Imaginary Numbers

Do you have a specific intent vermin , or is it just curiosity and general learning?

Maybe the people who hang out in Electricals may know a fair bit. They like to mess around with phase and so on , which drags into the icky world of i. Pure mathematicians use it , often as a sort of 'cheat' (that will get me shot). Dr Math seems to know a bit - you didn't say how advanced. Maybe it's a Linux thing.

I like playing with cubic equations but have no use for imaginary stuff that can be churned out.

Excellent!!!

Definition of imaginary number.

'The number women any bloke claims to have slept with!'

It's also true of women , but works on an opposite basis.

Dang - 8 minutes and you corrupted me Del ! I was going for a record. I am going to resist returning until I have something serious.

Back soon.

They are called complex conjugates :-)

Hey , thats good !

Hi Kris,

I just stumbed across that Falkirk Wheel thing. Fascinating bit of engineering from you lads over there. Ever took a ride on it?

Off topic I know vermin, but just had to ask about that thing. Wonder if any a+bi came up when they designed it?

It's very cool . This link has a video clip that I haven't viewed yet (I haven't seen the real deal yet , either ! ). The Anderton lift looks like one to keep on a list as well.

Stoke Bruere is great , I've been there lots but not for a long time . The inclined plane is here . I'm not sure where thet are with restoration plans. The whole are is great for narrow boat enthusiasts. The Pontcysyllte in Wales isn't for the faint hearted !

ps John ,

Off topic + vermin minding. ROFLMHAO ! Besides he already had some on-topic from me , he'll have to give feedback for more.

"imaginary and complex numbers"

Imaginary_number

Complex_number

• 6 Applications
• 6.1 Control theory
• 6.2 Signal analysis
• 6.3 Improper integrals
• 6.4 Quantum mechanics
• 6.5 Relativity
• 6.6 Applied mathematics
• 6.7 Fluid dynamics
• 6.8 Fractals

Courtesy of Wikipedia

Thanks, SS. I usually don't start there regarding things mathematical because it's often not very helpful, but this entry looks really good!

Much appreciated!

humph

Interesting link. My memory is of an experimental high school class in calculus (1963-4), in which one time we derived the proof that

e^iπ=-1. (that π is pi ... anybody for "transcendental" meditation?)

My "forgettory" suspects we used Fortran in grad. school with complex numbers and fourier transforms to calculate the spacial electron density distribution of crystalline molecules, to determine their 3-D structure.

Real world application of complex numbers is used to keep industrial plants from taking advantage of power companies by maintaining a low power factor. The power companies have figured out how to make watt-hour meters to measure VARS (imaginary power) as well as WATTS (real power), and since the power company has to produce the complex vector quantity, they want the industrial plant to pay for it. Years ago, they didn't . . .

and electric and magnetic field (link to #7)

quadrature modulation of two signal, like U, V signal in color TV.

AC calculate etc.

Vermin is an excellent computer programmer, he absolutely know the number. I think he must hope to attract most of us attation to creat a long long long... threads.

I vote and post the thread.

You are very perceptive cnpower. vermin probably wrote the authorative book on imaginary numbers , and is finding the answer to something else here. Who knows except vermin , he is mysterious.

to creat----to create(correct)

Have a look here cnpower . If the link near the bottom works you can have fun. A thread on CR4 is discussing "The Worlds most difficult problem".

Have fun !

visiting the web site, Its a relate to logic site, teach people how to thinking. some of them belong to simple symbolic logic. I forget almost! as well as logic, which studied by myself very long before. I thnk its very useful in my work. although its only simple formal logic.

Was the Chinese language bit near the bottom useful to you ?

One thing I did find of interest is through complex analysis there is a proof that the number of prime numbers that exist around any number is roughly equal to 1/ln(n). It's called the Asymptotic Proof of Prime Numbers. It seems to be fairly accurate. So, the larger the number, the fewer prime numbers are around it. Hmmmm.

Go play. It's quite a good site to explore.

"Go play. It's quite a good site to explore.Go play. It's quite a good site to explore."

'Priming' the 'pump' are we?

Did I mess up the link Stan ?

I think it is possible to work it out , but never mind eh. Soreee.

"Did I mess up the link Stan ?"

Intended to be a play on the words prime as in numbers and priming as in p. a pump.

Your original link leads to a URL, the copies to pdf's. ???

At least a break from the strain of complex numbers, etc.

Now the above link goes to the original URL! ???

I don't know which of us is more convoluted !

The original did , as you say , go to :http://primes.utm.edu/curios/includes/glossary.pdf , which will show at the address bar near the top.

If I link this bit http://primes.utm.edu/ you get to the root of things (so to speak). As you can see , this part can be cut and pasted from the longer bit above (to :http://primes.utm.edu/curios/includes/glossary.pdf) which displays in the browser when you click my link to the pdf. Curiously it got me to a higher level in that site than I had been looking at before. To add to confusion , I can paste the entire pink bit above behind a nice word such as repeat and it does exactly the same (you can ignore underlines in all this). You highlight a typed word in your own text , click on the globe shaped bit in the editor box , and paste the link into the URL space. You may well know all this , but I am just trying to discover how hopeless I am at explaining all of this. My conclusion is that it works , and I have not the slightest idea how. It is however good fun to play with. I am using IE7 as a browser. It has tabs for multiple windows unlike it's predecessor. There is nothing I know of to prevent opening whatever browser you use multiple times concurrently. Indeed I occasionally use Firefox and IE7 at the same time.

