By Bruce Director

``Can we deny that a warrior should have a knowledge of arithmetic?...

``.... It appears to me to be a study of the kind which we are seeking, and which leads naturally to reflection, but never to have been rightly used; for the true use of it is simply to draw the soul towards being...

``.... For, if simple unity could be adequately perceived by the sight or by any other sense, then, as we were saying in the case of the finger, there would be nothing to attract towards being; but when there is some contradiction always present, and one is the reverse of one and involves the conception of plurality, then thought begins to be aroused within us, and the soul, perplexed and wanting to arrive at a decision, asks, `Where is absolute unity?' This is the way in which the study of the one has a power of drawing and converting the mind to the contemplation of true being.

``And surely, he said, this occurs notably in the case of one; for we see the same thing to be both one and infinite in multitude?

``Certainly.

``And all arithmetic and calculation have to do with number?

``Yes.

``And they appear to lead the mind towards truth?

``Yes, in a very remarkable manner.

``Then this is knowledge of the kind for which we are seeking, having a double use, military and philosophical; for the man of war must learn the art of number or he will not know how to array his troops, and the philosopher also, because he has to rise out of the sea of change and lay hold of true being, and therefore he must be an arithmetician....

``... Then this is a kind of knowledge which legislation may fitly prescribe; and we must endeavor to persuade those who are to be the principal men of our State to go and learn arithmetic, not as amateurs, but they must carry on the study {until they see the nature of numbers with the mind only;} nor again, like merchants or retail-traders, with a view to buying or selling, but for the sake of their military use, and of the soul herself; and because this will be the easiest way for her to pass from becoming to truth and being....''

--Plato's {Republic} Book VII

Elementary considerations concerning prime numbers, directly present us with the fundamental questions to which Socrates refers in the above passage. To grasp this, however, one must confront, and overcome, any influence of Euler and the Enlightenment, in one's own thinking.

What are Prime Numbers?

Prime Numbers are integers, which are indivisible, by any other number, except one and itself. Composite numbers, are integers, which can be divided by another number. In the {Elements}, Euclid presents a proof that any composite number is divisible by some prime number. Since, any composite number by definition can be divided by another number, that other number is either another composite number or a prime number. If it is a prime number, the case is proved. If it is a composite number, that new number can be divided by another number, which is either a prime number or a composite number. By this method, you will eventually get to a prime number.

A method for discovering which integers are prime, was developed by Eratosthenes in approximately 200 B.C., known as the Sieve of Eratosthenes. This is a method for eliminating the composite numbers, from a group of integers, leaving only the primes.

List the integers from 1 to any arbitrary other integer A. Now, beginning with 2 (the first prime number after 1), strike from the list, all numbers divisible by 2, for they are composite numbers. Do the same with 3 (the next prime number), then 5, etc., until you come to the first or prime is first.

This raises the question, what happens when you try to construct all integers from the primes alone? First, you'd make all the integers composed only of 2. Then you'd make all the integers composed only of 3 and combinations of 2 and 3, and so forth with 5, etc. As you can see, this process would eventually generate all the integers, but in a non-linear way. Compare that process with constructing the integers by addition? Addition generates all the integers sequentially, by adding 1, but does not distinguish between prime numbers and composite numbers.

The unit 1 is indivisible, with respect to addition. With respect to division, the prime numbers are indivisible. Both processes will compose all the integers, but that result coincides only in the infinite. In the finite, they never coincide. The difference, is between the mental act of addition, and the mental act of division. Don't try resolve the matter, by asking if division is superior to addition. Instead, reflect on that which is different between the two processes, the ``in-betweeness.'' It is the relations between the numbers, which is the object of our thought, not the numbers in themselves.

This anomaly, is a reflection of the truth, that there exists a higher hypothesis which underlies the foundations of integers. An hypothesis, which is undiscoverable if limited to the domain of simple linear addition. By reflecting on this anomaly, we begin, as Socrates says, ``to see the nature of number in our minds only.'' Our minds ascend, as Socrates indicates, to contemplate the nature of true being. We ask, ``If the domain of primes is that from which the integers are made, what is the nature of the domain, from which the primes are made?''

