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BY THE NUMBERS

It was in 1972 that I sat down on the living room floor with my five year old son to write all the numbers from 0 to1000. I don't recall my motivation except that Michael was learning to count and I always spent Tuesday afternoon at home. We finished ( Michael went to sleep on the couch at about 400) and I began to wonder how many digits I had written on that long taped together scroll. I thought it should be easy to calculate.     A B C  AA AB AC  BA BB BC  CA CB CC etc . C to the first power for the single digit combinations, C to the second power for the two digit combinations and so on. So I came up with 10 single digit combinations and 100 two digit combinations. I quickly realized that calculation was incorrect and that is where this all started.

IN THE DAYS THAT FOLLOWED I CAME TO A REALIZATION WHICH MYSTIFIED ME. OVER THE YEARS I HAVE NOT BEEN ABLE TO FORGET OR EVEN TRIVIALIZE MY OBSERVATION. IN FACT IT IS A SELF EVIDENT TRUTH WHICH WILL EVENTUALLY RESULT IN THE ABANDONMENT OF OUR CURRENT COUNTING SYSTEM IN FAVOR OF ONE OR MORE ORDERLY SYSTEMS.


 So, where are we going with all this ?


It is obvious this guy is no mathematician !


COUNTING TODAY
THERE IS NO ARGUING THAT OUR COUNTING SYSTEM WORKS AND SERVES WELL FOR ALL TYPES OF COMPUTATIONS. THERE IS ALSO NO DOUBT THAT IT IS A DISORDERLY PROGRESSION OF DIGITS WHICH IS MADE MESSY BY THE INCLUSION OF ZERO AS A COUNTING DIGIT. A LOGICAL EXAMINATION OF THE PROGRESSION OF DIGITAL COMBINATIONS WILL REVEAL THIS TRUTH. FURTHER INSPECTION WILL SHOW THAT ZERO IS NECESSARY ON THE RIGHT SIDE OF THE DECIMAL POINT TO REPRESENT NAUGHT AND THAT IT IS USEFUL AS AN INDICATION OF NAUGHT IN COMPUTATION. IT IS NOT NECESSARY AS A PLACE VALUE NOTATION WHEN THE PROGRESSION OF DIGITS IS ORDERLY AND IT IS NOT NECESSARY TO COUNT WITH. ANY WHOLE VALUE MAY BE EXPRESSED WITH AN ORDERLY SYSTEM OF COUNTING DIGITS WITHOUT THE USE OF ZERO.

I know you are thinking "This guy is crazy." It won't take long to read. Just keep going.

SIMPLE AS ABC

LET US LOOK AT HOW A SERIES OF SYMBOLS MIGHT BE COMBINED IN AN ORDERLY PROGRESSION OF COMBINATIONS.
WE WILL START WITH THE FOLLOWING SYMBOLS:
A  B  C
THE NEXT STEP MIGHT BE TO CREATE TWO SYMBOL COMBINATIONS.
AA  AB  AC
BA  BB  BC 
CA  CB  CC
AND THE THREE SYMBOL COMBINATIONS.
AAA  AAB  AAC
ABA  ABB  ABC
ACA  ACB  ACC
AND THE NEXT VARIATION.
BAA BAB  BAC
BBA  BBB BBC
BCA  BCB  BCC
AND SO ON THROUGH CCC AND THEN AAAA  AAAB  AAAC ........

WE COULD ASSIGN QUANTITIES TO EACH OF THESE VARIATIONS AND USE THIS SYSTEM TO COUNT AND PERFORM COMPUTATIONS.
A=I   B=II  C=III    A+B=C
AAB = IIIII IIIII IIII       B=II
AAB DIVIDED BY  B = BA = IIIII  II

THIS IS A PLACE VALUE SYSTEM IN THE BASE C. THE FIRST PLACE INDICATES UNITS IN THE GIVEN QUANTITY. THE SECOND PLACE THE C'S ( III .) THE THIRD PLACE C X C. THE FORTH PLACE C X C X C  AND SO ON.

