Fundamental properties of decimal numbers. By: Bill Boas Numerology, the application of number analysis to everyday life, has fascinated mankind for ages. Unlike spoken and written languages, the symbols of number language are precise and can more consistently represent real quantities and relationships. Many numerological systems have been devised, but few seem to have a sound mathematical base. What follows is derived from numbers themselves, and can be used to evaluate numerological systems as well as natural and cultural numerical relationships. The number base 10, uses the integers 1,2,3,4,5,6,7,8,9 and 0. Other numbers are combinations of these integers, which will be shown to possess unique place values and inherent symbolism of their own. Two simple principles can apply to any number and provide a key to numerology. The first is integer reduction, the second is integer series addition or cross-summing. Reduction means adding the integers of any number to produce a sum which is a single digit. This is not a spurious process as will be shown. The integers of 1998, for instance, are reduced as follows: 1+9+9+8 = 27 = 2+7 = 9. The number 427 reduces to 4+2+7 = 13 = 1+3 = 4. As single digits, the numbers 1998 and 427 reduce to 9 and 4 respectively. Reduction is important to reveal a number's place value in what will be later demonstrated to be an infinitely repeatable pattern. The second principle is series addition, and involves adding up all the numbers that lead to a number. For instance, to arrive at the number 7 one has to `pass' the integers 1,2,3,4,5, and 6. Similarly, a person 50 years old lives each of the years that leads to 50 (50+49+48+...et.seq.). Thus, the `sum' of a person 50 years old is different from when that person was only 25 years old. The series addition principle is essential to the program formulas needed to generate anagrams by computer as shown by the addendum to this brief article. Series addition is applied after integer reduction, so that only the series sums of single digits need to be calculated. In the previous example, reduction of 1998 resulted in 9. Applying series addition to 9 yields 9+8+7+6+5+4+3+2+1 = 45 = 4+5 = 9. Thus, applying both principles to 1998 yields a final value of 9. The number 427 was reduced to 4. The series of 4 = 4+3+2+1 = 10 = 1+0 = 1. Reducing numbers to single digits by adding their place values, and then adding the series of the result generates a final integer that can represent any number's symbolic position in what will be seen as a universally valid numerical scheme. By this scheme, every 4th number from 1, inclusive, to infinity reduces to 1, which historically represents the idea of unity. Thus, the notion of unity can be derived from 4, 7, 10, 13, 16, 19, 22 ... etc. Thus 427, which reduces to 4+2+7 = 13 = 1+3 = 4, whose series 4+3+2+1 = 10 = 1+0 = 1, is another aspect of symbolic unity. Deriving the values of numbers to get a single digit is key to determining its place in the four number repeating series which may be conceptualized as: 1 (unity), 2 (its opposite), 3 (synthesis), and 4 (a new unity). Since all numbers reduce to the single integers 1,2,3,4,5,6,7,8, and 9, the series addition of each of these reveals that there are only four resulting numbers: 1,3,6, and 9. This is simply demonstrated by calculating the series sums of each: 1 = 1 2 = 2+1 = 3 3 = 3+2+1 = 6 4 = 4+3+2+1 = 10 = 1+0 = 1 5 = 5+4+3+2+1 = 15 = 1+5 = 6 6 = 