View the above result from the standpoint of Leibniz's {Monadology}:

``29. Knowledge of necessary and eternal truths, however, distinguishes us from mere animals and grants us {reason} and the sciences, elevating us to the knowledge of ourselves and of God. This possession is what is called our reasonable soul or {spirit}.

``30. By this knowledge of necessary truths and by the abstractions made possible through them, we also are raised to {acts of reflection} which enable us to think of the so-called {self} and to consider this or that to be in us. Thinking thus about ourselves, we think of being, substance, the simple and the composite, the immaterial, and even of God, conceiving what is limited in us as without limit in him. These acts of reflection furnish the principal objects of our reasoning.'' [bmd]

"The natural, sprouting origin of the rational art is number; indeed, beings which possess no intellect, such as animals, do not count. Number is nothing other than unfolded rationality. So much, indeed, is number shown to be the beginning of those things which are attained by rationality, that with its sublation, nothing remains at all, as is proven by rationality. And if rationality unfolds number and employs it in constituting conjectures, that is not other than if rationality employs itself and forms everything in its highest natural similitude, just as God, as infinite mind, in His coeternal Word imparts being to things. There cannot be anything prior to number, for everything other affirms that it necessarily existed from it....

"The essence of number is therefore the prime exemplar of the mind. For indeed, one finds impressed in it from the first trinity or the unitrinity, contracted in plurality. In that we conjecture symbolically from the rational numbers of our mind in respect to the real ineffable numbers of the divine Mind, we indeed say that number is the prime exemplar of things in the mind of the Composer, just as the number arising from our rationality is the exemplar of the imaginal world."

--Nicolaus of Cusa, "On Conjectures"

Last week, as Socrates says, we, began "to see the nature of number with the mind only." This week, we further unfold the essence of number, requiring our mind, to lift itself into a new higher domain. This journey may be difficult, at points, for the reader. For those with previous mathematical training, infected by the Euler-Lagrange-Cauchy fraud, you will find your previous training an annoying distraction. For those without such annoying distractions, you may find the lack of such, an annoying distraction in itself. To avoid these distractions, follow the proscription of Plato, Cusa and Leibniz, and see with your mind only. For this purpose, we continue our investigation using the principles of higher arithmetic, as developed by C.F. Gauss, which considers only the relations between whole numbers. Gauss, elaborated a visualization of the complex domain, translated by Jonathan Tennenbaum, in the Spring 1990 issue of Twenty First Century, under the title "Metaphysics of Complex Numbers," to which the reader is referred.

In the previous discussion, we discovered an hypothesis underlying the essence of number, through reflection on the prime numbers. By its extension, we will be lead to a new, higher hypothesis underlying the essence of number.

First, extend the idea of number, from positive whole numbers, to include their opposites, the negative whole numbers. Here, prime numbers, maintain their same relationship with respect to all numbers, with the exception of a change of direction from positive to negative. Whereas, positive whole numbers are formed sequentially by adding one, negative whole numbers are formed sequentially by subtracting one, or, adding -1. Positive composite numbers are formed by muliplying the prime factors, negative composite numbers are formed by multiplying the prime factors by -1. Think of positive and negative, not as position, but as directions, opposite one another. If positive is right, negative is left. If positive is up, negative is down. One dimension--two directions.

In short we have extended our concept of number, by conceiving of a two-fold unity: 1 and -1.

Once our concept of number is extended into the negative direction, an anomaly immediately appears. All positive and negative whole numbers can be squared to form a quadratic whole number. For example, 2 X 2 =4; 3 x 3 = 9; -2 x -2 = 4; -3 x -3 = 9. The nmber being squared is called, the square root, of the quadratic (square) number. Notice, however, that in this domain, all quadratic whole numbers are positive. A pradoxical question arises: "Can one form a negative quadratic number?" Or, conversely, "what is the square root of a negative number?"

The simplest case, which subsumes all others, is the case of the quadratic unity, 1. 1 x 1 = 1 -1 x -1 = 1. The square root of 1 = 1 or -1, as both numbers squared equal 1. What, then is the square root of -1?