I think I have fallen into some kind of literary equivalent of complex numbers. At any rate , I am possibly complex and almost certainly numb in the head ! Sorry if this is about as clear as custard but my mind-set is stuck in pen and paper days.

Why does every Englishman make references to custard. Do you have custard festivals in England.

"Oh, Hans! Quick! Bring the vat, it is Custardfest and we do not want to get caught with our Lederhosen down!!"

Does this have something to do with Wensel dancing?!

What's Wensel dancing ?

Custard is just inherently funny . There is a phrase 'Custard pie politics' which may make this clearer.

Sorry, what I meant to give you grief about was this little bit of English culture... Morris Dancing!!! Spelled with a capitol "D" for dork... What a bunch of bleeding' pansies!

If not pansies, just morons with no shame!!! And just where the hell are their sticks?!

You got from 'Morris' to 'Wensel' ? What a crouton.

Yes , Morris dancing is not exactly my bag either. On the other hand , I'm told it's good for pulling birds and getting bladdered. A Smurf like you would be confused by my idiom. Living on the West coast , you'll be off your head anyway.

And another thing , I saw Rikki Lake dancing so you may as well shut up.

Flippin' Canadians.

not very useful. its only for middle school level students.

I cannt open the chinese anguage page, because different encode. thanks at the same.

There is a very well written book "Visual Complex Analysis" that explores the history (briefly, but with further references) and the applications of "i".

From a philosophical point of view, "i" demonstrates that the symmetry of Nature extends beyond the +/- of simple objects or rules of the first power (say parameter "x"). Symmetry also applies to the higher powers (say parameter "x², x³..."). Euler's identity demonstrates the equivalence between the exponential and sinusoidal representations. This book shows very eloquently how the points "zero" and Infinity" are just two more points in a spherically structures universe.

Here is a link to the Website, where you can download parts of some chapters. http://www.usfca.edu/vca/

I couldn't help but notice that real numbers are a subset of complex numbers! (Leaving Descartes to his idea of imaginary numbers). All that needs to be done is setting the coefficient of the imaginary part equal to zero.

So, if a+bi = z, but if a+0i = z, then a = z and real numbers are a subset of complex numbers!!! QED (Well, maybe not that good).

Hi vermin,

To answer your question:

You are probably asking "What use is all this?" [the value of "i"] It is of course impossible to collect 'i' objects, nor can you walk 'i' metres, nor can I lend you 'i' dollars or pounds.

However, it turns out that complex numbers are capable of providing analogies with certain situations, and helping to solve them. For example, when fluids are flowing or swirling round in a complicated pattern, the movement of each particle may be represented by a complex number on an Argand Diagram. Known rules for manipulating complex numbers (as compact, single entities) are then applied to examine what changes would be expected in the flow under specified conditions.

This is useful in meteorology. Forced to use ordinary algebra, two separate equations of motion are necessary, one for north-south airflow, and another for east-west flow. Coordinating two such equations is laborious, but with complex numbers just one equation will suffice, which turns out to be much simpler and easier.

Electricity is another branch of science which employs complex numbers. An alternating current always surges up with its maximum voltage, then decreases and swings the other way under negative voltage. That process may easily be represented on an Argand Diagram by a rotating arm. One end of the arm is fixed to the centre; the height of the other end shows the instantaneous voltage.

When an induction coil is inserted in a circuit, it pushes the alternating voltage out of phase by a quarter of a wavelength, which is equivalent to a 90-degree turn on our modified Argand Diagram. A capacitor produces a similar phase change, but in the opposite direction. The overall result then depends on the relative strengths of all coils and capacitors present.

On a drawing of a rotating arm, there is an amazingly simple rule for swinging it through 90 degrees - you just multiply the complex number by 'i'. Thus, when an electric circuit is considered in terms of its corresponding Argand Diagram, all associated formulae suddenly become quite neat and compact, even if they do contain a sprinkling of i's.

Of course it is not possible to have an i-number of volts or amps or ohms, but the size and phase of the resulting voltage may easily be deduced from those formulae.

Incidentally, electricians write 'j' for Ö (-1) instead of 'i', because they use 'i' for current. Often, they will be dealing with an extended network of several interacting circuits: that would be a really formidable problem without complex numbers.

I was unfamiliar with Argand diagrams but the document covers them quite well. I, too, have wondered about practical applications for complex numbers and now have a better appreciation of them.

Regards,

-John

Thanks, Johnjohn! Looks like a good read. While posting on CR4 my interest in math started coming back to me, and I remembered how valuable complex numbers are. However, I couldn't remember exactly how they were used in everyday engineering and science. I do know there are some really neat relationships between Euler's number and complex numbers.

Well, guess it's time to get the old math books out again!

"While posting on CR4 my interest in math started coming back to me, and I remembered how valuable complex numbers are."

Same here! That's what I like about these threads. They're not only good for some side-bustin' humor, but sometimes they make you get down and really think. Usually, you think about things that you thought you'd forgotten to think about(?)

Anyway, regarding complex numbers check out the thread 1=2 if you haven't already. Some interesting thought providers regarding √-1. (Roger Pink, myself, and others have been discussing proofs, fallacies, absurdities, algebraic rules, etc.). Like to hear you weigh in.

Regards...

Will do.

Excellent post John !