# Radical Affinity and Variant Proportion in Natural Numbers

[N.b. the tabular data on this page is not well-formatted for printing purposes – please use the pdf version]

The term ‘Radical Affinity’ stems from an investigation into the properties of *natural numbers* – their tendencies to behave, according to characteristics of their radices (or ‘bases’), in ways previously unacknowledged in the analyses of quantitative systems. In the title page of this section I raised some concerns over conventional approaches to quantitative understanding, with respect to the definition of an ‘integer’, and to the principle of *rational proportionality* governing integers in the denotation of numeric value. This inquiry begins from an empirical comparison of values in exponential series across a limited range of diverse number radices (base-2 to base-9), in terms of the logarithmic ratios of sequential values in each exponential series, relative to the ratios of corresponding values in the decimal series. While the logarithmic ratios of sequential values in the decimal series are naturally consistent, and would produce graphs consisting of horizontal straight lines, in the case of each of the radical series reproduced here the distributions revealed are mostly irregular series of variegated peaks and troughs, displaying proportional inconsistency. In other pages in this section (and in the Analysis section below), this inquiry is treated more discursively; the following exercises attempt to explicate these concerns in basic empirical terms.

## Problematic

The following datasets are intended to explore comparisons between the decimal exponential series (10^{0}, [...], 10^{10}) with its corresponding series in a range of radices from binary to nonary (base-9). In what follows I have used the term ‘*z*’ to refer to the exponential index, and the term ‘*b*’ to refer to the radical index or base. The decimal series is represented by sequential values of *s*=10^{z}_{10} . Values in each respective corresponding radix are represented by sequential values of *s*=(*x ^{z}*)

*; i.e., for*

_{b}*x*=10

_{10};

*z*=(0, [...], 10);

*b*=(2, [...], 9). Generally,

*s*is equal to (

*x*)

^{z}*, and is employed here to represent the exponential*

_{b}*series*of any radix, whereas

*x*retains association with the initial decimal value (10

^{1}).

In the decimal series, for *z*=(0, [...], 10), *s*=(1, [...], 10000000000).

The following tables show distributions of values corresponding to *s*=10^{z}_{10} , in terms of *s*=(*x ^{z}*)

*for each of the respective radices:*

_{b}z |
s=10^{z}_{10} |
s=1010^{z}_{2} |
s=101^{z}_{3} |
---|---|---|---|

0 | 1 | 1 | 1 |

1 | 10 | 1010 | 101 |

2 | 100 | 1100100 | 10201 |

3 | 1000 | 1111101000 | 1101001 |

4 | 10000 | 10011100010000 | 111201101 |

5 | 100000 | 11000011010100000 | 12002011201 |

6 | 1000000 | 11110100001001000000 | 1212210202001 |

7 | 10000000 | 100110001001011010000000 | 200211001102101 |

8 | 100000000 | 101111101011110000100000000 | 20222011112012201 |

9 | 1000000000 | 111011100110101100101000000000 | 2120200200021010001 |

10 | 10000000000 | 1001010100000010111110010000000000 | 221210220202122010101 |

Table 1(a)

s=22^{z}_{4} |
s=20^{z}_{5} |
s=14^{z}_{6} |
s=13^{z}_{7} |
s=12^{z}_{8} |
s=11^{z}_{9} |
---|---|---|---|---|---|

1 | 1 | 1 | 1 | 1 | 1 |

22 | 20 | 14 | 13 | 12 | 11 |

1210 | 400 | 244 | 202 | 144 | 121 |

33220 | 13000 | 4344 | 2626 | 1750 | 1331 |

2130100 | 310000 | 114144 | 41104 | 23420 | 14641 |

120122200 | 11200000 | 2050544 | 564355 | 303240 | 162151 |

3310021000 | 224000000 | 33233344 | 11333311 | 3641100 | 1783661 |

212021122000 | 10030000000 | 554200144 | 150666343 | 46113200 | 20731371 |

11331132010000 | 201100000000 | 13531202544 | 2322662122 | 575360400 | 228145181 |

323212230220000 | 4022000000000 | 243121245344 | 33531600616 | 7346545000 | 2520607101 |

21110002332100000 | 130440000000000 | 4332142412144 | 502544411644 | 112402762000 | 27726678111 |

Table 1(b)

Clearly, within the terms of each respective series, the ratio: *s** _{n}*/

*s*

_{n}_{–1}is constant for each value of

*z*:

*s** _{n}*/

*s*

_{n}_{–1}= 10

_{10}= (

*x*)

_{b}and in a graphical representation with *z* as the horizontal axis, would produce horizontal straight lines at *y*=10_{10} , and *y*=(*x*)* _{b}* .