Within the concept of number as one-dimensional, (two directions), the concept of the square root of -1 remains paradoxical, and was given the unfortunate name of imaginary. (Just as the oligarchy attempted to limit human knowledge to one dimension, by naming the Lydian interval the "devil's" interval.) Euler and others, sought to limit progress of human knowledge, by giving the square root of -1 a purely formal, and therefore meaningless, definition. It was Gauss, who saw with his mind, in this paradox, a means of extending the concept of number, into a new domain--the complex domain. Instead of avoiding the paradox presented, by thinking of it as imaginary, or impossible, Gauss asked, in what higher domain, must such a magnitude exist? A shift in hypothesis, which his student, Riemann, would later designate as from n to n + 1 dimensions.

Gauss elaborates his hypothesis of the complex domain, in a section of the second paper on biquadratic residues in 1832, but, as he says, he developed the hypothesis, as early as 1799, while writing his original work on higher arithmetic, Disquisitiones Arithmeticae. He, says he was only waiting (over 30 years) for a suitable place, in which to announce his new hypothesis to the public.

Gauss approached the paradox of the square root of -1, by extending the hypothesis underlying the concept of number from one dimension (two directions) to two dimensions (four directions).

For purposes of brevity, the square root of -1 is denoted by the letter i. (Again, an unfortunate designation, associated with the term imaginary, owing to the fraudulent Euler.)

Now reflect on the properties of the complex domain, as investigated by Gauss.

In the complex domain of two dimensions, the square root of -1 is thought of as a different dimension, distinct from the dimension of simply positive and negative, but united in the complex domain. In the complex domain, all numbers are made up of two dimensions. One dimension is associated with positive and negative, the other dimension, is associated with +i and -i. The complex domain is ONE domain, indivisible, of two dimensions. A new hypothesis, under which, the positive and negative, i and -i are made congruent (harmonic) with each other. In this new domain, all numbers are of the form a + bi, where a (short for 1 x a) designates the positive-negative dimension, and bi designates the +i -i dimension, (1 + i dimensions). This is not a combination of two different numbers, but one number, with two parts. One dimensional numbers, those limited to positive and negative, such as integers, are the special case of complex numbers where b=0. Complex numbers where neither a nor b is 0 are called mixed complex numbers.

Reflect on the difference between the domain of one dimensional numbers and the complex domain.

In the domain of one dimension, unity is two-fold, 1 and -1. In the complex domain, unity is four-fold, 1 and -1, i and -i. In one dimension, each number is associated with its opposite, for example, 5 and -5, 2 and -2. Its associated number is formed by multiplying by -1. In the complex domain, with its four-fold unity, each number has four associates, found by multiplying that number by -1, i, -i. For example, a + bi is associated with -b + ai, -a - bi, b - ai. (The reader can confirm this for himself, by multiplying a + bi by i, -1, -i respectively.)

There is a special, unique, relationship in the complex domain-- the relationship between a number a + bi, and its conjugate, a - bi, that is, when the sign of i is reversed. The product of a number and its conjugate is a^2 + b^2 and is called its Norm. (Notice the similarity to the Pythagorean)

Gauss then investigates the nature of prime numbers in the complex domain. Just as in one dimension, all whole numbers are either prime or composite. However, not all one dimensional prime numbers, remain prime in the complex domain. For example, in the complex domain 2 = (1 + i)(1 - i); 5 = (1 + 2i)(1 - 2i), 13 = (3 + 2i)(3 -2i). In fact, Gauss showed, that all one dimensional prime numbers of the form 4n+1 are no longer prime in the complex domain, but all one- dimensional prime numbers of the form 4n+3 remain prime. (All one-dimensional prime numbers are either of the form 4n+1 or 4n+3. Not all numbers of this form are prime. The reader should verify this himself.)

There are also new kinds of prime numbers in the complex domain--mixed complex prime numbers. Gauss showed that mixed complex numbers are prime, if their Norm is a one-dimensional prime number. For example, 1 + 2i, is a mixed complex prime number, because its Norm, 1^2 + 2^2 = 5, which is a one dimensional prime number. So is 1 - 2i.

