The Yin and Yang of Prime Numbers: Finding Evidence of Unification Thought's Teachings on the Dual Characteristics in Prime Number Reciprocals

Adri de Groot
2007

The trick seems to be in how we look, not always in how much we already know. Often, the solutions are the simplest things, right in front of our eyes, which suddenly reveal themselves, once you wear the right kind of glasses.

Unification Thought (UT) submits one very specific and basic premise: God exists and is the Parent of all Mankind, and conceived of the creation of humans as His Children, for the sake of true love, the only thing He really needed. He conceived of humans before conceiving of the rest of Creation, even though it was created first.[i]

This notion of humanity envisioned first but created last, but vice versa for the rest of creation, proceeds like a mirror and involves seeming opposites. This is something we see also reflected in natural phenomena, like the Sun being closest to Earth in winter time (and yet we are cold) and furthest away in the summer (and yet we are warm). We also find it in the study of numbers, commonly called Number Theory.

According to UT, God created humans according to a dual pair system: humans are divided into male and female, but each member of this pair also has a mind and a body. Thus simply said, there are two types of pair systems, or two types of yang and yin: male/female and mind/body. The ancient Oriental philosophy of yang and yin does not differentiate between these two types of pair systems, and Rev. Sun Myung Moon has to be credited for making this further and far-reaching distinction.

In UT the internal invisible mind is called sungsang and the outer visible form is called hyungsang. The body of a human is its outward visible form, even though it has a larger invisible part hidden behind the outward skin. In other words, even the body reflects the ideas of sungsang and hyungsang already. Also keep in mind that the invisible mind of a person is made somewhat visible through actions, words, gestures, deeds and bodily expressions, and is especially visible in all kinds of works of art. Yet the vast treasures of the mind lay otherwise hidden.

Thus, the body has invisible parts and the invisible mind has visible manifestations. Furthermore, according to UT, even the overall sungsang of a person is divided into an invisible spiritual mind and a visible spiritual body (i.e., visible to those who are in the spiritual world).

It is this further subdivision of the notions of sungsang and hyungsang that I call to the reader's attention, because, surprisingly, I also found such distinctions of sungsang within hyungsang, and vice versa, to hold true for numbers: Something appears on the surface, but underneath it is a different and often partially or even totally hidden, although decipherable, structure. I found such "dual characteristics" (UT terminology) in the study of prime number reciprocals. As prime numbers are the fundamental building blocks of all numbers, we therefore concentrate on primes, even though my findings are also applicable to composite numbers.

In this paper I will first explain how I see the basic character of a prime number to already possess a fundamental sungsang and hyungsang structure, and then I will take a deeper look at some very fundamental arithmetic calculations and show that they clearly reveal these surprising affirmations of UT's dual pair system and dual characteristics.

Two Types of Reciprocals, Sungsang and Hyungsang

All numbers are derivatives of primes, so we have to start number theory with the study of primes (a prime is a number that cannot be subdivided any further). Let us look at the prime number 7. I will call its own apparent value, or its external form and outward appearance, namely 7, its hyungsang, and I will call its internal character, its sungsang which is normally invisible, what in traditional mathematics is called its reciprocal, in this case 1 ÷ 7 = 0.142857...[ii]

According to the standard mathematical definition (whereby "n" stands for any number value), the term "reciprocal" simply means: "1" (one) divided by any given number, or 1 ÷ n. However, something is intrinsically lacking in this approach. After all, n itself can only obtain its value by this definition: n ÷ 1. Thus, the question "How many ones are here?" is answered in the above example by "There are seven ones here!" The term reciprocal implies a relationship, and the relationship between n and "1" is indeed established in these two "mirror" formulas:

and we should add that these are "relational mirrors." We can also call them -- using UT language -- the dual characteristics of a number. Thus, there are two types of reciprocals:

Number theorists do not consider this way of looking at numbers with a dual structure, but in light of UT it is quite the correct way of doing so and sets the tone and stage for our further investigations.[iii]

I will continue to use the term reciprocal in the traditional standard definition way (1/n), as there is no need to write "regular" hyungsang numbers as n/1. We just keep these dual aspects in our mind.

