. (I was that kind of kid).
I'm increasingly certain that high order mathematical ability is evidence of serious neurological dysfunction. To help my understanding, I shall quaff more ale. Maybe I still have too many neurons left and they cancel each other out or something.
:D
Seriously. I do appreciate the help. I'm going to drag out the books. (Not for the first time).

[rwguinn]

Originally posted by Soapy Sam
69Dodge. Your post appeared while I was composing mine.
Thanks for that. I'm off to lie in a cold bath and think.
The comment that definitions can't be wrong is something I spent several years trying to get a math teacher to admit. It worried me deeply as a kid. (I was that kind of kid).
I'm increasingly certain that high order mathematical ability is evidence of serious neurological dysfunction. To help my understanding, I shall quaff more ale. Maybe I still have too many neurons left and they cancel each other out or something.
:D
Seriously. I do appreciate the help. I'm going to drag out the books. (Not for the first time).

Whaddya mean, serious neurological dysfunction??!! If you place ale in front of us, do we not drink it? No disfunctional person can do that!

btw- I was referring to transforming from time domain to frequency domain - it is another type of coordinate transformation.

Just as the idea that -1 mile North is actually 1 mile South takes a change in perspective from knowing that you cannot weigh -1 cow (although if someone owes you a cow, you may, in a fashion, actually possess -1 cows ;) ) , the complex numbers require a change in perspective, and new rules.
-1 mile North + 1 Mile East is, in truth, 1.4142... miles Southeast. This seems to violate the rules of arithmetic-but we are now talking 2 dimensions, not one- we are into geometry and trigonometry. There are new rules tacked on to the old ones. The arithmetic rules are still valid- but they now have restrictions on the circumstances in which they may be used.

Feel any better? I think I need a "Hatch-toberfest" brew about now...:D

RW

[jj]

Originally posted by rwguinn


That's where the circular argument breaks down.:) Nothing else can explain the phenomina! While something else might work for particular instances, only a value where i^2=(-1) fits ALL cases (lets stay with Newtonian physics, where x' (velocity) is much less than C, OK? I haven't checked it out in relativistic physics, and can't speak for that)!
Rather than go into a bunch of equations where I try to prove how smart I am and actually prove the reverse :D , let it suffice to say that when the equation of motion is converted from the time domain to the frequency domain, the only thing which allows the equations to reflect reality involves the square root of -1.

Rich, how about, say, the DCT domain? The DCT is a frequency transform, and the "outputs" are real values.

I would suggest that we could say that 'i', construct or not, is successful in modeling a great lot of things.

It's like the variable in the Schroedinger Wave Equation. It represents nothing concretely physical, but interpreting it's modulus has proven stunningly useful, none the less.

To Soapy...

Mathematics IS an abstraction. The "existance" of something isn't really a germane question, because you're tying math to physics. Mathematics is a language that is used to describe what happens in physics, but it's not physics.

The use of 'i' to develop explainations, theories, etc, does not require 'i' to be anything other than abstract. There is no reason for it to "exist" or "not exist" physically, if, by using it and the properties that it can be shown to have, you can develop a set of equations that (for instance) model harmonic motion, etc.

[jj]

Originally posted by Soapy Sam
I wonder how many engineers understand imaginary numbers because they have an understanding of circuit behaviour to exemplify it, rather than the process operating the other way around?


I think you've just pointed out a very important difference between a mathemetician and an engineer. The engineer may indeed have a really solid intuitive understanding of how a+bi applies to a circuit design, impedence, what-have-you.

The mathemetician has a STRONG ABSTRACT UNDERSTANDING. Mathematics is abstract. Mathematics is a system wherein assertions can be made, evaluated, etc, inside the system. In "standard algebra" one can construct, if one uses the rules carefully, an infinite number of correct assertions, and know that they are correct just because they are constructed correctly.

One can also write down assertions that *may* be correct but can NOT be constructed (Goedel's theorem here), or assertions that can be shown to be wrong.

But mathematics is abstraction, plain and simple. As I think I said somewhere else the variable in Schroedinger's wave equation is another example. It, by itself, represents nothing particularly interesting, but its modulus (a^2 + b^2) is remarkably useful.

There are many similar things, but the Guinness is already flowing, I fear. :)

[Soapy Sam]

Yup. The ability to think (or perceive) in that abstract manner is one I lack and envy. I'm a verbal thinker and secretly believe everyone else is too, but there's a conspiracy to make me feel inferior. You're all kidding. Admit it.

(I nearly said it's like "Mornington Crescent", but I don't want to open that kettle of steam.).

I have an early plane to catch tomorrow.
Evenin' all.

[jj]

Originally posted by Soapy Sam
Yup. The ability to think (or perceive) in that abstract manner is one I lack and envy.

Actually, your questions suggest that you can think abstractly rather well. Now, nobody can "percieve" this, as it's abstract, so maybe you're trying to do something that is generally considered impossible.

??

[Soapy Sam]

Was it the Red Queen who said to Alice she often thought of six impossible things before breakfast?

I just bought a book on the value of impossibility in science. I'm just back in Kazakhstan after 3 days with about six hours sleep, so reading it is clearly one of the impossibilities.

Maybe I'll find what I'm looking for therein.

I admit I have gone through much of life in the vague belief that there was a critical lesson I missed in High School- about second year- when they gave out all the REAL answers.
(If this belief has no name in psychology, it should have. I reckon quite a few of us share it!)