Reflect further on the difference between the one dimensional domain of positive and negative numbers, and the complex domain. The complex domain, is not simply the one-dimensional domain, in two directions, as in the fraud perpetrated by Cauchy. IT IS AN ENTIRELY DIFFERENT DOMAIN, lawfully connected, but distinct from the one-dimensional domain. In the complex domain, the universal characteristic is changed. Fundamental singularities, such as prime numbers, are re-ordered. Some are changed, some are unchanged, and new ones are created. It is the domain, which determines the singularities. Is 5 a prime number? Yes and No. It depends on the domain. How do you know what domain you're in? Through the creative powers of your mind. Analysis situs. If 5 is a prime number, you're in one dimension, if not, you may be in the complex domain, but, maybe not. Gauss speculates about the possibility of numbers of higher dimensions than two. What happens to prime numbers in domains of more than two dimensions?

Isn't this the key to improving performance in sales and intelligence?

CAN YOU SOLVE THIS PARADOX

Over the previous two weeks, we've demonstrated, by reliving a discovery made by the young Carl Friedrich Gauss when he was 10 years old, how numbers are creations of the mind. Once this basic principle is understood, the mind is no longer a slave to formal rules concerning numbers. Problems such as adding all the numbers from 1 to 100, which at first appear tedious and perhaps even difficult, are easily solved, once the mind breaks the formal rules, and re-orders the numbers according to a new, higher principle.

This week, we'll look more deeply into the nature of numbers, and in doing so, we'll gain an increased mastery over our minds' creative process.

What Are Prime Numbers?

Among the whole numbers, there exist unique integers known as Prime Numbers, which are distinguished by the property that they are indivisible by any other number except themselves and 1. Thus, 2, 3, 5, 7, and 11 are all examples of prime numbers. Composite Numbers are integers which can be divided, not only by themselves and 1, but by some other number.

It had already been discovered by the ancient Greeks, and written down by Euclid (flourished c. 300 B.C.) in his ``Elements,'' that all numbers are either prime or composite, and that any composite number is divisible by some prime number.

You can prove this for yourself, in the following way. Any composite number can by definition be divided by some other number, and that other number is either another composite number or a prime number. If it is a prime number, we need go no further. If it is a composite number, then that new composite number can be divided by another number, which is either a prime number or a composite number, and so on. By this method, you will eventually get to a prime number divisor.

For example, 30 is a composite number, and can be divided into 2, a prime number, and 15, a composite number. In turn, 15, can be divided into 3, a prime number, and 5, also a prime number. So the composite number 30 is made up of, and can be divided by, prime numbers 2, 3, and 5.

A method for discovering which integers are prime, was developed by Eratosthenes in approximately 200 B.C. This approach, known as the Sieve of Eratosthenes. is a method for eliminating the composite numbers from a group of integers, leaving only the primes.

List the integers from 1 to any other arbitrary integer A. Now, beginning with 2 (the first prime number after 1), strike from the list all numbers divisible by 2, for they are composite numbers. Do the same with all numbers divisible by 3, the next prime number; then those divisible by 5, etc., until you come to the first prime number whose square is greater than A. (If A = 100, then you only need do this procedure with primes less than 11.) See Figure 1.

This method allows us to find the prime numbers which are smaller than any arbitrary number--but is there some large number after which there are no prime numbers? In other words, are there an infinite number of prime numbers?

That the answer is yes, is easily proved by showing that, no matter how many prime numbers are found, there can always be found one more. First, find all the prime numbers less than any arbitrary number, a, b, c, ..., z. Now multiply all these prime numbers together, and add 1 to the product [(a x b x c x d x ... x z) + 1]. Call this new number A. If this new number A is a prime number, then you've found another prime, which was not known before. If it is a composite number, then it must be divisible by some prime number. But, that prime number cannot be one of those already known, as the known prime numbers (a, b, c, ..., z) will always leave a remainder of 1, when divided into A.

