The Complex CircleBy Liona Fan-Chiang By the conclusion of Gauss' first treatise on biquadratic residues, brilliant Gauss was finally able to algebraically arrive at a representation of specific primes into the form p=a^{2}+b^{2},^{1} and thus conclude that these particular primes, and therefore their moduli, are suited to representation on the complex plane, namely as relations between two complex numbers, (a+ib) and (a-ib). However, is the ability to express prime numbers of the form 4n+1 in the algebraic form p=a^{2}+b^{2}, and therefore p=(a+ib)(a-ib), a necessary prerequisite to the demonstration that these numbers are then suited to be represented on the complex plane?^{2} Or, do the nature of moduli and number themselves presuppose the ability to investigate their relations on the complex plane? That is, is the complex plane, alternately, a required concept for the generation of these numbers?
The answer to this question, is another question: what defines a number?
The number 1 has been extensively investigated by Nicolas of Cusa and others, such that, if I were to ask, "What is 1?", the answer might be "1 is unity", or "1 is the first number" etc. If, in this context, I were to then ask, "What is 2?", the answer might be "2 is many", or "2 is the unfolding of oneness".^{3}
However, what if I were to first present a series?:
and then proceeded to ask, "What is 2?" What would be the answer be then? One might say, "2 is the second of counting numbers."
Or, how about again after this series?:
"2 is the first of the even numbers."
Alternately, what if 2 were first presented in the following progression?:
What kind of number is this number 2 in the above case?
Is the number 2 above even the same as that which appears in the following progression?
In each of the above cases, the number 2 does not change its clothing. The number consistently looks the same, and is pronounced the same way, "two." However, the answer to the same question yielded wildly varying results. For example, in the last two cases, 2 is first presented as a square number, namely 2=Ö2^{2}, after which, it then becomes a square root, or a line. If these two 2's were the same, would it not then have to be admitted that a square is the same as a line? Therefore, an apparent investigation of number, or even a number, already shows itself to require an initial investigation into the process which is generating the number, which, in turn, itself will demand the inquiry: what principles shape the domain in which the process is being generated?
What role does 2 play in a modulus of 5?
If the reader has not already done so, play around with modulus 5 until some of its properties are intimately familiar. Add the residues. Multiply them. Add congruent numbers etc.
Modulus 5 For example, here are the multiplicative properties of modulus 5:
Next to every residue are pairs of factors, which, when multiplied, result in a product which is congruent to those corresponding residues. In investigation of the modulus, the characteristics of the modulus cannot be known by simply staring at numbers themselves, but rather, to know the personality of the modulus and its residues, you must first investigate the relations or proportions that the residues have with one another, within the context of their respective relationships to 5, i.e. the first investigation is of proportions between proportions, as stated by Gauss in 1809:
The general representation [Vorstellung] of things, where each only has one proportion of inequality to two [things], are points in a line. After the initial dancing about with proportions, the next question is, how do these proportions act to define the characteristic of the whole? Or inversely, how does the whole bound and determine the unique potential for action of each of its parts?
Let us look again at the multiplicative characteristics shown above. Notice that the multiplication of each successive number 1, 2, 3, 4 by 4 gives the numbers 4, 3, 2, 1, respectively. Thus, multiplying by the residue 4, inverts the entire modulus, so that the modulus runs clockwise rather than counterclockwise. What type of action does the number 2 have on the modulus? Multiplying each number by 2 maps each number of modulus 5 to 2, 4, 1, 3; a sequence which may at first appear to the mind to have no order. Multiplying by 3 gives 3, 1, 4, 2. Even though the series resulting from multiplication by 3 again appears to have no ordering principle, a quick comparison between the series generated by multiplication of 2, and that generated by multiplication of 3 shows that the two are inversions of each other! In general, these relations can be summarized thus:
How can the particular characteristics of each number, which can be seen by the effect each number has on the modulus when the modulus is acted upon by the number, be more easily investigated? Can the above relations be represented geometrically? As quoted above, since we are dealing with relations of relations, Gauss' proposal is to apply a planar representation, namely, the complex plane. As can be seen, the first three relations give qualities of action which can be easily mapped as congruent complex numbers, i.e. numbers that hold the same potential for action: The number 0 with respect to modulus 5 takes every number and upon multiplication, maps them to 0, in exaclty the same way as does the origin of the complex plane. Therefore, it can be said that 0 mod 5 is congruent with the origin of the complex plane. In the same vein, 1 is congruent to unity, and 4, inverting the entire modulus, is congruent with -1. What about 2 and 3? To place these, take a look above at a very singular relation, namely that besides 1·4 º 4, both 2·2 and 3·3 are congruent to 4, thereby satisfying 2·2 º 4 and 3·3 º 4 or 2·2 º -1 and 3·3 º -1. What does this relation say about 2 and 3?
