Editorial Note
Introduction to Fekete's Stepnumbers monograph (c. 2010), setting out the theoretical motivation for a number system with infinitely many digits and explaining its relationship to the binary system and Cantor's transfinite ordinals.
- The rule of enumeration endows the binary number system with a natural order of magnitude.
The value of a binary number is just its ordinal under the natural order.
The table on the next page displays the first one hundred consecutive binary numbers in lexicographic order. The arrangement in ten columns helps finding their value quickly. For example, we have 1111 = 15 because 1111 stands in the first row and fifth column (the numbering of rows and columns starts with zero); 100000 = 32 as it stands in the third row, second column. The binary numbers 1, 10, 100,... are just the powers of 2: 10n = 2n.
Table of the first one hundred consecutive binary numbers
1000
1001
1010
1011
1100
1101
1110
1111
10000
10001
10010
10011
10100
10101
10110
10111
11000
11001
11010
11011
11100
11101
11110
11111
100000
100001
100010
100011
100100
100101
100101
100110
100111
101000
101001
101010
101011
101100
101101
101110
101111
110000
110001
110011
110100
110101
110110
110111
111000
111001
111010
111011
111100
111101
111110
111111
1000000
1000001
1000010
1000011
1000100
1000101
1000110
1000111
1001000
1001001
1001010
1001011
1001100
1001101
1001110
1001111
1010000
1010001
1010010
1010011
1010100
1010101
1010110
1010111
1011000
1011001
1011010
1011011
1011100
1011101
1011110
1011111
1100000
1100001
1100010
1100011
The table can also be used for performing addition and subtraction of binary numbers, following the Slide Rule Principle. For example, in calculating the sum 10011 + 1001111 we locate the larger number in the table and move forward through a number of steps that corresponds to the smaller number 10011 = 19 to get 10011 + 1001111 = 1100010. To check, we write: 79 + 19 = 98 =
1100010, the binary number in row 9 and column 8. In calculating the difference 1001000 – 11100 we move backwards from 1001000 through 11100 = 28 steps to get 101011. To check we write:
72 – 28 = 44 = 101011, the binary number in row 4 and column 4.
There is a natural isomorphism between the binary number system and the graded lattice of finite subsets. In more details, consider
= {1, 2, 3,..., n,...}, the set of natural numbers, and its subsets n = {1, 2, 3,..., n}. n form a lattice for every n = 1, 2, 3,... under inclusion. The union of these is a graded lattice with grading furnished by n, the number of digits. A binary number of at most n digits can be interpreted as a characteristic function of n , provided that we add a sufficient number of 0 digits up front to make it a binary number of exactly n digits (which, of course, will not affect its value). This is a one-to-one correspondence between binary numbers and finite subsets. It can be extended to a lattice isomorphism. It does not depend on arbitrary choices. In enumerating binary numbers we actually construct all finite subsets. Thus binary numbers can be used to count the number of subsets. The great utility of the binary number system is due mostly to this fact, which is not widely recognized. The binary number system owes its naturality to the existence of this natural isomorphism.
* * *
In order to be able to count in the binary number system we introduce names for the two digits: 0 ala, 1 ale. We also introduce names for the numbers 10n called ‘milestones’ as follows. We adopt a variation of the paradigm million, billion, trillion, etc. modified and simplified by using permutations of the five vowels a, e, i, o, u. Using the names of the milestones on the following page we can count in the binary system on the same principle as we do in the decimal system, thus: 0 ala, 1 ale, 10 mala, 11 mala ale, 100 bala, 101 bala ale, 110 bala mala, 111 bala mala ale, 1000 trala, 1001 trala ale, 1010 trala mala, 1100 trala bala, 1101 trala bala ale,
1110 trala bala mala, 1111 trala bala mala ale, 10000 quadrala, 10001 quadrala ale, etc.