Primes Are Nonlinear

After we have found a large number of primes, it is obvious that, in the small, there is no regular, or linear, pattern of distribution of the prime numbers. Gauss showed, however, that over a large interval, the distribution of the primes is approximated by a logarithmic curve; that is, the larger the prime numbers become, the more spread apart they tend to be. This approximation breaks down increasingly, the smaller the interval. A nightmare for Leonhard Euler--nonlinearity in the small. (See Figure 2.)

{Again, what are Primes?} It is easy to see that any composite number can be decomposed into prime numbers, by division. For example, 12 can be decomposed into 2 x 2 x 3, or 2^2 x 3. The number 504 can be decomposed into 2 x 2 x 2 x 3 x 3 x 7, or 2^3 x 3^2 x 7.

Gauss was the first to prove Disquisitiones Arithmeticae, Article 16) that a composite number can be decomposed into only one combination of prime numbers. In the above examples, no combination of prime numbers other than 2 x 2 x 3 will equal 12. Likewise for 504, or any other composite number.

This remarkable result, which Gauss says was ``tacitly supposed but had never been proved,'' provokes a fundamental question concerning the nature of the universe. The fact that Gauss was the first to consider this result important enough to prove, is another indication of his genius.

With Gauss's proof, and the preceding discussion, it is shown that prime numbers are that from which all other numbers are composed. The primes are primary. The word the ancient Greeks used for ``prime,'' was the same word they used for ``first'' or ``foremost.''

This raises the question, what happens when you try to construct all integers from the primes alone? First, you'd make all the integers composed only of 2, such as 4, 8, 16, .... Then you'd make all the integers composed only of 3, and of combinations of 2 and 3, such as 6, 9, 12, ..., and so forth with 5, etc. As you can see, this process would eventually generate all the integers, but in a nonlinear way.

Compare that process with constructing the integers by addition. Addition generates all the integers sequentially, by adding 1, but does not distinguish between prime numbers and composite numbers.

The unit 1 is indivisible, with respect to addition. With respect to division, the prime numbers are indivisible. Both processes will compose all the integers, but that result coincides only in the infinite. In the finite, they never coincide. The difference is between the mental act of addition, and the mental act of division. Don't try to resolve the matter, by asking if division is superior to addition. Instead, reflect on that which is different between the two processes, the ``in-betweenness.'' It is the relations between the numbers, which is the object of our thought, not the numbers in themselves.

This anomaly is a reflection of the truth that there exists a higher hypothesis which underlies the foundations of integers--a hypothesis which is undiscoverable if limited to the domain of simple linear addition. By reflecting on this anomaly, we begin, as Socrates says, ``to see the nature of number in our minds only'' (from Plato's ``Republic''). Our minds ascend, as Socrates indicates, to contemplate the nature of true Being. We ask, ``If the domain of primes is that from which the integers are made, what is the nature of the domain from which the primes are made?''

View the above result from the standpoint of Leibniz's ``Monadology'':

``29. Knowledge of necessary and eternal truths, however, distinguishes us from mere animals and grants us {reason} and the sciences, elevating us to the knowledge of ourselves and of God. This possession is what is called our reasonable soul or {spirit.}

``30. By this knowledge of necessary truths and by the abstractions made possible through them, we also are raised to {acts of reflection} which enable us to think of the so-called {self} and to consider this or that to be in us. Thinking thus about ourselves, we think of Being, Substance, the Simple and the Composite, the Immaterial, and even of God, conceiving what is limited in us as without limit in Him. These acts of reflection furnish the principal objects of our reasoning.''

Next week: When is 5 not a prime number.

 CAN YOU SOLVE THIS PARADOX?

``The natural, sprouting origin of the rational art is number; indeed, beings which possess no intellect, such as animals, do not count. Number is nothing other than unfolded rationality. So much, indeed, is number shown to be the beginning of those things which are attained by rationality, that with its sublation, nothing remains at all, as is proven by rationality. And if rationality unfolds number and employs it in constituting conjectures, that is not other than if rationality employs itself and forms everything in its highest natural similitude, just as God, as infinite mind, in His coeternal Word imparts being to things. There cannot be anything prior to number, for everything other affirms that it necessarily existed from it....