Now, to check if this representation is accurate, try filling in the rest of the plane. Do the additive properties hold?
Let's put this one to the test. If 3 is equivalent to 1+i, then it should satisfy all the multiplicative properties of (1+i), for example, (1+i)^{2}:
In this case, (1+i)^{2} º (1+2)^{2} º 4, while (1+i)^{2} is also equivalent to 2i which in this case, since i º 2, º 2·2 º 4! By doing this mapping, you will see a certain pattern growing, namely, that there is repetition on the plane, and if you connect all numbers divisible by 5, i.e. which mod 5 = 0, you will see a new geometry form, which exactly corresponds to modulus 5!
Shown above is the "right handed" mapping of modulus 5, where the number 2 was chosen to be represented by +i and 3, as -i. The result of this mapping is a grid whose unit is 1+2i. Or, is the unit 2-i, -2+i, or -1-2i? That is, which side of any square is equivelent to the unit 1, and which to -1, i or -i? In addition, remember that the only property which characterized the residues 2 and 3 were that they had to be opposite and that they both satisfied the condition x^{2}=-1. Therefore, their positions on the complex plane can be exchanged, resulting in the formation of a similar but mirrored grid.
It is highly suggested that the reader become a co-conspirator and do this mapping his or herself!
Keep in mind that since both of the above representations are consistent with the properties of the space defined by modulus 5, they should both be accurate mappings of modulus 5, that is, there are two unique, yet absolutely consistent representations of the one and the same modulus.
What about for other moduli? Does this transformation hold for other moduli? Will the numbers 2 and 3 play the same role in, for example, modulus 13? For example, which of the 12 residues of modulus 13, when squared, result in a number which is congruent to -1, or 12 in this case?
By mapping the properties of modulus 5 onto the complex plane, you can already see how the complex plane gives you an ability to see, manipulate and function in the domain of relations of action, thereby illuminating the actual quality of interval represented by each residue within the modulus.
Finally, where do the unit lengths which bound the modulus intersect the real number line? As you might have expected, the properties of the modulus on the real number line are preserved throughout this doubly extended mapping. For example, for the complex representation of modulus 5 above, the real number line reads 1, 2, 3, 4, 0, 1, 2, ¼ How does the unit of the complex modulus, in this case 1+2i, relate to the modulus itself, 5?
Try this with your mapping of modulus 13 as well.
By beginning with an initial mapping of the modulus onto the complex plane, we arrive at 5=(1+2i)·(1-2i), or more generally, p=(a+bi)·(a-bi), just as derived algebraically at the end of Gauss' 1828 First Treatise on Biquadratic Residues. Therefore the question is again posed: Did Gauss actually begin with an algebraic derivation of the form p=a^{2}+b^{2}, from which he concluded that these numbers could then be represented on the plane, or did he begin with the end? Which came first?
Footnotes:^{1}This is the result toward the end of Gauss' First Treatise on Biquadratic Residues, published in 1828
^{2}If these primes can be represented as p=(a+ib)(a-ib), i.e. a product of two numbers, can they be called prime? In the expanded field of doubly extended numbers, many prime numbers lose their citizenship as primes!
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On 05 Mar 2008, 19:07. |