However, for the non-decimal series, if we calculate *s** _{n}*/

*s*

_{n}_{–1}dividing the figures according to base-10 rules (i.e., treating them

*as if*they were decimal values) instead of base-

*b*rules, in each case the resulting series becomes inconsistent above a certain (variable) value of

*z*.

Tables 2 & 3 below display the resulting distributions of values of (*s** _{n}*/

*s*

_{n}_{–1})

_{10}for each series

*s*=(

*x*)

^{z}*(the expression ‘(*

_{b}*s*

*/*

_{n}*s*

_{n}_{–1})

_{10}’ is used here simply to imply that the sequential radical values of (

*x*)

^{z}*are divided*

_{b}*as if*they were decimal values):

(s_{n}/s_{n}_{-1})_{10} |
||||
---|---|---|---|---|

z |
[ s=1010^{z}_{2} ] |
[ s=101^{z}_{3} ] |
[ s=22^{z}_{4} ] |
[ s=20^{z}_{5} ] |

0 | - | - | - | - |

1 | 1010 | 101 | 22 | 20 |

2 | 1089.207920792 | 101 | 55 | 20 |

3 | 1010 | 107.930693069 | 27.454545455 | 32.5 |

4 | 9010.072000655 | 101 | 64.121011439 | 23.846153846 |

5 | 1098.781452499 | 107.930686774 | 56.392751514 | 36.129032258 |

6 | 1010.008080065 | 101.000589126 | 27.555447702 | 20 |

7 | 9010.720064805 | 165.161950272 | 64.054313251 | 44.776785714 |

8 | 1010.000000001 | 101.003496315 | 53.443411218 | 20.049850449 |

9 | 1097.912088781 | 104.846159379 | 28.524266590 | 20 |

10 | 9017.207279337 | 104.334590762 | 65.313129759 | 32.431626057 |

Table 2

(s_{n}/s_{n}_{-1})_{10} |
||||
---|---|---|---|---|

z |
[ s=14^{z}_{6} ] |
[ s=13^{z}_{7} ] |
[ s=12^{z}_{8} ] |
[ s=11^{z}_{9} ] |

0 | - | - | - | - |

1 | 14 | 13 | 12 | 11 |

2 | 17.428571429 | 15.538461538 | 12 | 11 |

3 | 17.803278689 | 13 | 12.152777778 | 11 |

4 | 26.276243094 | 15.652703732 | 13.382857143 | 11 |

5 | 17.964536025 | 13.729928961 | 12.947907771 | 11.075131480 |

6 | 16.207086510 | 20.081882857 | 12.007320934 | 11 |

7 | 16.676027065 | 13.294115285 | 12.664634314 | 11.622932272 |

8 | 24.415732638 | 15.415932157 | 12.477130193 | 11.004828431 |

9 | 17.967452971 | 14.436710488 | 12.768596866 | 11.048259227 |

10 | 17.818855798 | 14.987188276 | 15.300084870 | 11 |

Table 3

If we examine the ratio *s** _{n}*/

*s*

_{n}_{–1}logarithmically, we can more simply employ subtraction rather than division in determining the series.

Generally, log* _{b}x* is given by: log

_{10}

*x*/log

_{10}

*b*.

As it is conventional to derive radical logarithms from decimal logarithms, we may do so for the values of *s*=(*x ^{z}*)

*given in Table 1 above, which allows us to express the ratio*

_{b}*s*

*/*

_{n}*s*

_{n}_{–1}in terms of:

*r* = (log_{b}*s** _{n}*) – (log

_{b}*s*

_{n}_{–1})

The following 8 subsections display tables showing the values for log_{b}*s* [i.e., log* _{b}*(

*x*)

^{z}*] and*

_{b}*r*for each radical data series (binary to nonary) given in Table 1, i.e., for the initial decimal value of

*x*=10 (the linked pdf document: Radical Affinity etc. repeats the same exercises for the starting values of

*x*from 2 to 9 – see pp.21-101). Graphical representations of the tabular data in terms of

*r*against

*z*are displayed as vertical and horizontal axes respectively in the following graphs (n.b. the vertical axes in these graphs are not at a constant scale).