``The essence of number is therefore the prime exemplar of the mind. For indeed, one finds impressed in it from the first trinity or the unitrinity, contracted in plurality. In that we conjecture symbolically from the rational numbers of our mind in respect to the real ineffable numbers of the divine Mind, we indeed say that number is the prime exemplar of things in the mind of the Composer, just as the number arising from our rationality is the exemplar of the imaginal world.''

--Nicolaus of Cusa ``On Conjectures''

Last week, we investigated the nature of prime numbers. This week, we further unfold the essence of number, requiring our minds to lift themselves into a new, higher domain. This journey may be difficult, at points, for the reader. For those with previous mathematical training, infected by the Euler-Lagrange-Cauchy fraud, you will find your previous training an annoying distraction. For those without such annoying distractions, you may find the lack of such to be an annoying distraction, in itself. To avoid these distractions, follow the prescription of Plato, Cusa, and Leibniz, and see with your mind only. For this purpose, we continue our investigation using the principles of higher arithmetic as developed by Carl Friedrich Gauss, which considers only the relations among whole numbers.

In the previous discussion, we discovered, through reflection on the Prime Numbers, a hypothesis underlying the essence of number. By its extension, we will be led to a new, higher hypothesis underlying the essence of number.

First, extend the idea of number from positive whole numbers, to include their opposites, the negative whole numbers. Here, prime numbers maintain their same relationship with respect to all numbers, with the exception of a change of direction from positive to negative. Whereas positive whole numbers are formed sequentially by adding one, negative whole numbers are formed sequentially by subtracting 1, or adding -1. Positive composite numbers are formed by multiplying the prime factors, negative composite numbers are formed by multiplying the prime factors by -1. Think of positive and negative, not as position, but as directions, opposite to one another. If positive is right, negative is left. If positive is up, negative is down. One dimension--two directions.

In short, we have extended our concept of number, by conceiving of a twofold unity: 1 and -1.

An Anomaly Appears

Once our concept of number is extended into the negative direction, an anomaly immediately appears. All positive and negative whole numbers can be squared to form a quadratic whole number. For example, 2x2=4; 3x3=9; -2x-2=4; -3x-3=9. The number being squared is called the square root of the quadratic (square) number. Notice, however, that in this domain, all quadratic whole numbers are positive. A paradoxical question arises: ``Can one form a negative quadratic number?'' Or, conversely, ``What is the square root of a negative number?''

The simplest case, which subsumes all others, is the case of the quadratic unity, 1: 1x1=1, and -1x-1=1. The square root of 1=1 or -1, as both numbers squared equal 1. What, then, is the square root of -1?

Within the concept of number as one-dimensional (two directions), the concept of the square root of -1 remains paradoxical, and was given the unfortunate name of imaginary. (Just as the oligarchy attempted to limit human knowledge to one dimension, by naming the Lydian interval the ``devil's interval.'') Euler and others sought to limit the progress of human knowledge, by giving the square root of -1 a purely formal, and therefore meaningless, definition. It was Gauss who saw with his mind, in this paradox, a means of extending the concept of number, into a new domain--the complex domain. Instead of avoiding the paradox presented, by thinking of the square root of -1 as imaginary, or impossible, Gauss asked, in what higher domain must such a number exist? In other words, a shift in hypothesis, which his student, Bernhard Riemann, would later designate as changing from {n} to {n}+1 dimensions.

Gauss elaborated his hypothesis of the complex domain, in a section of the second paper on biquadratic residues in 1832, but, as he says, he developed the hypothesis as early as 1799, while writing his original work on higher arithmetic, Disquisitiones Arithmeticae. He says he was only waiting (over 30 years) for a suitable place in which to announce his new hypothesis to the public.

Extending The Hypothesis

Gauss approached the paradox of the square root of -1, by extending the hypothesis underlying the concept of number from one dimension (two directions) to two dimensions (four directions).

For purposes of brevity, the square root of -1 is denoted by the letter {i.} (Again, an unfortunate designation, associated with the term imaginary, owing to the fraudulent Euler.)