Prime Number Reciprocals: Overlooked in Mathematics

Mathematical dictionaries and encyclopedias make no particular fuss over reciprocals (also called "multiplicative inverse"), or have no entry on the topic at all except to note that a reciprocal is 1/n and perhaps give an example like 1/5 = 0.2. I have come to conclude over the past five years of research that this remains an overlooked area of mathematics.

The conventional method of resolving fractions like 1/8 + 1/9 as 2/72, instead of as 0.125000... + 0.111111111... = 0.326111111..., definitely looks neater and more presentable, but this convention is nevertheless also rather superficial. In order to see the intricate inner structures of reciprocals, it is best to have a calculator or computer capable of hundreds, even millions of digits.[iv]

Sungsang and Hyungsang in Prime Number Reciprocals

The first prime number with a beautiful sungsang and hyungsang in its reciprocal value is the number 7. Its reciprocal value is simply 0.142857... repeated forever. It is actually quite easy to see that this number starts with a multiplication series from left to right, starting with 14, doubling to 28, doubling to 56, then 112, 224, etc. Since these values are allotted only two digit spaces each, and progress by two digits at a time, the value 112, which has three digits, must share its first digit by overlapping it on top of the previous 56, turning it into a 57 (and so on). The final result of all these overlapping digits is then the beautiful and steady repeat of 142857.

It is quite remarkable: all those seemingly random values as the result of continuously multiplying 14 by 2 and then layering all the results on top of each according to a 2-digit advance for each next value, still results in the same 142857 repeat. It reveals a very orderly phenomenon.

The important point I want to make here is about what we see: an external result which is quite different in appearance than its internal value structure! The external form seems fixed: 142857..., like a rigid application of a fixed pattern. The internal form is also fixed, even though it occurs as ever increasing values which are constricted by a fixed allotment of digits. But we cannot change this.

It is precisely this internal, invisible structure, namely the multiplication series running from left to right (or the divisions running from right to left), which I will call the invisible sungsang structure. That is the inherent nature of the number that has become visible as its hyungsang (and as a totally different looking numerical reciprocal repeat value). Thus the simple looking 142857 represents a far greater hidden reality: hyungsang hides sungsang.

And it is precisely this different looking value, the external visible form as the very result of the internal sungsang, which becomes the tangible expression of the reciprocal value -- in the above case, 0.142857... The latter should be called the hyungsang aspect of the reciprocal value. Keep in mind that above I called the reciprocal as a whole the sungsang (and in my example, the number 7 itself, its hyungsang). Thus, we see here in a prime number reciprocal, just as in the mind and body of a person as explained above, further divisions of sungsang and hyungsang.

The reciprocal value is so fixed and so absolute, that even when we add 1/7 to 1/7, the values of 142857 remain in place:

1/7 = 0.142857142857...

2/7 = 0.285714285714...

3/7 = 0.428571428571...

4/7 = 0.571428571428...

etc.

Allow me to briefly demonstrate this absoluteness also with another reciprocal value, the prime number 19, whereby the reoccurring same digits have been aligned vertically:

1/19 = 0.052631578947368421052631578947368

2/19 = 0.10526315789473684210526315789474

3/19 = 0.15789473684210526315789473684211

4/19 = 0.21052631578947368421052631578947

etc.

The Remarkable Number 81

Even the simplest and most fundamental number series, namely the multiplication table of the number 1, which we all learned when we were children, shows an amazing internal sungsang and hyungsang structure. This table simply starts with 1, and continues with 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, etc. Simple enough. But, if I were to write this number like a reciprocal, allotting only one digit space per next appearing number, I have to make adjustments: the 10 that follows the 9 forces this 9 to become a 10, of which I can only write the zero, and forces the 8 before it into a 9, while the 7 before that is not affected. Let me show this in a simple table:

We discover that the sequence of zero to infinity in single digit spacing will write out as: 0.012345679012345679012345679... (introducing a decimal point). In this case, the hyungsang or the "what we see" is 0.12345679..., while the sungsang or the "what we don't see" (but can decipher) is 0-1-2-3-4-5-6-7-8-9-10-11-12-13-14-15-16-17-18-19-20-21-22-etc. (hyphens mine).

The most remarkable thing is that 0.012345679... happens to be the reciprocal of 81. The number 81 thus unlocks all the numbers in the universe from 0 to __ (infinity) in its reciprocal value.

