What do you not fully understand?

[YADD]

Thanks for your comments, Khoa and hanguyen.

I deleted my recent post as it sounds too metaphysical an answer for a quant question/forum. My apologies.

Hanguyen, you point out the wonders descended from the concept of complex numbers. I surely missed them.

To repost my previous thought, basically what has struck me is the apparent inevitability of complex numbers. Are complex numbers merely a technical devise of the Creator or do they faithfully reflect His mind?

For a long time I subscribed - indeed I was not alone - into a belief that complex numbers served as a neat bookeeper; that mathematical/physical formalisms all are recastable in terms of real numbers. Take quantum mechanics as an example. Say, the Schrodinger eqn may acquire a complex phase but it can be decomposed in two real components thru exp(ix) = cos(x) + i*sin(x).

The (complex) Pauli matrices shattered my belief. They are 2x2 matrices corresponing to the smallest non-trivial representation of the isometry group of our 3-dim space. Remarkably, they describe spin-1/2 particles - electrons, quarks. With only real number, the representation would be 3x3 matrices or spin-1 particles.

Half-spin particles are bizzare objects that only become identical after two full rotations! Their existence - not describable in the real number realm, at least as of now - is a testimony for the power of complex numbers. Complex numbers likely are Truth itself.

[Hung.Q.Ngo]

This is a very interesting topic.

I'm personally torn between two lines of thoughts:

1. Complex numbers are inevitable, as YADD very eloquently put above.

2. Complex numbers are not inevitable. They only seem inevitable because of the limitation of our knowledge and IQ. We do not have (fundamentally) better tools to model the physical world, so we naturally make use of the best one available. They play the role of the abacus at an era where digital computers are not yet invented.

I'd like to pose a simlar metaphysical question: is linearity inevitable?

[hanguyen]

Complex numbers are natural extention of real numbers. If you work with polynomials of real coefficients, you'll find the extention to complex numbers necessary to solve their roots. This is not the case for complex numbers, since every N-order polynomial of complex coefficients has N roots. Hence, we stop at complex numbers, no need to go further. If real numbers are inevitable, in the sense that they are inherent attribute of nature, so are complex numbers.

Linearity is also inevitable, but only for humans. I believe it's not something inherent to nature. The difference linearity vs complex numbers is more like engineering vs science. We humain beings tend to perform linear approximations because it's convenient and produces reasonable results. I don't know anything of nature that is strictly linear. If tools using quadratic approximations are equally simple and efficient, we might have seen "quadraticity" everywhere. Of course you can always argue that it is nature's credit that linearity approximations are universally acceptable .

[Hung.Q.Ngo]

If you work with polynomials of real coefficients

Well, I feel this is circular reasoning. By "complex number" we (or at least I) do not really mean the a+bi numbers per se, but also all the motivations and by-products around it, including finding roots of real polynomials. In other words, to show that complex numbers are inevitable using the above reasoning, you'll have to show that "finding roots of real polynomials" are inevitable too.

Kronecker once said: "God created the integers, all else is the work of Man". The oldie had a point! This discussion also reminds me of Hamilton and the quaternions.

Oh, by linearity I meant the spirit of linear algebra (of which complex number is just a special case).

[Khoa Tran]

I also view the difference complex numbers vs. linearity similar to engineering vs. science. But my analogy is quite different from yours.

Linearity is inherent in the sense that it's a mathematical concept that describes ... linear structures: linear transformations, linear operators, groups, fields, etc. Linear structures are prevalent in our life, if we notice. The concept of linearity is not just about linear transformations.

On the other hand, complex numbers is a pure conceptual invention. The concept wasn't born to describe what already there (*). The invention of was born either to satisfy some crazy demand of mathematicians (**) or to ease the way people do mathematics. The latter is probably the real motivation at first (***) as the former is quite meaningless.

Nevertheless, in my opinion, complex numbers only became "inevitable" when Euler defined the famous identity:

This definition is miraculously consistent with the firmly-established mathematics and the invention (why is that?). It remarkably unifies the notion of elliptic functions (sinudoids) and hyperbolic functions (exponentials). Applications would follow naturally. I believe that Euler's identity has revolutionized the theory of complex analysis. In Mathematics, nothing is a miracle but I really see this consistency miraculous (I'm waiting for some explanations on this).

In short (the following just reflects what I view), complex numbers is a remarkable encoding machinery that helps make Mathematics much cleaner. Why such a crazy invention can be that powerful is still hanging on my mind.



(*) It may depend on the perspective of each person. Some philosophers may view the imaginary part of a number as "real" as the real part. I view it differently: the imaginary part of a physical quantity does represent something but using complex numbers is just a way of encoding information in 1 box.
(**) Such crazy demand may be "I don't feel happy when a half of the real line doesn't have square root" or "every polynomial of degree n should have n roots".
(***) The following gives a reliable story to support my suspect that it was born merely to ease mathematical manipulations.

