# Square root of minus one

sonhouse
Posers and Puzzles 27 Oct '07 13:46
1. 27 Oct '07 18:56
Originally posted by wolfgang59
I didnt know that!

But isnt it a matter of definition?

What if I define the 'size' of a complex number C where C = R +iI as

sqrt(I^2 + R^2) ... does that have a use?

🙄
Yes. That's the modulus of a complex number.

It comes in useful because you can write any complex number as
r e^(i theta),
where r is the modulus.

If you consider the complex numbers as lying on a graph, with the real numbers as the x-axis and the imaginary numbers as the y-axis, and you draw a line from the origin to the number, then r is the length of the line and theta is the angle between the line and the x-axis.
2. genius
Wayward Soul
27 Oct '07 19:18
Originally posted by sonhouse
But one peculiarity of the complex number is that no complex number is larger or smaller than any other number. Another one is that negative infinity is the same as positive infinity.

I follow everything but that one. It sounds like you are saying all complex numbers are equal?? And that (+inf) + (-inf)=0? and (+inf) * (-inf)= (-inf)^2?
Have you come across the different 'levels' of infinity? Countable and uncountable, and all the stuff inbetween that doesn't actually matter?

I haven't heard anything about -inf=+inf for complex numbers though. I may take a quick peek at that sometime soon.
3. 27 Oct '07 21:47
Originally posted by sonhouse
But one peculiarity of the complex number is that no complex number is larger or smaller than any other number. Another one is that negative infinity is the same as positive infinity.

I follow everything but that one. It sounds like you are saying all complex numbers are equal?? And that (+inf) + (-inf)=0? and (+inf) * (-inf)= (-inf)^2?
If you have to complex numbers z1 and z2, Then they are either equal or not equal. If they are not equal, there is no way to tell wich one is larger or lesser, and no need to tell either.

Your example of (+inf) + (-inf)=0 doesn't work in the real numbers neither. The absolute of a complex number can only grow to inf, not +inf or -inf.
4. sonhouse
Fast and Curious
27 Oct '07 22:21
Originally posted by FabianFnas
If you have to complex numbers z1 and z2, Then they are either equal or not equal. If they are not equal, there is no way to tell wich one is larger or lesser, and no need to tell either.

Your example of (+inf) + (-inf)=0 doesn't work in the real numbers neither. The absolute of a complex number can only grow to inf, not +inf or -inf.
But if you have say, 2+3i v 4+3i, why isn't the second value larger?
or 2 + 3i vi 2 + 4i, why isn't the second example larger?
It isn't larger on the real number line of course but on its own number line situated 90 degrees from the real number line the two should be considered one larger than the other.
Another issue I have thought about, if the complex number line is 90 degrees to the real number line, could there be further second generation complexities at 90 degrees to the first complex line? Could anything mathematical be gained from this arrangement? I was thinking along the lines of annotating jerk (changing acceleration patterns) or some such.
5. 28 Oct '07 05:421 edit
Originally posted by sonhouse
But if you have say, 2+3i v 4+3i, why isn't the second value larger?
or 2 + 3i vi 2 + 4i, why isn't the second example larger?

It isn't larger on the real number line of course but on its own number line situated 90 degrees from the real number line the two should be considered one larger than the other.
Another issue I have thought about, if the co ...[text shortened]... as thinking along the lines of annotating jerk (changing acceleration patterns) or some such.
But if you have say, 2+3i v 4+3i, why isn't the second value larger?
or 2 + 3i vi 2 + 4i, why isn't the second example larger?

Which one is larger? 4+3i or 3+4i? Or this one: 1 or i? In fact, there is no ordering property among the complex numbers. Only equal or non equal.

... could there be further second generation complexities at 90 degrees to the first complex line?

There is something called hyper complex number with 3 complex components along with the real one. Called quartenions or something. Usable in certain applications but abandoned by the matematicians because there is other ways to solve those problems with standard methods.
6. wolfgang59
Mr. Wolf
28 Oct '07 10:07
Originally posted by sonhouse
But if you have say, 2+3i v 4+3i, why isn't the second value larger?
or 2 + 3i vi 2 + 4i, why isn't the second example larger?
It isn't larger on the real number line of course but on its own number line situated 90 degrees from the real number line the two should be considered one larger than the other.
Another issue I have thought about, if the comple ...[text shortened]... as thinking along the lines of annotating jerk (changing acceleration patterns) or some such.
What about if we define 'size' of a complex number as its distance from the line R=-I in the complex plane?

Real numbers can thus be ordered as can purely imaginary numbers.

Of course there will be an infinite amount of numbers of any size but we get by with saying the size of +1 = size of -1

The infinities are still difficult though 🙁
Baby Gauss
28 Oct '07 10:23
Originally posted by wolfgang59
What about if we define 'size' of a complex number as its distance from the line R=-I in the complex plane?

