Essays On Imaginary Numbers

Until now, we have been dealing exclusively with real numbers. This chapter introduces a new topic--imaginary and complex numbers. Complex numbers are numbers of the form a + bi, where i = and a and b are real numbers. They are used in a variety of computations and situations. Complex numbers are useful for our purposes because they allow us to take the square root of a negative number and to calculate imaginary roots.

The first section discusses i and imaginary numbers of the form ki. Here, the reader will learn how to simplify the square root of a negative number.

The focus of the next two sections is computation with complex numbers. Section two explains how to add and subtract complex numbers, how to multiply a complex number by a scalar, and how to multiply a complex number by another complex number. Section three introduces the concept of a complex conjugate and explains its use in dividing a complex number by another complex number.

Complex numbers are useful in a variety of situations. They appear frequently in almost every branch of mathematics. We will use them in the next chapter when we find the roots of certain polynomials--many polynomials have zeros that are complex numbers.

Imaginary Numbers, Note 3

The exception to algebraic commutation creates some problems for determining values involving i, j, & k. Thus, if we begin with ij = k and multiply both sides by i, we get i2j = ik = -j. But if we begin with ijk = -1 and divide both sides by j, we get ijk/j = -1/j = ik = j.

Note that the reciprocal of j, like that of i, is equal to the negative of the simple number, i.e. 1/i = -i and 1/j = -j. This property of i is something I have never seen mentioned in any mathematics textbooks, though I'm sure that someone must have noticed it.

In any case, the result we get is both ik = -j and ik = j. This does not look good. In fact, we can get j as well as -j from ij = k just by altering where we insert the multiplying i:  iji = -ji2 = ik = -j(-1) = j. Since the first result of ik = j was derived using division rather than multiplication, this is perhaps the correct result; and it means that, with the exception to commutation, we must introduce restraints on how multiplication is used. If we are going to multiply both sides of ij = k by i, we cannot just insert i anywhere. Evidently, as we see here, it must come at the end of the expression. However, this means that the result is actually ki = j (!). Does commutation apply to ki = ik? That needs figuring out. Either way, the basic violation of commutation is a real headache.

I am now informed by Martin Ondracek, at the Institute of Physics of the Academy of Sciences of the Czech Republic, about the convention employed to avoid the contradictions just described. He says, "either multiply both sides of the equation from the left (insert "i" to the front of the expression) or multiply both side[s] of the equation from the right (append "i" to the end of the expression)." This also means, "you never get to cancel something in the middle of a product, as you did in ijk/j." If this is enough to preserve consistency, then the convention would be logically justified (salve veritate). He also mentions, "this is the same with matrix multiplication, by the way."

So if we wish to multiply ijk = -1 by j, we get either jijk = -j or ijkj = -j, but we cannot evaluate either jijk or ijkj. Similarly, if we wish to divide ijk = -1 by j, which means multiplying by -j, we get either -jijk = j or -ijkj = j, but we cannot evaluate either -jijk or -ijkj. We cannot evaluate them directly, of course, but we can do some substitution. Given that ij = -ji, we can go -jijk = ijjk = ij2k = -ik = j, which looks like something we already accept.

There is an interesting aside from Dr. Ondracek:

It surprises me (and even bothers a little bit, as it is not what I would expect from a philosopher) that you were not curious enough to consult some mathematician to find out what mathematics has to say about this equation and what answer it provides.

While I am always "curious enough" to consult mathematicians, I don't know any at the moment; and my experience with inquiries to academics I do not know, including physicists and mathematicians, is often to get a brush off rather than an answer. The questions I ask often seem to be annoying, perhaps because I am curious about things that they are not. With pages on the Internet, however, which express precisely my own curiosity, I can always hope that they attract the attention of someone like Dr. Ondracek, who can give a succinct, ad rem answer to questions I have. He has done this. Even Dr. Ondracek, however, cautions that he does not speak for his institution. Perhaps the Czech Academy of Sciences would not want to endorse restrictions on the manipulation of quaternions.

He also seems to miss the point of this imaginary numbers web page, saying, "if I understand it correctly that Godel's incompleteness theorem might be somehow used to argue that real numbers are metaphysically real, then why not complex numbers by the same token?" Of course, it is Roger Penrose who argues that Gödel's Proof implies that mathematical expressions have meaning and reference -- q.v.. Why that argument would not work in the same way for imaginaries depends on the foundational question what the derivation and basis is of such numbers. That mathematics in general has meaning and reference does not necessarily mean that imaginaries, for all their meaning, also have reference. "The same token" is not logically transitive there. If Dr. Ondracek has no curiosity about that, then, as I say above, "So, I'm sorry if some of you are determined not to worry about any of this. Don't bother reading the essay."

There is a curious feature to the violation of commutation where ij = -ji. As Dr. Ondracek notes, this is also characteristic of matrix multiplication. Now, Werner Heisenberg formulated quantum mechanics as "matrix mechanics," using matrices. This turned out to be mathematically equivalent to Schrödinger's Equation, but it too possessed this curious violation of commutation.

Wolfgang Pauli derived a physical attribute of great significance from this curiosity. It turns out, the violation of commutation is the mathematically basis of the Pauli Exclusion Principle, which is that more than one particle with half-integer spin cannot possess the same quantum numbers in the same physical system, such as an atom. The violation of commutation violates the Bose-Einstein Statistics, in which commutation is allowed but never makes any physical difference because of the absolute identity of identical particles. Now, only particles with integer spin, or "bosons," obey the Bose-Einstein Statistics, while the other particles with half-integer spin obey the Fermi-Dirac Statistics and consequently are called "fermions." The structure of the chemical elements and ordinary matter absolutely depends on the difference between bosons and fermions; for, if electrons were bosons, in the atoms they would all collapse to the lowest state of energy and angular momentum, making all atoms chemically identical.

The diagram illustrates the Bose-Einstein Statistics in the way they were explained by Albert Einstein himself in a 1925 letter to Irwin Schrödinger [cf. A. Douglas Stone, Einstein and the Quantum, The Quest of the Valiant Swabian, Princeton, 2013, p.239]. Ordinarily, if we toss two coins in the air, half the time (2/4) we will get one coin heads and the other tails. We get this result in two out of four tosses because one coin is one way and the other the other, but which is which can be switched. Now, Einstein decided that the "switched" cases are not going to be different in quantum mechanics because the identity of the particles means, not only is there no way to distinguish them, but that in principle they cannot be reckoned as distinct. The cases where one is one way and the other is the other are consequently the same case as far as quantum mechanics is concerned. The means that the probability of tossing the coins where one is heads and one is tails only happens one out of three times rather than two out of four times. This extraordinary identity of identical particles is only broken for fermions by the extraordinary violation of commutation that we see in matrices and quaternions. On the other hand, the mixing of identities, where it looks like one can be either heads or tails, as long as the other is the opposite, is an example of the mixing or "superposition" of particles, which is the result of adding their wave functions.

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