Now reflect on the properties of the complex domain, as investigated by Gauss.

Gauss conceived of the complex domain as a domain of two dimensions, in which the square root of -1 is thought of as a different dimension, distinct from the dimension of simply positive and negative, but united in the complex domain. If the domain of positive and negative numbers is thought of as a one-dimensional series, the complex domain can be thought of as a series of one-dimensional series. (See Figure 1.) Movement within a series is associated with the concept of positive and negative. Movement from one series to the next, is associated with +{i} and -{i}.

In the complex domain, all numbers are made up of two dimensions. One dimension is associated with positive and negative; the other dimension is associated with +{i} and -{i}. The complex domain is {one} domain, indivisible, of two dimensions. A new hypothesis, under which the positive and negative, {i} and -{i}, are made congruent (harmonic) with each other. In this new domain, all numbers are of the form {a}+{bi}, where {a} (short for 1x{a}) designates the positive-negative dimension, and {bi} designates the +{i}-{i} dimension, (1+{i} dimensions). This is not a combination of two different numbers, but one number, with two parts. One-dimensional numbers, those limited to positive and negative, such as integers, are the special case of complex numbers where {b}=0. Complex numbers where neither {a} nor {b} is 0, are called mixed complex numbers.

One Dimension, and Complex Domain

Reflect on the difference between the domain of one-dimensional numbers and the complex domain.

In the domain of one dimension, unity is twofold, 1 and -1. In the complex domain, unity is fourfold, 1, -1, {i,} and -{i}. In one dimension, each number is associated with its opposite, for example, 5 and -5, 2 and -2. Its associated number is formed by multiplying by -1. In the complex domain, with its fourfold unity, each number has four associates, found by multiplying that number by -1, {i,} -{i}. For example, {a}+{bi} is associated with -{b}+{ai}, -{a}-{bi}, {b}-{ai}. (The reader can confirm this for himself, by multiplying {a}+{bi} by {i}, -1, -{i}, respectively.)

There is a special, unique relationship in the complex domain--the relationship between a number {a}+{bi}, and its conjugate, {a}-{bi}, that is, when the sign of {i} is reversed. The product of a number and its conjugate is {a}2+{b}2, and is called its Norm.

Gauss then investigated the nature of prime numbers in the complex domain. Just as is the case in one dimension, all whole numbers are either prime or composite, in the complex domain. However, not all one-dimensional prime numbers remain prime in the complex domain. For example, in the complex domain 2=(1+{i})(1-{i}); 5=(1+2{i})(1-2{i}), 13=(3+2{i})(3-2{i}). In fact, Gauss showed that all one-dimensional prime numbers of the form 4{n}+1 are no longer prime in the complex domain, but all one-dimensional prime numbers of the form 4{n}+3 remain prime. (All one-dimensional prime numbers are either of the form 4{n}+1 or 4{n}+3. But, not all numbers of this form are prime. The reader should verify this himself.)

There are also new kinds of prime numbers in the complex domain--mixed complex prime numbers. Gauss showed that mixed complex numbers are prime, if their Norm is a one-dimensional prime number. For example, 1+2{i} is a mixed complex prime number because its Norm, 12+22=5, is a one-dimensional prime number. So is 1-2{i}.

Reflect further on the difference between the one-dimensional domain of positive and negative numbers, and the complex domain. The complex domain is not simply the one-dimensional domain in two directions, as in the fraud perpetrated by Cauchy. {It is an entirely different domain,} lawfully connected, but distinct from the one-dimensional domain. In the complex domain, the universal characteristic is changed. Fundamental singularities, such as prime numbers, are re-ordered. Some are changed, some are unchanged, and new ones are created. It is the domain which determines the singularities.

Is 5 a prime number? Yes and No. It depends on the domain. How do you know what domain you're in? Through the creative powers of your mind. If 5 is a prime number, you're in one dimension; if not, you may be in the complex domain. But, maybe not, as Gauss speculates about the possibility of numbers of dimensions higher than two.

For further reading see, in English translation, Gauss's ``The Metaphysics of Complex Numbers,'' 21st Century Science and Technology magazine, spring 1990.