Real numbers can thus be ordered as can purely imaginary numbers.

Of course there will be an infinite amount of numbers of any size but we get by with saying the size of +1 = size of -1

The infinities are still difficult though 🙁
http://www.cut-the-knot.org/do_you_know/complex_compare.shtml

http://library.wolfram.com/infocenter/MathSource/4802/
8. wolfgang59
Mr. Wolf
28 Oct '07 10:49
Thanks for the link Adam ... and I now have a new word TRICHOTOMY 😀
Baby Gauss
28 Oct '07 10:531 edit
Originally posted by wolfgang59
Thanks for the link Adam ... and I now have a new word TRICHOTOMY 😀
When I was on my first year at college I talked with my teacher about you order complex numbers and he told me that it can be done but it is a little bit useless even for mathematicians. I think I once even saw how it can be done in a book but I don't remember which book it was. 🙁

Edit: And trichotomy is a pretty wild concept in mathematics. Back in early 20th century a war waged between mathematicians because not all of them thought that it was a valid concept.
10. 28 Oct '07 11:40
Originally posted by wolfgang59
What about if we define 'size' of a complex number as its distance from the line R=-I in the complex plane?
(
With this definition we can have two complex numbers with different real and complex coeffiences yet be neither larger nor smaller, but equal in size? No, it doesn't work.
11. 28 Oct '07 12:25
Originally posted by FabianFnas
With this definition we can have two complex numbers with different real and complex coeffiences yet be neither larger nor smaller, but equal in size? No, it doesn't work.
Well, it "works", but you don't get all the properties of ordering that actually make it a useful concept. The links Adam posted seem to explain it quite well.
12. 28 Oct '07 14:18
Originally posted by mtthw
Well, it "works", but you don't get all the properties of ordering that actually make it a useful concept. The links Adam posted seem to explain it quite well.
Well, works and works... Matematicians doesn't use this property, as they do in the real numbers.

A question:
Fysicists tend to discard solution of equations that are complex. I know, of course that complex number are not unknown by them, but it seems that the complex number system is merely a tool of which real answer can be deducted.
Does anyone know somewhere in nature where complex numbers can be an answer? Like a complex distance, mass, time, or energy, or anything?
13. wolfgang59
Mr. Wolf
28 Oct '07 14:44
Dont understand it but complex numbers are used in electricity. I know they use 'j' rather than 'i' for imaginary numbers to avoid confusion with the standard I for current.

http://regentsprep.org/REgents/mathb/2C3/electricalresouce.htm
Baby Gauss
28 Oct '07 15:37
Originally posted by FabianFnas
Well, works and works... Matematicians doesn't use this property, as they do in the real numbers.

A question:
Fysicists tend to discard solution of equations that are complex. I know, of course that complex number are not unknown by them, but it seems that the complex number system is merely a tool of which real answer can be deducted.
Does anyone ...[text shortened]... complex numbers can be an answer? Like a complex distance, mass, time, or energy, or anything?
It depends on what you mean by use complex number. If you mean that a final answer of a physical number is given by a complex number than the answer is no. Because on the physical world everthing is best described by using the real numbers. But we physicist use complex numbers a lot. In electirc theory they are used more as a tool to solve problems more easily. The thing is that by using complex numbers you can transform differential equations into algebraic equations and they are real easy to solve.

But another field were complex numbers are really in the mix is quantum mechanics. Even though the answers depend one way or the other on the square of the modulus of the wave equation complex numbers are really a necessity and not just a commodity. Modern Quantum-Mechanics, a book by Sakurai, adresses this issue on a very clear way on his first chaper when he is discussing Stern-Gerlach type experiments
15. sonhouse
Fast and Curious
28 Oct '07 16:041 edit
Originally posted by FabianFnas
Well, works and works... Matematicians doesn't use this property, as they do in the real numbers.

A question:
Fysicists tend to discard solution of equations that are complex. I know, of course that complex number are not unknown by them, but it seems that the complex number system is merely a tool of which real answer can be deducted.
Does anyone ...[text shortened]... complex numbers can be an answer? Like a complex distance, mass, time, or energy, or anything?
q
THAT I know: Complex numbers are inherently involved with alternating current and RF, you have to use complex #'s to solve problems in current flow and voltages, say on an RF open wire feedline for instance, the current sine wave and the voltage sin waves are not in sync and require complex #'s to solve the real energy exchange, absorption or emission of RF from an antenna for instance. Without complex numbers, you would be left guessing, for instance, how much power is actually being transmitted. Fortunately all that math was done a century ago and now meters calibrated for that show the true powers, standing wave voltages, absorption, emission, reflection, return loss, etc., and we usually don't need to do complex math unless you are taking a ham license test or engineering test or designing new circuitry, but for the pro's, they just input parameters into Mathcad or other software pacs and don't have to do the math by hand any more.