Ponder for a moment at the fact that the universe is constructed from 81 stable elements, who, according to German chemist and number theorist Peter Plichta[v] can be arranged, according to their inherent numerical structure (based on their atomic number and the number of isotopes, and which seemingly reflects a sungsang again), into 4 columns of 1 + 19 elements, with one element sitting on top of this chart:

Ponder also that 81/19 = 4.2631578..., whereby the first four digits coincide with the order of these columns. It is furthermore quite remarkable that the number 19 forms the top of the chart, as 100 - 81 = 19, following a decimally-based approach.

The number 19, in turn, in its reciprocal value, actually provides a clue to unlocking the numbers involved in human/animal ancestral lineage: 0.052631578947368421.... When the reciprocal is read from right to left, we see the numbers 1 (me), 2 (number of parents), 4 (number of grandparents), 8 (number of great-grandparents), 16 (etc.), and we see how the hyungsang of this number appears when analyzed this way.

In my analysis of prime number reciprocals, I discovered that each and every one can be constructed from such a logical number series, or more correctly said, an (often hidden) multiplication/division table (i.e. with multiplication running from left to write in our writing convention and a division table running in the opposite or mirror direction.) We have just used the multiplication table of 1 to derive the reciprocal of 81, and we have noticed a division table hidden in the reciprocal of 19.

The Decimal System at Work in Reciprocal Values

We just saw that 19 is related to 81 via the number 100, using a decimally-based calculation. We all know that 10 - 7 = 3. We also saw above that the issue of digit allotment is the key to unraveling the inner sungsang structure of reciprocal values. As the number 3 is decimally related to the number 7, and since we have been working according to decimal system so far, what would happen if I construct a reciprocal based on the series 0,1,2,3,4,5,6,7,8, etc. that we just discussed, and by the number 3? Thus, I am combining two approaches.

The way I will do this is rather simple: I will introduce a single digit allotment per each new value established, and then discover what value the overlapping digits would produce, and of which number this could possibly be the reciprocal. The series is simply: 3⁰, 3¹, 3², 3³, 3⁴, 3⁵, 3⁶, etc. To facilitate my table, I have introduced a decimal point (note that each new value is written one more digit to the right than the previous value):

0	0.10000000
1	0.03000000
2	0.00900000
3	0.00270000
4	0.00081000
5	0.00024300
6	0.00007290
	Etc.,
Total:	0.142857...

Taking the sum of all the terms in the series we find the value 0.142857142857... As we already know, this is, perhaps still surprisingly, the reciprocal of the number 7.

Thus, there are at least two ways to arrive at the value of 0.142857:

With further research, I found that I could construct reciprocals of any prime number via this decimally-based approach. Thus, as hinted at above, I can construct the reciprocal of 81 with the number 19, as 100 - 81 = 19. And so on. As I have not found any other method to establish the same reciprocal value, I preliminarily assume that these two methods are also linked to the dual characteristics mentioned in UT.

It is very important to note the fundamental role of the decimal system in deciphering the sungsang and hyungsang structure of prime number reciprocals. Only the decimal system allows these two different methods just described to be applied in order to establish the same reciprocal value. I am saying this, because many mathematicians insist that the decimal system is a mere human invention, based on the "convenience" of us having ten fingers.[vi]

Dual Characteristics Inside Simple Calculations

Allow me to make an interesting observation of the kinds of things we seem to take for granted in basic arithmetic: In the handwritten figure at left, I have revealed the simple calculation of the reciprocal of 7, which we already saw is 0.142857...

	0.700000
				0.280000
										0.014000
																				0.005600
																																			0.000350
																																																								0.000049
																																																																																				Total: 0.999999...

Also, if I take the non-circled numbers from each of which is being deducted, namely 1(0), 3(0), 2(0), 6(0), 4(0), 5(0), and we split this series in half, we have: 1-3-2 and 6-4-5, and added together become 7-7-7.

The idea behind splitting a series of even numbers in half is directly inspired by UT's emphasis on the pair system, rooted in the ancient Chinese philosophy of yang and yin, whereby each member is seen as each being half of a whole.

Let's return to my handwritten calculation above once again. The solution to 1/7 = 142857... Keep in mind that the number of repeat digits of a prime number reciprocal always follows the formula of n = 1. Thus, the number 7 has 6 digits in its reciprocal repeat section, as 7 - 1 = 6.