^{1}

*x*=10

## Binary

z |
s=1010^{z}_{2} |
log_{b}s |
r |
---|---|---|---|

0 | 1 | 0 | - |

1 | 1010 | 9.980139578 | 9.980139578 |

2 | 1100100 | 20.069203241 | 10.089063663 |

3 | 1111101000 | 30.049342819 | 9.980139578 |

4 | 10011100010000 | 43.186665738 | 13.137322919 |

5 | 11000011010100000 | 53.288354486 | 10.101688748 |

6 | 11110100001001000000 | 63.268505605 | 9.980151119 |

7 | 100110001001011010000000 | 76.405932289 | 13.137426684 |

8 | 101111101011110000100000000 | 86.386071867 | 9.980139578 |

9 | 111011100110101100101000000000 | 96.486618692 | 10.100546825 |

10 | 1001010100000010111110010000000000 | 109.625083660 | 13.138464968 |

log_{10}2 = 0.301029995664

*r*=(log_{2}*x*^{z}) – (log_{2}*x*^{z–1}), for *x*=1010_{2}

## Ternary

z |
s=101^{z}_{3} |
log_{b}s |
r |
---|---|---|---|

0 | 1 | 0 | - |

1 | 101 | 4.200863730 | 4.200863730 |

2 | 10201 | 8.401727460 | 4.200863730 |

3 | 1101001 | 12.663002651 | 4.261275191 |

4 | 111201101 | 16.863866381 | 4.200863730 |

5 | 12002011201 | 21.125141519 | 4.261275138 |

6 | 1212210202001 | 25.326010559 | 4.200869040 |

7 | 200211001102101 | 29.974535395 | 4.648524836 |

8 | 20222011112012201 | 34.175430634 | 4.200895239 |

9 | 2120200200021010001 | 38.410313290 | 4.234882656 |

10 | 221210220202122010101 | 42.640743808 | 4.230430518 |

log_{10}3 = 0.4771212547197

*r*=(log_{3}*x*^{z}) – (log_{3}*x*^{z–1}), for *x*=101_{3}

## Quaternary

z |
s=22^{z}_{4} |
log_{b}s |
r |
---|---|---|---|

0 | 1 | 0 | - |

1 | 22 | 2.229715809 | 2.229715809 |

2 | 1210 | 5.120395666 | 2.890679857 |

3 | 33220 | 7.509882226 | 2.389486560 |

4 | 2130100 | 10.511244865 | 3.001362639 |

5 | 120122200 | 13.419963781 | 2.908718916 |

6 | 3310021000 | 15.812096612 | 2.392132831 |

7 | 212021122000 | 18.812708520 | 3.000611908 |

8 | 11331132010000 | 21.682678615 | 2.869970095 |

9 | 323212230220000 | 24.099737559 | 2.417058944 |

10 | 21110002332100000 | 27.114388128 | 3.014650569 |

log_{10}4 = 0.602059991328

*r*=(log_{4}*x*^{z}) – (log_{4}*x*^{z–1}), for *x*=22_{4}

## Quinary

z |
s=20^{z}_{5} |
log_{b}s |
r |
---|---|---|---|

0 | 1 | 0 | - |

1 | 20 | 1.861353116 | 1.861353116 |

2 | 400 | 3.722706232 | 1.861353116 |

3 | 13000 | 5.885722315 | 2.163016083 |

4 | 310000 | 7.856362447 | 1.970640132 |

5 | 11200000 | 10.085150978 | 2.228788531 |

6 | 224000000 | 11.946504094 | 1.861353116 |

7 | 10030000000 | 14.308626795 | 2.362122701 |

8 | 201100000000 | 16.171526676 | 1.862899881 |

9 | 4022000000000 | 18.032879792 | 1.861353116 |

10 | 130440000000000 | 20.194587325 | 2.161707533 |

log_{10}5 = 0.698970004336

*r*=(log_{5}*x*^{z}) – (log_{5}*x*^{z–1}), for *x*=20_{5}

## Senary

z |
s=14^{z}_{6} |
log_{b}s |
r |
---|---|---|---|

0 | 1 | 0 | - |

1 | 14 | 1.472885940 | 1.472885940 |

