discussion: are there any places that complex numbers are necessary?

[DGMP]

I mean that every scientific sector using them only for comfort treatment with the data. but you can do the hard calculations, formulas & all the things without the complex numbers. all the treat are like this just because the data is behave just like these numbers are behave...

for example, I found an equation that you must use complex to solve:
x = cosx

what do you think?

 

[csanders]

Laplace Transform

 

[goaowonk]

Partial differential equations.

Sent from my Deltaglider using Potatolk.

 

[Urwumpe]

They are also important for analysis of electromagnetic systems and alternating current systems.

 

[DGMP]

They are also important for analysis of electromagnetic systems and alternating current systems.

but they're there just for comfort...

 

[Urwumpe]

but they're there just for comfort...

That can be said of any mathematical concept. Why do we need prime numbers? Just for comfort? Why linear algebra, when trigometric solutions also exist? And can't we just solve everything iteratively?

 

[BLANDCorporatio]

but they're there just for comfort...

I'm an Engineer. With a little background in deeper math than what Engineering typically requires. What do you mean by the text quoted, explain yourself please.

That can be said of any mathematical concept.

Exactly. We use all kinds of math because it behaves, to the extent we care about, exactly as the thing we model with said math.

There is one situation I can think of where a particular mathematical notion is used only "for comfort", despite us having an exact model available, and it is quaternions to represent rotations. Technically, the group of rotations and the group of unit quaternions are not even the same mathematical object. So it's important to note, this is not a mismatch between a model and reality; we know mathematically what we want to model- rotations-, we have an exact model for it (special orthogonal matrices) and nonetheless use another mathematical model for it!

Quaternions have a nice relation to rotations, they -are- very convenient to use (more convenient in practice than matrices), and the difference between them and rotations matters nowhere in most practical cases.

 

[malisle]

but they're there just for comfort...

Formally yes, but keep in mind that in electrical systems the order of the differential equation is dictated by the number of energy containers - capacitors ,inductances and memristances. Even for a basic circuit that can become very high order differential equations that would be extremely impractical or even impossible to solve analytically. That makes phasor transformation somewhat non optional.

Also nobody mentioned quaternion transformation.

 

[Urwumpe]

Yes, mathematics is simply said, just a big toolbox and also the research for improving this toolbox with new better tools (which you do by proving that they solve a mathematical problem)

Yes, we use some tools more often for some problem, because it is more comfortable to solve them that way. But that does not mean that you should only know your few special tools.

The biggest fun in mathematics is digging in your own toolbox for solving a problem with multiple tools, that you are not able to solve by the tools you already know. It must not be the higher art of mathematics, no group theory or what the other dark magic mathematicians love. Even a simple differentiation at the right point can help finding a better solution sometimes.

 

[DGMP]

I'm an Engineer. With a little background in deeper math than what Engineering typically requires. What do you mean by the text quoted, explain yourself please.

you can represent the data just as points on plane, with axis' system.
you can express the relationship among the details with just real formulas,
but because that relationship is like the behavior of complex algebra, we use the complex numbers to simplify the calculations...

 

[RisingFury]

In quantum Physics, the wave function is by its very nature complex. It consists of a real and imaginary part.

 

[Spacethingy]

Crikey, do you even realise the number of maths jokes that are dependent on complex numbers? Without them, maths-based humour would be seriously limited in scope.

xkcd.com

 

[asbjos]

Yeah, I can't even imagine how it would be without them!

Sorry for the bad joke, I'll be quiet now...

 

[Urwumpe]

you can represent the data just as points on plane, with axis' system.
you can express the relationship among the details with just real formulas,
but because that relationship is like the behavior of complex algebra, we use the complex numbers to simplify the calculations...

So you mean, you can calculate with complex numbers?

And no, just by "points on plane" you can't get much joy of complex numbers, since you could use complex numbers in mixed vectors....

 

[DGMP]

Crikey, do you even realise the number of maths jokes that are dependent on complex numbers? Without them, maths-based humour would be seriously limited in scope.

xkcd.com

not only the humor...

 

[BLANDCorporatio]

you can represent the data just as points on plane, with axis' system.
you can express the relationship among the details with just real formulas,
but because that relationship is like the behavior of complex algebra, we use the complex numbers to simplify the calculations...

I'm still not quite sure what you mean here. That's like asking, why use, say, matrices at all, after all they're just blocks of real numbers, but because the relationships between the matrix components, the transformed from quantities, and the transformed to quantities, are like linear algebra, that's what we use ...

No. We don't use linear algebra or complex numbers to simplify calculations. We use them to know what calculations to make, to see what structures there are. (And yes, sometimes as quoted here also to get at easier solutions to a problem, as in Laplace transforms of differential equations).

 

[Linguofreak]

you can represent the data just as points on plane, with axis' system.
you can express the relationship among the details with just real formulas,
but because that relationship is like the behavior of complex algebra, we use the complex numbers to simplify the calculations...

AFAIK, the use of complex numbers in relativity is pretty much unavoidable.

 

[Gumok]

And don't forget that summing physical vectors in space is in fact the same thing as summing complex numbers. When you pick the first two vectors to sum, the plane when they are in is actually a gaussian plane with two complex numbers on it.

 

[DGMP]

And don't forget that summing physical vectors in space is in fact the same thing as summing complex numbers. When you pick the first two vectors to sum, the plane when they are in is actually a gaussian plane with two complex numbers on it.

so you can just use vectors algebra, why to put complex numbers, that they called as they are? complex can be existed only on plane, when 3D vectors can be also in all the space, & even 4D & more if you need to use the data like that...

 

[Gumok]

But vectors are working with coordinates separately unlike complex numbers that have inbuilt them right into their algebra. Complex numbers are not only in plane - if you want space, use quaternions (Wiki: "number system that extends the complex numbers. They were first described [...] and applied to mechanics in three-dimensional space.") or higher dimension numbers such as octonions.

As malisle pointed...

Also nobody mentioned quaternion transformation.