NAMES OF MILESTONES mala bala trala quadrala pentala hexala heptala octala novala
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059 halla malla balla tralla quadralla pentalla hexalla heptalla octalla novalla
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019 hale male bale trale quadrale pentale hexale heptale octale novale
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069 halle malle balle tralle quadralle pentalle hexalle heptalle octalle novalle
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029 hali mali bali trali quadrali pentali hexali heptali octali novali
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079 halli malli balli tralli quadralli pentalli hexalli heptalli octalli novalli
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039 halo malo balo tralo quadralo pentalo hexalo heptalo octalo novalo
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089 hallo mallo ballo trallo quadrallo pentallo hexallo heptallo octallo novallo
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049 halu malu balu tralu quadralu pentalu hexalu heptalu octalu novalu
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099 hallu mallu ballu trallu quadrallu pentallu hexallu heptallu octallu novallu
Second order milestones are the stepnumbers 1010n . Their names are as follows:
Names Of Second Order Milestones
1010
1020
1030
1040
1050
1060
1070
1080
1090 hale hali halo halu halla halle halli hallo hallu
10500
10510
10520
10530
10540
10550
10560
10570
10580
10590 halya halye halyi halyo halyu hallya hallye hallyi hallyo hallyu
10100
10110
10120
10130
10140
10150
10160
10170
10180
10190 hela hele heli helo helu hella helle helli hello hellu
10600
10610
10620
10630
10640
10650
10660
10670
10680
10690 helya helye helyi helyo helyu hellya hellye hellyi hellyo hellyu
10200
10210
10220
10230
10240
10250
10260
10270
10280
10290 hila hile hili hilo hilu hilla hille hilli hillo hillu
10700
10710
10720
10730
10740
10750
10760
10770
10780
10790 hilya hilye hilyi hilyo hilyu hillya hillye hillyi hillyo hillyu
10300
10310
10320
10330
10340
10350
10360
10370
10380
10390 hola hole holi holo holu holla holle holli hollo hollu
10800
10810
10820
10830
10840
10850
10860
10870
10880
10890 holya holye holyi holyo holyu hollya hollye hollyi hollyo hollyu
10400
10410
10420
10430
10440
10450
10460
10470
10480
10490 hula hule huli hulo hulu hulla hulle hulli hullo hullu
10900
10910
10920
10930
10940
10950
10960
10970
10980
10990 hulya hulye hulyi hulyo hulyu hullya hullye hullyi hullyo hullyu
For example, we have: 10401 mula, 10402 bula, 10403 trula, 10404 quadrula, 10405 pentula,
10406 hexula, 10407 septula, 10408 octula, 10409 novula; 10410 hule, 10411 mule, 10412 bule, etc.
Table of the first three hundred consecutive stepnumbers
1000
1001
1002
1010
1011
1012
1020
1021
1022
1023
1100
1101
1102
1110
1111
1112
1120
1121
1122
1123
1200
1201
1202
1203
1210
1211
1212
1213
1220
1221
1222
1223
1230
1231
1232
1233
1234
10000
10001
10002
10010
10011
10012
10020
10021
10022
10023
10100
10101
10102
10110
10111
10112
10120
10121
10122
10123
10200
10201
10202
10203
10210
10211
10212
10213
10220
10221
10222
10223
10230
10231
10232
10233
10234
11000
11001
11002
11010
11011
11012
11020
11021
11022
11023
11100
11101
11102
11110
11111
11112
11120
11121
11122
11123
11200
11201
11202
11203
11210
11211
11212
11213
11220
11221
11222
11223
11230
11231
11232
11233
11234
12000
12001
12002
12003
12010
12011
12012
12013
12020
12021
12022
12023
12030
12031
12032
12033
12034
12100
12101
12102
12103
12110
12111
12112
12113
12120
12121
12122
12123
12130
12131
12132
12133
12134
12200
12201
12202
12203
12210
12211
12212
12213
12220
12221
12222
12223
12230
12231
12232
12233
12234
12300
12301
12302
12303
12304
12310
12311
12312
12313
12314
12320
12321
12322
12323
12324
12330
12331
12332
12333
12334
12340
12341
12342
12343
12344
12345
100000
100001
100002
100010
100011
100012
100020
100021
100022
100023
100100
100101
100102
100110
100111
100112
100120
100121
100122
100123
100200
100201
100202
100203
100210
100211
100212
100213
100220
100221
100222
100223
100230
100231
100232
100233
100234
101000
101001
101002
101010
101011
101012
101020
101021
101022
101023
101100
101101
101102
101110
101111
101112
101120
101121
101122
101123
101200
101201
101202
101203
101210
101211
101212
101213
101220
101221
101222
101223
101230
101231
101232
101233
101234
102000
102001
102002
102003
102010
102011
102012
102013
102020
102021
102022
102023
102030
102031
102032
102033
102034
102100
102101
102102
102103
102110
102111
The question arises whether there exist other natural number systems beside the binary. The decimal number system is clearly not natural as it depends on the arbitrary choice of the base 10. If we want to change it, we can go in either one of two directions. If we lower it, then fewer digits will suffice at the price of having to cope with longer strings of digits. If we increase it, then we get shorter strings of digits at the price of having to cope with a larger inventory of digits. In exercising the first choice we go to the extreme and get the simplest number system using only two digits: 0 and 1. This is the natural number system of the binary numbers. However, simplicity comes at a price: the string of digits has maximal length. All other number systems are more economical in that they work with shorter strings of digits. As one may expect, further development of science will necessitate the use of ever larger numbers. At one point the price we pay for simplicity may become prohibitive. The length of strings of digits may outpace the memory and manipulative capabilities of the best computers for very large numbers. The binary number system may well become obsolete. We should not be too complacent in this regard.