2 | 244 | 3.068028002 | 1.595142062 |

3 | 4344 | 4.675042050 | 1.607014048 |

4 | 114144 | 6.499318847 | 1.824276797 |

5 | 2050544 | 8.111365354 | 1.612046507 |

6 | 33233344 | 9.665953809 | 1.554588455 |

7 | 554200144 | 11.236461588 | 1.570507779 |

8 | 13531202544 | 13.019752124 | 1.783290536 |

9 | 243121245344 | 14.631889245 | 1.612137121 |

10 | 4332142412144 | 16.239391402 | 1.607502157 |

log_{10}6 = 0.7781512503836

*r*=(log_{6}*x*^{z}) – (log_{6}*x*^{z–1}), for *x*=14_{6}

## Septenary

z |
s=13^{z}_{7} |
log_{b}s |
r |
---|---|---|---|

0 | 1 | 0 | - |

1 | 13 | 1.318123223 | 1.318123223 |

2 | 202 | 2.727909971 | 1.409786748 |

3 | 2626 | 4.046033194 | 1.318123223 |

4 | 41104 | 5.459584413 | 1.413551219 |

5 | 564355 | 6.805781229 | 1.346196816 |

6 | 11333311 | 8.347382756 | 1.541601527 |

7 | 150666343 | 9.677002975 | 1.329620219 |

8 | 2322662122 | 11.082721287 | 1.405718312 |

9 | 33531600616 | 12.454713875 | 1.371992588 |

10 | 502544411644 | 13.845937269 | 1.391223394 |

log_{10}7 = 0.8450980400143

*r*=(log_{7}*x*^{z}) – (log_{7}*x*^{z–1}), for *x*=13_{7}

## Octal

z |
s=12^{z}_{8} |
log_{b}s |
r |
---|---|---|---|

0 | 1 | 0 | - |

1 | 12 | 1.194987500 | 1.194987500 |

2 | 144 | 2.389975000 | 1.194987500 |

3 | 1750 | 3.591046402 | 1.201071402 |

4 | 23420 | 4.838484485 | 1.247438083 |

5 | 303240 | 6.070033515 | 1.231549030 |

6 | 3641100 | 7.265314311 | 1.195280796 |

7 | 46113200 | 8.486225483 | 1.220911172 |

8 | 575360400 | 9.699963563 | 1.213738080 |

9 | 7346545000 | 10.924806260 | 1.224842697 |

10 | 112402762000 | 12.236628843 | 1.311822583 |

log_{10}8 = 0.9030899869919

*r*=(log_{8}*x*^{z}) – (log_{8}*x*^{z–1}), for *x*=12_{8}

## Nonary

z |
s=11^{z}_{9} |
log_{b}s |
r |
---|---|---|---|

0 | 1 | 0 | - |

1 | 11 | 1.091329169 | 1.091329169 |

2 | 121 | 2.182658339 | 1.091329170 |

3 | 1331 | 3.273987508 | 1.091329169 |

4 | 14641 | 4.365316677 | 1.091329169 |

5 | 162151 | 5.459743807 | 1.094427130 |

6 | 1783661 | 6.551072976 | 1.091329169 |

7 | 20731371 | 7.667472316 | 1.116399340 |

8 | 228145181 | 8.759001215 | 1.091528899 |

9 | 2520607101 | 9.852322719 | 1.093321504 |

10 | 27726678111 | 10.943651889 | 1.091329170 |

log_{10}9 = 0.9542425094393

*r*=(log_{9}*x*^{z}) – (log_{9}*x*^{z–1}), for *x*=11_{9}

## Proportional Graphs

The first graph below shows the distributions represented individually above, in a single graph with a proportional vertical axis for the full range *b*=(2, [...], 9). The three graphs in the subsequent section show the relationships between the distributions for the range *b*=(3, [...], 9), with expanded vertical scales (*r*):

*r*=(log_{b}*x*^{z}) – (log_{b}*x*^{z–1}), for *x*=(10_{10})_{b}

### Graphs to show relations of close sequential groups

Comparison of ternary, quaternary, & quinary series.