Originally Posted by wiki

Complex numbers became more prominent in the 16th century, when closed formulas for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (see Niccolo Fontana Tartaglia, Gerolamo Cardano). It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. For example, Tartaglia's cubic formula gives the following solution to the equation x³ − x = 0:

At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation z³ = i has solutions −i, and . Substituting these in turn for in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of x³ − x = 0.

[hanguyen]

Originally Posted by Hung.Q.Ngo

In other words, to show that complex numbers are inevitable using the above reasoning, you'll have to show that "finding roots of real polynomials" are inevitable too.

Exactly, anh Hung! If you work with numbers and arithmetic operators (+-*/) to solve real problems (such as measuring, allocating resources), finding roots of polynomials are inevitable. This is why rational numbers were born from integers, real numbers were born from rationals/integers, and complex numbers were born from real numbers.

Again arises the issue whether arithmetic operators (+-*/) are inherent of human or nature. If they are human, then Kronecker is right, as you suggested.

Khoa: could you elaborate why linearity is inherent of nature? I think linearity is purely artificial.

[YADD]

Whoa, shocking to see how many "aspiring quants" have turned metaphysicist!

Originally Posted by hanguyen

This not the case for complex numbers, since every N-order polynomial of complex coefficients has N roots. Hence, we stop at complex numbers, no need to go further.

Good point! A stronger case can be made: indeed not only are polynomial eqns - involving multiplication (power-raising, that is) and addtion/subtraction – ‘closed’ in the complex set, but every (am I correct???) transcesdental eqn conceiveable so far is! [The transcendentality is not trivial since it involves an infinite Taylor expansion.]

Perhaps this is the (practical and potentially theoretical) reason why no further extension of the numerics concept have been called for.

[Hamilton – renowed by then – spent the last 22 yrs of his life finding applications of quarternions in the physical realm. I guess the closest call would have been a (meta-physical) explanation why space-time is 3+1 dim(?) But this attempt should fail since space-time is curved anyway.]

Quarternion, octonion, the SU(5) version of the grand unification theory are abundantly clear situations in which mathematically consistent objects have no physical correspondence. [The 10-dim "string" speculation may someday end up sharing the same fate. Let's wait and see how supersymmetry fares next year at LHC.]

Back to the main story, even though complex numbers help ‘close’ all arithmetic manipulations, I find it so mind-boggling – given the failed examples above – that these incredibly imperceptible objects can dictate how Nature should behave. (See excerpt from Khoa’s post below.)

Originally Posted by Khoa Tran

In short (the following just reflects what I view), complex numbers is a remarkable encoding machinery that helps make Mathematics much cleaner. Why such a crazy invention can be that powerful is still hanging on my mind.

[Khoa Tran]

Originally Posted by hanguyen

Khoa: could you elaborate why linearity is inherent of nature? I think linearity is purely artificial.

We coined the term linearity to describe an inherent property is ... almost everywhere. Some common examples of linear structures:

* Linear space (or more general, groups): they are every where, e.g. R^3
* Linear system: a lot of physics can be described by a linear system: either system of linear equations or linear differential equations (e.g. diffusion, convection, wave propagation, etc.)
* Linear operator: matrix is a linear operator; differential operator (i.e. taking derivatives) is linear; the expectation of a random variable is also linear, ...


Let me pose a simple question that is related to some of the posts above. That is: how do you understand the derivative of function from to ? (For interviews, I would prefer asking conceptual questions rather than tricky puzzles . But well, they overlap many times, such as some stability questions here.)

[hanguyen]

Hi Khoa,

You yet have to explain to me why linear space and linear operator are inherent to nature.

The case of linear system seems more convincing to me. But it's still questionable if physical systems are truly driven by linearity, or they are just approximately so. What if it turns out that the string theory is right .

[ngtridung]

To answer the question about inevitability, I would start answering the following question:

1) Could science/technology advance without complex number?
2) Is there an alternative for using complex number?
3) If the answers to 1) and 2) are No, is complex number all what we need? Is this possible that at some point in time we could invent another number ‘j’ which is even more general and more convenient to use than complex number?

My answers would be:
1) Yes, I believe they could
2) Yes, there is. If someone would have defined operator g(k, f) as the kth root of a function f and if this operator can be used in all the places where complex number ‘i’ is used, then complex number are not inevitable. The math might be more complicated though. (another operator is h(a, b) = a^b. here i = (-1)^(1/2)) is just one special case.
3) I believe there is a more convenient type of number than complex number. For example, how could we represent quantities such as: (-1)^i, i^i

So, I think complex number is not inevitable. If someone could show me an application where we can’t help using complex number or if someone could show me any function of complex number f(i) has the unique representation a+b*i, my view would change a bit

Anyway, I think it would be fruitful if our discussion goes this way: One group finds the applications where complex number plays an essential role (this would be helpful, at least for me). Another group shows the alternatives.