Complex Analysis: Finding Better Numbers for Math Problems

[alemsalem]

Complex analysis gives us theories about functions that u can't get without the complex algebra, could there be an extension to complex numbers that might solve important problems in mathematics..

Thanks to all..

 

[yenchin]

How about http://plus.maths.org/content/curious-quaternions" ?

 

[mathwonk]

this is a good question and has occurred to many people for a long time. As suggested, quaternions help solve certain problems, as in Herstein's topics in algebra where he uses integral quaternion i believe to solve the 4 squares problem.

There are also theorems telling us that only certain number systems exist having certain nice properties. In general you give up more properties as you expand your search. E.g. complex numbers are nice but they are not ordered. Quaternions are not even commutative. The only other real division algebra, the Cayley octonians, is not even associative. So there is a limit to how many useful number systems one can hope to find, if "useful" includes having familiar properties.

In recent decades theoretical physics has provoked discovery of quantum cohomology theories which I know very little about, but seem to use power series as coefficients in place of ordinary number systems.

 

[AlephZero]

I agree this is a good question (and not one that you will find answered in most textbooks!).

Part of the answer is that unlike the real numbers, complex numbers are "complete" in the sense that any polynomial with complex coefficients can be fully factorized using only complex numbers. Or to put it a different way, all the roots of a polynomial with complex numbers are complex, not something "more complicated".

So if you want to use numbers to represent something about "the real world", the most obvious place to start would be the integers, but then you quickly find you need to extend you idea of "numbers" to include rationals, reals, and complex numbers to avoid lots of special cases that don't have any solutions - but once you have got to complex numbers, you don't need to go any further.

Quaternions etc are useful for certain types of problems, but in fact you can represent them in terms of complex numbers, so in that sense they are not really anything "new", just a convenient notation.

 

[Mute]

To add to the discussion of quaternions: invented by Hamilton, he originally advocated their use in mechanics problems. One could formulate mechanics problems in terms of quaternions: a + bi + cj + dk. However, they were reportedly rather cumbersome, and when Gibbs came along and invented vectors, it pretty much spelled the death of the use of quaternions in mechanics. If you've ever wondered why the spatial unit vectors in 3d were labeled i j and k, it's because of the original use of quaternions to do mechanics problems which vectors were eventually used for.

 

[lurflurf]

Much like people each number system has its good and bad parts. Complex numbers happen to be very well balanced. If we start trying to add good or remove bad we end up losing good parts and gaining bad parts.
The Cayley-Dickson Construction
The Cayley-Dickson Construction

 

[lavinia]

Complex analysis gives us theories about functions that u can't get without the complex algebra, could there be an extension to complex numbers that might solve important problems in mathematics..

Thanks to all..

While the other posts show other number systems that are important in mathematics and physics there is something especially important about the complex numbers. That is that they are the largest field containing the real number. There is no extension of them to a large field.

 

[Tac-Tics]

What is a number?

If you consider groups, function spaces, vectorspaces, manifolds, topologies, etc to be abstract variations of numbers, then mathematicians have spent a long time coming up with just about every kind you could imagine.

 

[Bacle]

Maybe you can also reasonably think of matrices or tensors as multi-dimensional numbers, i.e., as extensions on the concept of 1-d numbers.

 

[Monocles]

To add to the discussion of quaternions: invented by Hamilton, he originally advocated their use in mechanics problems. One could formulate mechanics problems in terms of quaternions: a + bi + cj + dk. However, they were reportedly rather cumbersome, and when Gibbs came along and invented vectors, it pretty much spelled the death of the use of quaternions in mechanics. If you've ever wondered why the spatial unit vectors in 3d were labeled i j and k, it's because of the original use of quaternions to do mechanics problems which vectors were eventually used for.

This is a historical anecdote not terribly relevant to the thread, but it was actually Grassmann who invented almost all of basic linear algebra (including vectors). Unfortunately he was too far ahead of his own time and his work was never really appreciated until after he died - people continued to insist on using Hamilton's quaternions until the early 1900's, in part because of Hamilton's influence and fame. But the point is that it actually took most of a century for vectors to spell the death of quaternions in mechanics problems.