Comparison of senary & septenary series.

Comparison of octal & nonary series.

## Variation Factors

To measure the degrees of variation in proportion exhibited in the values of *r* in each radical series, I have taken as a baseline the value of *r** _{u}* given for

*z*=1, then calculated the increase factor for

*r*

*at*

_{v}*z*=7 (as this appears as a frequent point of high elevation): (

*r*

*/*

_{v}*r*

*); and, if*

_{u}*r*

*>*

_{max}*r*

*, the increase factor for*

_{v}*r*at

_{max}*z*=

*n*: (

*r*

_{ma}*/*

_{x}*r*

*):*

_{u}b |
r_{u} |
r_{v} |
r_{max} |
r_{v}/r (_{u}z=7) |
r_{max}/r (_{u}z=n) |
---|---|---|---|---|---|

2 | 9.980139578 | 13.137426684 | 13.138464968 | 1.316 | 1.316 (z=10) |

3 | 4.200863730 | 4.648524836 | – | 1.107 | - |

4 | 2.229715809 | 3.000611908 | 3.014650569 | 1.346 | 1.352 (z=10) |

5 | 1.861353116 | 2.362122701 | – | 1.269 | – |

6 | 1.472885940 | 1.570507779 | 1.824276797 | 1.066 | 1.239 (z=4) |

7 | 1.318123223 | 1.329620219 | 1.541601527 | 1.009 | 1.170 (z=6) |

8 | 1.194987500 | 1.220911172 | 1.311822583 | 1.022 | 1.098 (z=10) |

9 | 1.091329169 | 1.116399340 | – | 1.023 | – |

*r*=(log_{b}*x*^{z}) – (log_{b}*x*^{z–1}), for *x*=(10_{10})_{b}

These variation factors are represented as percentage-increase as the vertical axis in the graphs below. The horizontal axis represents the value *b*. It is clear that the values for *r** _{v}*/

*r*

*and*

_{u}*r*

_{ma}*/*

_{x}*r*

*are virtually identical for the radices represented by*

_{u}*b*=(2, 3, 4, 5, and 9), i.e., excepting

*b*=7, and the two adjacent radices

*b*=6 and

*b*=8.

*r*_{v}/*r _{u}* +% (

*z*=7)

*r*_{max}/*r _{u}* +% (

*z*=

*n*)

## Analysis

Much of what has been written across the other pages in this section was influenced by the findings of an earlier version of the pdf document Radical Affinity etc., which focused initially on a comparison of values in the decimal exponential sequence: 10^{z}_{10}, for *z*=(0, [...], 10), with their *octal* correspondents: 12^{z}_{8} (see graph of the octal series). That document was then extended to similar comparisons for the series of numerical radices from binary to nonary, for values in each radix corresponding to 10 in decimal. Those exercises (presented above) reveal that, while the ratios between successive values in a progressive exponential sequence (e.g. 10^{z}_{10} or 12^{z}_{8}) calculated by division, according to the rules of the specific radix, will be constant:

(*x ^{n}*)

*/(*

_{b}*x*

^{n}^{–1})

*=(*

_{b}*x*)

*, and would be represented graphically by a horizontal straight line at*

_{b}*y*=(

*x*)

*(i.e., with*

_{b}*z*as the horizontal axis); the treatment of the same ratios rendered through the logarithmic function (which derives all values from common logarithms – log

_{10}) results in series of ratios that are

*no longer constant*– excepting of course those of base-10 itself. This is true for each of the numerical radices from binary to nonary, which are shown to be inconsistent with decimal, as well as being inconsistent with each other, and is evidenced by the graphs represented above, which in each case display a series of variegated peaks and troughs. Those exercises were then further extended to consider the same series of radical correspondents for the decimal values of

*x*between 2 and 9 (see pp.21-101 of Radical Affinity etc.).

The same calculations performed for any value of *x* in decimal (without involving a derived radical logarithm), result in horizontal straight lines at *y*=Log*x*. In the exercises above there are no instances of horizontal straight lines; however, in later exercises, for values of *x* other than 10_{10}, horizontal straight lines in general result only where the decimal value of *x* is equal to the value *b* (see for example p.22 of Radical Affinity etc., where *x*=2, *b*=2). There are three exceptions to this in the result sets: occurring where *x* is equal to *b*^{2} or *b*^{3} (see *Comments* on p.31 & p.51).