6+5+4+3+2+1 = 21 = 2+1 = 3 7 = 7+6+5+4+3+2+1 = 28 = 2+8 = 10 = 1+0 = 1 8 = 8+7+6+5+4+3+2+1 = 36 = 3+6 = 9 9 = 9+8+7+6+5+4+3+2+1 = 45 = 4+5 = 9 Note that the numbers 1, 3, 6, or 9 reduce only among themselves, with 3 and 6 being complementary to each other as 3 evolves into 6, and 6 into 3 to wit: 3 = 3+2+1 = 6 = 6+5+4+3+2+1 = 21 = 2+1 = 3. This 1-3-6-9 pattern seems to fit natural archetypes. If 1 is said to be unity and a point of initiation, then 3 and 6 are `tools of synthesis' which lead to 9 as completion. The essence of 3 may be said to be an early assembly, with 6 representing a more advanced ordering before completion. To verbalize the analogy: an idea is (1) born, (3) explored, (6) ordered, and (9) completed. The 1-3-6-9 pattern reveals itself as some kind of primal relationship by the same principles by which it was derived here: (1+3+6+9 = 19 = 1+9 = 10 = 1+0 = 1), or unity. Interestingly, the square root of 1369 is the whole number 37, which is 3+7 = 10 = 1+0 = 1, also representing unity, but this seems to be a special case. However, all higher whole powers of a number that represents unity also represents unity, as does the aggregate sum of the series of any unity number. Thus, 1369 to the power of 3 is 2,565,726,409, which reduces to 1. The same unity will result from 1369 calculated to the powers of 4, 5, 6, 7, et seq. A computer will show the sum of the series of numbers of 1369 (1369+1368+1367+.et.seq.) as 937,765, which also reduces to 1. The number 427 cited previously has a series sum of 91,378 = 28 = 10 = 1 and therefore is consistent with unity. A short program in BASIC for any computer that can run a BASIC compiler or interpreter is appended to this article to demonstrate this or the series sum of any other number. Since these principles demonstrate their own proofs, they can also reveal which numerological systems have some basis in mathematical reality and which might be flights of someone's fancy. Using these techniques, much light can be shed on many esoteric doctrines of ancient and modern science. An anomaly is that both 8 and 9 reduce to 9. Verbally, this might be expressed as things are `initially complete' at 8, yet can be finally tweaked to become definitely complete at 9. At this point one can only conjecture the meaning of some prominent numbers in the ancient literature. In the `Judeo-Christian' cosmology's `Genesis' the creation is described as occurring over 7 days. Are we to infer, therefore, that creation itself is a unity? In a chapter called `Revelation' from the same source, we are presented the number 666 as that of an apocalypse. Since 666 reduces to 9, could it not simply mean the `completion' of an era before a new order begins? Lately, new research indicates that this number is actually 616 instead of 666. If that's the case, 6+1+6 = 13 = 1+3 = 4, the series sum of which is 10 = 1+0 = 1, or unity! Similarly, in the tarot deck the 13th card is that of `death'. Is the proverbial glass half-empty or half-full? Put another way is what's beyond `death' a new beginning or `unity' as the much-maligned yet unity number 13 reveals about itself? The unity and completion symbolism inherent in the numbers 4, 7, 8 and 9 respectively might explain their veneration by many ancient cosmologies. Creative use of these two tools of number analysis will reveal endless fascinating relationships of life's many past and present number symbols. Addendum BASIC programs to generate anagrams