When I divide these six numbers into two groups of three digits each, I have 142 and 857. Added together they become 999. If I then analyze the even and oddness of the digits involved of these two halves, I get: O - E - E, followed by E - O - O. We have "mirrors" once again.

The idea behind the notion of yang and yin is that they are like opposites, or better yet, that they are complementary of each other. In ancient Chinese philosophy yang is a single solid line, while yin is a broken line (a small break in the middle). I like to see these two symbols also as mirrors.

What's fascinating here is that apart from the fixed hyungsang of 142857... and the eternally growing value of the sungsang as 14-28-56-112-224-etc., underneath it we also simultaneously witness a pair system and dual characteristics. The sungsang and hyungsang seem more fundamental, as the yang and yin can only be derived from them and cannot appear by themselves.

Let us now consider the order of the values taken out (subtracted), which are: 7, 28, 14, 56, 35 and 49 (circled in the illustration above), before the calculation process repeats itself all over again. Lined up properly we get:

Due to space limitations, suffice it to say that many similar analyses are possible of this 1/7 calculation when seen from various angles, which all result in the same confirmation of these dual characteristics.

Perfect Plus and Minus Order

Before finishing up this paper, allow me to share one more confirmation of these dual characteristics. In this exercise we will analyze the differences between the consecutive digits of prime reciprocals.

Again, we turn to the reciprocal of 7, or 0.142857... and analyze the plus or minus difference from digit to digit: from 1 to 4 is a difference of +3, and from 4 to 2 is a difference of -2, etc., which gives me this result, written again as yang and yin halves:

-6 +3 -2

+6 -3 +2

When one value is "plus", the one in the next half of the group is "minus".

The same "plus and minus pairs" also show up in an analysis of the values subtracted, as seen in my handwritten calculation above of 1/7. First of all, we see these numbers lined up: 7, 28, 14, 56, 35, 49, 7, etc. The differences between these numbers are; +21, -14, +42, followed by the "mirror" of -21, +14, and -42. Feel free to apply these ideas to calculations of any other prime number, and you will discover the same kind of absolute patterns and phenomena.

A Final Unification Thought Perspective

What then can we conclude about the entire matter so far? My personal observation has been and remains a deeply mysterious and mind-boggling astonishment when staring at a prime number reciprocal: how is it possible that so many perfectly orderly structures can all be true at the same time? And yet also be so fundamentally simple? There are no adequate words to describe this sense of awe.

Let me sum up that we have witnessed so far in prime number reciprocals:

1. Prime numbers are the fundamental building blocks of all numbers, so we need to analyze them in depth.

2. Each prime number has two types of values: n/1 and 1/n, hyungsang and sungsang, respectively.

3. In a prime number reciprocal we discovered a hidden multiplication table running from left to right and a hidden division table from right to left (due to space considerations we could not dwell on this much). This I have labeled the sungsang aspect of the reciprocal.

4. Reciprocal values can be constructed in two different ways, whereby the decimal system plays a pivotal role.

5. A visible reciprocal value that will repeat itself and whose length of digits is determined by the n-1 formula (thus prime number 19 has 18 reciprocal digits), can be called the hyungsang aspect of the reciprocal.

6. When reciprocal values are added to each other, they show the original digits repeating themselves in different locations, as the digits of the reciprocal only shift and ‘dance around" but are not altered.

7. The two halves of a reciprocal repeat section (which is always even numbered) always adds up to a series of 9's (for some types of reciprocals, this is not immediately obvious, but can be achieved by two or more steps).

8. These two halves show inner Odd and Even mirrors.

9. The differences between consecutive digits of reciprocals, as well as the fundamental reciprocal calculations involved all show inner Even and Odd mirrors as well as Plus and Minus mirrors.

Rev. Moon has stressed more than once that God is very orderly. This brief and still partial study of the reciprocal structures of prime numbers has made it clear that at the heart of primes is a perfect order. This perfect order shows itself both at the sungsang and at the hyungsang level, as well as at their sublevels. This order for each prime number reciprocal is a fixed absolute, confirming Rev. Moon's favorite catchphrase: "Absolute Values."

We could also metaphorically say: The number soldiers always stay true to their marching orders. God told Moses: "I am who I am" (Exodus 3:12), stressing this immutable absoluteness. Likewise, numbers simply are what they are. There can be no chaos or randomness in prime reciprocal values and structures, only absolute order.