The logarithmic function was developed in the 17th century by John Napier (and later by Leonard Euler) on the premise of *invariant* proportion between integers, and only on this basis can the derivation of radical logarithms from ‘common’ logarithms be assumed to be unproblematic. For example, in the example of the octal series, the deviation from the horizontal affecting the values *z*=3 onwards is the result of applying logarithms to the octal series; i.e., the essential *proportionality* between integers on which the logarithmic function is based is lost. If we represent these ratios without the use of logarithms, i.e., by dividing successive exponentials in the series 12^{z}_{8} along base-8 rules, we of course end up with a horizontal straight line at *y*=12_{8}. It is the derivation of the octal logarithmic values from ‘common’ decimal logarithms that introduces these deviations. This undermines the role of the logarithmic function in expressing *common ratios of proportion* across diverse number radices, and suggests that the rules of proportionality between integers apply only in a restricted sense, according to the particular range, or group characteristics, of the select digits at our disposal. This failure inherent in the logarithmic function undermines the accepted principles of rational proportionality pertaining between diverse number radices and indicates that rationality operates effectively under formally circumscribed limits, where previously no such limits had been perceived. To account for this it is necessary to consider exactly what it is that defines an ‘integer’, as the behaviour of the values in the octal series, as well as in other series, suggests that integers are unable to fulfil their customary role as stable indices of *intrinsic* value.

A comparison of the set of all eight series may help eliminate some erroneous or misleading explanations. The most striking comparison to make in the shapes of the various distributions is that between the ternary and nonary series, as both series feature significant peaks at *z*=7, as well as featuring comparable smaller peaks to either side of *z*=7:

*r*=(log_{3}*x*^{z}) – (log_{3}*x*^{z–1}), for *x*=101_{3}

*r*=(log_{9}*x*^{z}) – (log_{9}*x*^{z–1}), for *x*=11_{9}

A comparison of the two elevations at *z*=7 in absolute terms shows the scale of the increase in the base-9 series is only ≈5.6% of that in the base-3. However, in relative terms (i.e., taking into account the difference in the baseline value at *z*=1) the difference is between a ≈10.7% increase in the ternary series, and a ≈2.3% in the nonary, so that the latter increase is proportionally ≈21.6% of the size of elevation in the ternary series.

This high individual peak is not a characteristic that is repeated in the other series however, with the exception of the septenary and octal series – in the former the peak occurring at *z*=6, with larger adjacent peaks than those in the previous examples. The binary, quaternary, quinary, and senary series all feature more regular variations, with some signs of patterning, particularly in those of the binary and quaternary series. The octal series shows quite a unique distribution, especially in view of its elevated peak at *z*=10.

The statistical assessment of the variation factors with respect to the maximum (*r** _{max}*) and minimum (

*r*

*) variations exhibited in each distribution, and their comparison with the variation factor at*

_{u}*z*=7 (

*r*

*), shows that in four of the eight series (*

_{v}*b*= 2, 3, 5, and 9)

*r*

*is found to occur at*

_{max}*r*

*(for*

_{v}*b*=2, an equivalent maximum value recurs at

*z*= 4, 7, and 10 – the value of

*r*at

*z*=10 is only ≈0.008% higher than that at

*z*=7, which I have assumed to be negligible); while in the quaternary series (

*b*=4) the increase factor at

*r*

*is only ≈0.5% lower than its maximum, which occurs at*

_{v}*z*=10. The effect of this is that the values for

*r*

*/*

_{v}*r*

*and those for*

_{u}*r*

_{ma}*/*

_{x}*r*

*are identical across 4/8 of the series, and almost identical across 5/8 – the exceptions being*

_{u}*b*=7 and its two adjacent radices

*b*=6 and

*b*=8. This similarity is displayed in the two graphs above showing percentage-increase factors for