The other choice is to increase the base. As we do, the string of digits becomes shorter, but at the price that we shall need ever more digits. Can we minimize the string of digits for very large numbers? Does it make sense to talk about infinitely many digits? It turns out that the answer to these questions is “yes”. There is another natural number system at the far end of the rainbow: that of the stepnumbers. It minimizes the string of digits and so it is ideally suited for calculations with very large numbers.
The same conventions will be used for the stepnumbers as we have introduced for the binary numbers. In particular, we also print them in bold-face type: 1, 10, 11, 12, 100, 101, 102, 110, 111, 112, 120, 121, 122, 123, 1000,... Thus we may write 5 = 100 ≠ 100, 123 = 14, 14 + 1 = 15 = 1000.
The convention for subscripts applies, e.g., 102 = 100, 1203 = 12000, 13 = 111, 1322 = 11122.
We shall also write k! = 123...k where k is the kth digit for every natural number k.
Remember that n! ≠ n! We also use the notation k!0n – k for the n-digit stepnumber 123...k00...0 where the number of 0 digits is n – k. By convention 0! = 0. Note that k! can be characterized as the largest stepnumber with k digits, whereas the smallest one with k digits is 10k – 1 = bk , the kth Bell number, see Chapter 4. The consecutive Bell numbers are: b1 = 1, b2 = 10 = 2, b3 = 100 = 5, b4 = 1000 = 15, b5 = 10000 = 52, b6 = 105 = 203, b7 = 877, b8 = 4140, b9 = 21,147, b10 = 115,975, b11 = 678,570, b12 = 4,213,597,… A table of the Bell numbers can be found at the end of this book.
In Chapter 1 we shall continue the sequence of the ten decimal digits by adding infinitely many digits, the so-called rainbow digits. They are the decimal digits written in the colors of the rainbow. The table on the previous page lists the first three hundred consecutive stepnumbers in lexicographic order. Stepnumbers, just as binary numbers, can be introduced heuristically by enumerating them under the lexicographic order subject to the following restrictions.
(1) The Stepnumber Principle states that no digit can be higher than the highest digit to its left plus 1;
(2) the Overflow Principle states that in increasing a maximal digit by 1 we replace it with 0 and add 1 to the adjacent digit to the left.
The extreme case is furnished by the Overflow Formula: n! + 1 = 10n which is triggered by the stepnumbers 1, 12, 123,..., e.g., 123 + 1 = 1000.
The rule of enumeration endows the stepnumber system with a natural order of magnitude. The value of a stepnumber is just its ordinal under the natural order. The arrangement in ten columns is designed to help finding the value of stepnumbers in the table quickly. For example, we have 1111 = 29 because 1111 stands in the second row and ninth column (the numbering of rows and Just as for binary numbers, the table for stepnumbers can also be used for performing addition and subtraction, following the Slide Rule Principle. For example, in calculating the sum 100 + 10000 we locate the larger number and from there move forward through 100 = 5 steps to get: 100 + 10000 = 10012. To check we write 5 + 52 = 57 = 10012. In calculating the difference 101233 – 12233, in the table we move backwards from 101233 through 12233 = 176 steps to get: 101233 – 12233 = 11100.
To check we write: 276 – 176 = 100 = 11100.
The stepnumbers form another number system that is natural in the sense that there is a natural isomorphism between the stepnumber system and the graded lattice of finite quotient sets. In more details, consider = {1, 2, 3,..., n,...}, the set of natural numbers and its subsets n = {1, 2, 3,..., n} of n elements. The quotient sets of n form a lattice under rarefaction, the opposite of refinement (see Appendix). The union of the lattices n for all n is a graded lattice. Grading is furnished by the number n of elements of n , i.e., the number of digits. Every stepnumber of at most n digits can be interpreted as a characteristic function of n , provided that we add a sufficient number of 0 digits up front to make it a stepnumber of exactly n digits (this of course will not affect its value). There is a one-to-one correspondence between the set of stepnumbers and the graded lattice of finite quotient sets. The former is endowed by a lattice structure under this correspondence, and we shall refer to it as the natural isomorphism between the stepnumber system and the graded lattice of finite quotient sets. It is also natural in that it does not depend on arbitrary choices such as, e.g., the basis of the number system.