Since numbers are immutable, the question may come up in our minds as to whether numbers "pre-existed" the Creation, and whether God had any numerical consciousness before the Creation. In other words: were numbers created, or are they simply an integral, inherent part of God?

Rev. Moon sometimes uses the language of something "preceding" God, such as "conscience." What I understand he means to say is that certain attributes of God cannot be any different from what they are. Even God has no choice and cannot alter them; they are the way things simply are. I am personally inclined to think that numbers belong to this realm.

One thing I have not covered so far is to use the UT language of "subject and object" or "give-and-receive action" with reference to numbers. It is probably a safe assumption to call the internal sungsang the "subject" of a prime number reciprocal value, and to label the visible hyungsang as its "object," but this needs further investigation, as give-and-receive action between a subject and an object, according to UT, brings about a new result. Are numbers symbolic static things, or are they the result of a process? It would fall beyond the scope of this paper to delve into this matter, which I also wish to develop further.

If indeed the universe is created with the inclusion of numerical structures, which several researchers are confirming and deciphering more and more,[vii] this has far-reaching implications for such disciplines as physics, biology, chemistry, science and astronomy, to name just a few -- all of which today reject the idea of a created and planned universe in favor of politically correct Darwinism.[viii]

Rev. Moon has said that the ultimate proof of God's existence lies with science, not with religion. It is my contention that in this effort, the dialogue between Unification Thought, scientific inquiry and the study of numbers will not only prove to be extremely helpful, but will actually be indispensable.

I hope that this small glimpse into the world of numbers has provided the reader with a sense of awe and mystery and that it has helped to show that even at the level of numbers, we see a divinely rooted harmony, yin and yang, sungsang and hyungsang. We can all agree with Unification Thought that all these dual structures are created only for one purpose: harmony.

Perfection is the Foundation for Numbers

Numbers are the foundation for Order

Order is the foundation for Beauty

Beauty is the face of Goodness

Goodness is the face of God's True Love

And True Love is always Perfect

Notes

[i] New Essentials of Unification Thought: Head-Wing Thought (Tokyo, Japan: Unification Thought Institute, 2005).

[ii] Throughout this paper the three dots indicate that all the digits appearing after the decimal point repeat themselves indefinitely.

[iii] It is interesting to note here that the sidereal period of the Moon, 27.32 days (a full circle around the Earth) has a reciprocal value of 0.03660..; a value that points to the number of Practical Days the Earth and Moon together go around the Sun. In my upcoming book, an entire chapter is devoted to correlative and repetitive numbers present in our solar system.

[iv] It is posted by a data processing firm, not by a university department: comptune.com/calc.php

[v] See Peter Plichta, God's Secret Formula: Deciphering the Riddle of the Universe and the Prime Number Code (Rockport, MA: Element Books, 1997).

[vi] Although we have no space for this argument here, anyone who reads Dr. Plichta's or my forthcoming book will more clearly see that the decimal system is inherent in the structure of any prime number, and is not a mere human invention.

[vii] The following books are good starters to discover the other side of the Darwinian fence: Richard Heath, The Matrix of Creation (St. Dogmaels: Bluestone Press, 2002); Robin Heath, Sun, Moon, & Earth (New York: Walker & Company/Wooden Books, 1999); Priya Hemenway, Divine Proportion: Ф (phi) in Art, Nature and Science (New York: Sterling, 2005); John Martineau, A Little Book of Coincidence (New York: Walker & Company/Wooden Books, 2001); and Scott Olsen, The Golden Section: Nature's Greatest Secret (New York: Walker & Company/Wooden Books, 2006).

[viii] UT rejects the basic Darwinian evolutionary premise that man (and the rest of creation for that matter) is a non-planned "accidental result" of an unforgiving biological process with common ancestry, by emphasizing that in God's Mind, humans came first, not last. Darwinism, despite the claims of its defenders, is inconsistent with the scientific evidence -- and the inconsistencies are mounting, according to Jonathan Wells, who furthermore points out that the pro-Darwinian movement has more to do with politics today than with objective science (see his The Politically Incorrect Guide to Darwinism and Intelligent Design (Washington, D.C.: Regnery, 2006).