copyright © July 2009 by Tienzen (Jeh-Tween) Gong

Among many natural languages, there are three important numerals still widely in use today. And, I will discuss them first before the PreBabel numerals.

The purpose of any numeral is to describe the numbers. Thus, it must include the followings:

- the concept of number
- the representation of any number, and this requires the followings:
- the expression of a number
- the symbols (glyphs) for making up that expression

One: Roman Numerals -- its main interest was to indicate *dates*. Thus, it did not need the concept of *zero* nor the organization of the positional representation of a number. Yet, its glyph design was both straightforward and genius.

- it started with a vertical rod, I as 1, II as 2, III as 3.
- for bigger numbers, it created V (as 5), X (as 10), L (50), C (100), D (500), M (1,000), 一 (a horizontal bar over the above glyphs) meant to multiply the number by 1,000. 一 over V = 5,000.
- Obviously, the above glyphs are not enough to represent all whole numbers, such as, 4 and 9. Then an arithmetics was added to the glyphs system.
- rule 1: any smaller number in front of any larger number -- indicates subtraction, such as IV (is 5 - 1) = 4. XL ( 50 - 10 ) = 40. IX ( 10 - 1) = 9.
- rule 2: a smaller number after any larger number -- indicates addition, such as VII ( 5 + 2) = 7, etc.
- rule 3: the number is by adding up its glyphs without any consideration of the positional-value
- front right to left, such as CCC is 300
- the smaller number (glyphs) is always on the right, such as, CXIX is 119, not 121.

Two: Arabic (Hindu-Arabic) numerals -- by including the glyph of *zero*, Arabic numeral is able to express any number with a *positional notation in a decimal system*. For the following two reasons, it becomes a universal numerals today.

- it is a base 10 system with 10 glyphs, including 0.
- a number is represented with positional notation, including decimal position.

Three: Chinese numerals -- the introduction of *zero* in Hindu-Arabic numeral was to satisfy the philosophical and religious needs of expressing the reality of *emptiness* in the Hindu religion. On the contrary, every glyph of Chinese numeral is reflecting the ideas of Chinese cosmology.

- 一 (1), it is not a counting rod but signifies the first creation (Heaven) from the nothingness.
- 二 (2), it represents Earth, the second creation.
- 三 (3), it represents man, the third creation.

In fact, the top 一 is Heaven, the second the man, the third the Earth. I (the vertical line) represents the fully expressed energy. So, 王 depicts the state that Heaven, Earth and man are united. Anyone who is able to unite those three is the 王 (king). - 四 (4), it is made of 八 (dividing) and 囗 (circled wall, also universe). That is, the universe is divided into 4 directions.
- 五 (5), it is 王 plus a short I (energy). After the creation of the direction (coming out from chaos) and after the union of the great three (Heaven, man and Earth), the engine of the universe comes alive, and it is the five forces.
- 六 (6), it is made of ぞ (Heaven) and 八 (divide). That is, the signs of Heaven are given with the hexagram of Yijing.
- 七 (7), it is made of 一 (heaven) and 乙 (weak energy), the energy of the universe is still weak.
- 八 (8), the division. The division is the force of the universe.
- 十 (10), the combination of 一 (first creation) and I (fully expressed energy) means perfection.
- 九 (9), it is composed of 十 (perfection) and 乙 (still weak). 九 (9) is a bit weaker than 十 .

As all natural languages are dialects of the PreBabel, those numerals above are also vocabulary of the PreBabel. However, for the PreBabel proper, we do want to have a set of PreBabel numerals.

As the main interests of those three numerals are different, does PreBabel (PB) numeral have its own metaphysical or ontological interest? Or, it simply has some glyphs to represent the 10 digits? The goal of PB numerals is for having the capability to ** mark** every number. Can the current Arabic numerals accomplish this task? According to the current mathematics, it cannot.

For any two numbers a and b, the current mathematics states that there are infinite numbers between them. Therefore, there is no way to mark those infinite numbers with any numeral system. The above statement is the result of the concept of *completeness* of the *real number*. And, it is the consequence of the concept of *continuity* in mathematics. Continuity is defined in two steps in mathematics:

- the concept of
*limit*: x is a number, f(n) is a number sequence approaching to x. When n goes to infinite, f(n) = x.When n goes to infinite, f(n) = x. - the concept of
*continuity*: z is an arbitrary small number. If we can always find a "n" to ensure that [f(n) - x] < z, then the segment [f(n) - x] is continuous.

- there are three
*zero*in PB numerals:- 0(1), -- nothing ever existed and will never come to existence
- 0(2), -- something existed but is
*now*nothingness - 0(3), -- there is nothing
*now*, yet it will come into being in the future

- there are two cases for the equation x - y = 0
- identity issue -- x and y are the same number. So, there is no dispute of any kind for x - y = 0
- distance issue -- x, y are two different numbers but they are touching each other with the distance between them to be zero. Do such numbers exist? Modern mathematics says no. However, in the process of x becoming y, x was different from y all the way before it becomes y. We might not have the ability to catch
*the moment*that x turns into y. However, there is such a moment. That is, we could and ought to name that*dark moment*with a numeral glyph, regardless of that moment is a single number or a bucket of numbers. Thus, every number has two numbers associate with it, the coming in bucket and the going out bucket.

Yes. in PB numerals, there are two more numbers (cx = y, xg = z) to every number x which is identified with the Arabic numerals while y and z are defined with the following equations.

- y (n) is a sequence of number and every y (n) =< x; y is a number of y (n) and is not x, but x - y = 0, then y is a dark moment number of x or the coming in x.
- z (n) is a sequence of number and every z (n) >= x; z is a number of z (n) and is not x, but z - x = 0, then z is a dark moment number of x or the going out x.

Thus, the PB numerals need two more glyphs, the coming in, the going out. For the number 5.01, there are three numbers

- 5.01, five point zero one
- 5.01 , coming in five point zero one
- 5.01 , going out five point zero one

However, in addition to the above modified Arabic numerals, we do need names for those PB numeral glyphs. After reviewed many numeral systems, I believe that the Chinese numerals provide the best metaphysical and ontological foundation. Thus, I will simply encode Chinese numbers with the PB roots together with some Biblical stories as the names for the PB numerals.

- zero (no, one)

- one , creation of the Heaven (the time)

- two , creation of the universe (the space)

- three , creation of the man

- four (divide, direction), creation of order from chaos

- five , (wood, water, fire), creation of elements

- six (engineering, complete) completion of creation

- seven (house, time), day of rest

- eight (new, one)

- nine (near, completion)

- ten (complete, complete), perfection

The last but not the least, we also need some names for the big numbers. In Chinese, the large number is marked with 10,000 increment while it is 1,000 in increment in English. I will encode them with English system.

- hundred (big step, ten)

- thousand (rice, rice)

- million (bushes, bushes)

- billion (hair, hair)

- trillion (man, billion)