and series sums. Separate the individual programs as ASCII files, name them as shown in "quotes", and copy them to any C:\ directory. Open and run them with GWBASIC or QBASIC under the respective program instructions. Line 109 of each program is illustrative of the series sum principle. The line reads: "I(x) = X - " where X is the series sum of x in all cases. No other X will run the program properly. Thus in the program of 5 elements, 15 must be the number used after I3, not 13 or 14. In the program of 10 elements X must be 55 the series sum of 10, no other number will do. In line 109 of any program change X and prove it for yourself. Note: When a program is run the prompt "?" will appear. Enter any ascii character or several, separated by a coma until the requisite number is entered, e.g. for "Anag5.Bas" enter, ? 1,2,3,4,5 (enter) and the program will generate all permutations of 1,2,3,4,and 5. Any ASCII string can substitute for integers. Input can be words, ascii graphic symbols, even whole sentences, as long as the string is separated by comas. BASIC computer program for calculating series sums: 1 rem "Series.bas" 5 REM BASIC program to calculate the series sum of any number 10 DEFDBL S 20 INPUT "Number"; X 30 S = 0: FOR I = 0 TO X: S = S + I: NEXT I 40 PRINT , "Series Sum = "; S: PRINT : GOTO 20 BASIC computer programs for calculating anagrams: 6 REM "ANAG5.BAS" a BASIC Program to make anagrams of 5 elements. 10 CLS 20 INPUT L$(1), L$(2), L$(3), L$(4), L$(5) 40 FOR I1 = 1 TO 5 50 FOR I2 = 1 TO 5 60 IF I2 = I1 THEN 130 70 FOR I3 = 1 TO 5 80 IF I3 = I1 OR I3 = I2 THEN 120 91 FOR I4 = 1 TO 5 92 IF I4 = I1 OR I4 = I2 OR I4 = I3 THEN 119 109 I5 = 15 - (I1 + I2 + I3 + I4) 111 PRINT , L$(I1); L$(I2); L$(I3); L$(I4); L$(I5); 119 NEXT I4 120 NEXT I3 130 NEXT I2 140 NEXT I1 6 REM "ANAG6.BAS" a BASIC Program to make anagrams of 6 elements. 10 CLS 20 INPUT L$(1), L$(2), L$(3), L$(4), L$(5), L$(6) 40 FOR I1 = 1 TO 6 50 FOR I2 = 1 TO 6 60 IF I2 = I1 THEN 130 70 FOR I3 = 1 TO 6 80 IF I3 = I1 OR I3 = I2 THEN 120 91 FOR I4 = 1 TO 6 92 IF I4 = I1 OR I4 = I2 OR I4 = I3 THEN 119 95 FOR I5 = 1 TO 6 96 IF I5 = I1 OR I5 = I2 OR I5 = I3 OR I5 = I4 THEN 118 109 I6 = 21 - (I1 + I2 + I3 + I4 + I5) 111 PRINT , L$(I1); L$(I2); L$(I3); L$(I4); L$(I5); L$(I6); 118 NEXT I5 119 NEXT I4 120 NEXT I3 130 NEXT I2 140 NEXT I1 6 REM "ANAG10.BAS" a BASIC Program to make anagrams of 10 elements. 10 CLS 20 INPUT l$(1), l$(2), l$(3), l$(4), l$(5), l$(6), l$(7), l$(8), l$(9), l$(10) 40 FOR I1 = 1 TO 10 50 FOR I2 = 1 TO 10 60 IF I2 = I1 THEN 130 70 FOR I3 = 1 TO 10 80 IF I3 = I1 OR I3 = I2 THEN 120 91 FOR I4 = 1 TO 10 92 IF I4 = I1 OR I4 = I2 OR I4 = I3 THEN 119 95 FOR I5 = 1 TO 10 96 IF I5 = I1 OR I5 = I2 OR I5 = I3 OR I5 = I4 THEN 118 97 FOR I6 = 1 TO 10 98 IF I6 = I1 OR I6 = I2 OR I6 = I3 OR I6 = I4 OR I6 = I5 THEN 117 99 FOR I7 = 1 TO 10 100 IF I7 = I1 OR I7 = I2 OR I7 = I3 OR I7 = I4 OR I7 = I5 OR I7 = I6 THEN 116 101 FOR I8 = 1 TO 10 102 IF I8 = I1 OR I8 = I2 OR I8 = I3 OR I8 = I4 OR I8 = I5 OR I8 = I6 OR I8 = I7 THEN 115 103 FOR I9 = 1 TO 10 104 IF I9 = I1 OR I9 = I2 OR I9 = I3 OR I9 = I4 OR I9 = I5 OR I9 = I6 OR I9 = I7 OR I9 = I8 THEN 114 109 I10 = 55 - (I1 + I2 + I3 + I4 + I5 + I6 + I7 + I8 + I9) 111 PRINT , l$(I1); l$(I2); l$(I3); l$(I4); l$(I5); l$(I6); l$(I7); l$(I8);l$(I9); l$(I10); 114 NEXT I9 115 NEXT I8 116 NEXT I7 117 NEXT I6 118 NEXT I5 119 NEXT I4 120 NEXT I3 130 NEXT I2 140 NEXT I1 Comment or questions to: wboas@nyx.net