YOU CAN SEE THAT NO QUANTITY WOULD BE OMITTED IN THE SYSTEM AND THAT ZERO IS NOT USED. IN COMPLICATED COMPUTATION ZERO IS USEFUL TO EXPRESS NAUGHT AND TO HOLD PLACES OPEN.

B0 = AC      B00 = ABC    A0A = CA

ZERO IS NEVER NECESSARY TO EXPRESS ANY WHOLE QUANTITY AND IT IS POSSIBLE TO PERFORM ALL TYPES OF COMPUTATION WITHOUT USING ZERO AS A COUNTING DIGIT OR AS A PLACE HOLDER WHEN EXPRESSING WHOLE VALUES.

I just checked my abacas. No zero !

But this is ridiculous ! I'm not going to count with A B C . Talk about "messy."


LET'S TAKE A CLOSE LOOK
!
WE CAN NOW TAKE A LOOK AT OUR POPULAR COUNTING SYSTEM. THE FIRST THING WE WILL DO IS REPLACE THE NUMERALS WITH LETTERS SO WE CAN SEE THE PATTERNS OBJECTIVELY.

                                0     1    2     3    4    5     6     7    8    9
       A    B    C    D    E    F    G    H    I    J  

THE NEXT STEP IN AN ORDERLY PROGRESSION MIGHT BE TO CREATE  TWO DIGIT COMBINATIONS.
                                        
                                       AA    AB   AC    AD    AE    AF    AG    AH    AI    AJ   
                                       00     01    02     03    04     05     06     07    08    09   

THESE COMBINATIONS ARE NOT USED IN OUR POPULAR COUNTING SYSTEM.
DISORDER !

LET US SEE WHAT IT LOOKS LIKE IF WE OMIT THE ZERO.

                                                                 1    2     3     4     5     6    7    8    9   
                                                                 A    B    C     D    E     F    G   H    I

AND OUR TWO DIGIT COMBINATIONS

11   12   13   14   15   16    17   18   19   
21   22    23    24   25   26   27   28   29  
 
CONTINUING TO 99 WITH TWO DIGITS AND ON TO 111. WITHOUT THE USE OF ZERO WE HAVE A BASE NINE SYSTEM AND NEW QUANTITIES WILL NEED TO BE ASSIGNED TO NUMBERS GREATER THAN 9.

15 =  IIIII IIIII IIII
29 = IIIII IIIII IIIII IIIII IIIII II

 
AT SOME POINT ZERO WAS GIVEN A PLACE IN OUR COUNTING SYSTEM. THIS RESULTED IN A BASE TEN PLACE VALUE SYSTEM IN WHICH WE HAVE NO SINGLE DIGIT DESIGNATION FOR TEN. THE RESULT IS A COUNTING SYSTEM WHICH IS PART BASE TEN AND PART BASE NINE AND IN THE END IT IS DISORDERLY.


SO WHAT ?
 THERE IS NO ARGUING THAT OUR COUNTING SYSTEM WORKS. YOU COULD ALSO BUILD A HOUSE USING A RULER WHICH IS A MIXTURE OF INCHES AND CENTIMETERS AS LONG AS YOU ARE CONSISTENT AND DON'T CHANGE RULERS.

WE COULD CONTINUE TO USE OUR CURRENT COUNTING SYSTEM WITHOUT DIRE EFFECTS. HOWEVER, JUST AS WE PREFER TO VISUALIZE THE SUN AS THE CENTER OF THE SOLAR SYSTEM, WE WILL EVENTUALLY ABANDON OUR DISORDERLY COUNTING SYSTEM IN FAVOR OF ONE OR MORE ORDERLY SYSTEMS. INTELLECTUALLY WE ALWAYS PREFER THE SIMPLER AND MORE ORDERLY ARRANGEMENT OF INFORMATION. THE QUESTION IS NOT IF. THEQUESTION IS WHEN ?



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