Thus the stepnumber system is dual to its analogue, the binary number system. In enumerating stepnumbers we actually construct all finite quotient sets. For example, there are 15 quotient sets of a set of 4 elements, and there are 15 stepnumbers of at most 3 digits:
0, 1, 10, 11, 12, 100, 101, 102, 110, 111, 112, 120, 121, 122, 123
In writing them in up-front-0 notation, they become uniformly 4-digit stepnumbers:
0000, 0001, 0010, 0011, 0012, 0100, 0101, 0102, 0110, 0111, 0112, 0120, 0121, 0122, 0123 exhibiting the quotient sets of the set of the four places of 4-digit stepnumbers, where the classes are marked by the digit 0, 1, 2, 3.
Stepnumbers can be used to count the number of finite quotient sets. This new number system is also expected to have great utility, especially in information technology. Indeed, it may well be the number system of the future to encode information by quantum computers. Unlike the present generation of computers which are based on classical physics, quantum computers are based on quantum mechanics. In particular, the state of an electron of the atom is the new bit, or unit of information, corresponding to a stepnumber digit. The variable admissible orbits of electrons in an atom, which are discrete and ordered according to increasing radius, epitomize the variable positional values of digits in the stepnumber system.
The stepnumber system overcomes the handicap of the binary of being unwieldy due to the inordinate length of the strings of digits for very large numbers. Writing a sufficiently large number in stepnumber form will take fewer digits than it does in the binary or in any other number system with base n, however large n may be. This property is expressed by saying that the stepnumber system enables one to write very large numbers in their most compact form.
In the decimal system we are forced to do rounding at the expense of accuracy every time an extra large number presents itself. Rounding is made unnecessary, and there is no loss of accuracy, if we are using the stepnumber system.
In order to be able to count in the stepnumber system we introduce names for the digits. For the first eleven, these are:
0 ala, 1 ale, 2 ali, 3 alo, 4 alu, 5 alla, 6 alle, 7 alli, 8 allo, 9 allu, 10 mala
This already suggests that, for the Bell numbers, we shall retain the names for the milestones and also that of the secondary milestones (see pp 3, 4):
10 mala, 102 bala, 103 trala, 104 quadrala, 105 pantala, 106 hexala, 107 heptala, 108 octala,… 1010 hale, 1020 hali, 1030 halo, 1040 halu, 1050 halla, 1060 halle, 1070 halli, 1080 hallo, 1090 hallu,… In Chapter 1 we shall see the rule how to obtain the names of the rest of the (infinitely many) rainbow digits. Here, once again, the permutations of the five vowels a, e, i, o, u will play a role. Counting in the stepnumber system uses a different principle from that used in the decimal and the binary systems. The reason for this is the variable place value of the digits, a feature of stepnumbers unknown for the decimals and the binary numbers. To count, we read out the names of the digits from left to right, including 0 ala, and repeating if necessary, thus:
0 ala, 1 ale, 10 mala, 11 ale ale, 12 ale ali, 100 bala, 101 ale ala ale, 102 ale ala ale,
110 ale ale ala, 111 ale ale ale, 112 ale ale ali, 120 ale ale ali, 122 ale ali ali, 123 ale ali alo, 1000 trala, 1001 ale ala ala ale, 1002 ale ala ala ali, 1010 ale ala ale ala, 1011 ale ala ale ale,…
*
*
*
One of the great remaining mysteries in mathematics is the question whether a formula exists whereby consecutive prime numbers can be calculated. The conjecture is still outstanding that such a formula does not exist in the same sense as the set of all sets doesn’t. If this is the case, then the best one can hope for is a rapid development of various algorithms to find ever larger prime numbers. This raises the possibility of decoupling between substance and form. Finding the substance, namely, ever larger prime numbers, may outpace the form in which the new information can be put. Information technology, in particular the binary number system, may be an obstruction. The physical limitations of computer memory could defeat our efforts to forge ahead with the theory, in want of a better facility to handle very large numbers. The quantum computer reinforced with the stepnumber system fitting as a glove opens up new perspectives in computing.
Philosophy has been grappling with the dichotomy of substance and form for thousands of years. It tacitly assumes that they are inseparable as guardians of the depository of knowledge.
Decoupling would most certainly create a crisis of the first magnitude. The discovery of stepnumbers opens a new chapter in philosophy in that it extends the tenure of the partnership of substance and form in storing, transmitting, retrieving, or otherwise manipulating and organizing information.