*r*

*/*

_{v}*r*

*and*

_{u}*r*

_{ma}*/*

_{x}*r*

*(the horizontal axis represents the value for*

_{u}*b*in these two graphs). For instance, where

*b*=4 the value of

*r*

*is a ≈34.6% increase on the value of*

_{v}*r*

*.*

_{u}The identity of shape across the greater part of these two distributions (see graphs above) gives empirical proof of the qualitative uniqueness of the integer ‘7’ (i.e., within the context of this particular series starting from the decimal *x*=10)^{2}, and which is irreducible to rational or quantitative principles. A further observation is that in 3/8 of the series (*b*=2, *b*=4, and *b*=8) *r** _{max}* occurs at

*z*=10 (n.b., in

*b*=2 and

*b*=4,

*r*

*and*

_{max}*r*

*are virtually identical), so that both*

_{v}*z*=7 and

*z*=10 appear to be ‘potentiated’ in comparison to other values of

*z*. It is noticeable also that the distribution of peaks where

*b*=2 (i.e., at

*z*=4,

*z*=7, and

*z*=10) is also a characteristic found where

*b*=4, and where

*b*=8, although in the latter case the peaks do not exhibit the regularity of elevation found in the former two cases.

Viewing the set of distributions as a whole, it is noticeable that there is an absence of close correspondence or consistency between any two distributions (where they exhibit similarities of shape for instance, they disagree in scale). However, the statistical analysis reveals the frequency of *z*=7 as the point at which a half of the distributions reach maximum variation, and at which an additional one is very nearly at its maximum. The recurrent significance of *z*=7 and *z*=10 therefore seems to exclude any explanation of variation in terms of random or chaotic factors – an explanation which is also resisted by evidence of patterning in some of the distributions.

In the notations we have employed in the foregoing there are two values – *z* and *b* – which represent series of progressive whole numbers. In the conventional understanding of the meaning of an ‘integer’ (i.e., an *entity* which is self-contained, qualitatively neutral, and whose value is determined intrinsically) we should expect such series to behave proportionally. If we limit our calculations to decimal notation, there might never be an indication that integers behave in any other way (this possibly explains how these phenomena have escaped the attention of mathematicians during the 400 years since the development of Napier’s method). However, in the comparisons outlined above, there are exposed proportional inconsistencies both in the sequence *z*=(0, [...], 10), and in the sequence *b*=(2, [...], 9) – the former by comparing the logarithmic differences between the sequential values in any individual radical sequence, and the latter in the absence of formal consistency between the shapes of the graphs of successive radical distributions (compare, for instance, the distributions for *b*=6 & *b*=7).

### Numbers: Transcendental Objects or Symbolic Constructs?

I have discussed elsewhere some of the theoretical problems with the concept of integers (see: Somatic Inscription & Integers & Proportion). Integers are conventionally understood to represent marks of stable intrinsic value, on the basis that any integer can be broken down, proportionally, into its constituent single units. The integrity of the unit ‘1’ is fundamental to this understanding – it is taken as an objective unit of quantity, capable of expressing any larger quantity unambiguously by simple addition or multiplication. The problem with this understanding is that the unit ‘1’ is not a phenomenal object with measurable properties in itself, but a symbolic construct, which ‘stands-in’ by a sort of tacit mental agreement as an index for value. As such it is a character that lacks a stable substantial basis, unless, that is, we assert that certain mental constructs possess transcendental objectivity. If we consider the unit ‘1’ in the context of binary notation, for instance, we perceive that in addition to its quantitative value it is also invested with the *quality* of ‘positivity’, it being the only alternative character to the ‘negative’ ‘0’^{3}. Can such qualitative properties be contained within the transcendental objectivity of the unit ‘1’, considering that they do not similarly apply to ‘1’ in decimal? In this case clearly not, as the property arises only as a condition of the restrictive binary relationship between the two digits. Hence it appears as a necessary conclusion that there are dynamic, context-specific attributes associated with particular integers which are not absolute or fixed (intrinsic), but variable, and which are determined extrinsically, according to the *relative frequency* of individual elements within the restricted range of available characters circumscribed by the terms of the current working numerical radix.

The results that have been elaborated above in the comparisons of diverse number radices add empirical weight to this critique. Numbers, it appears, are subject to *behaviours* according to their relative position within a limited series of available digits; that is, according to their relative values. It makes a difference to an instance of ‘1’, in terms of its relative frequency, or its *dispositional* value, whether it is 1 in base-3 or 1 in base-10, for instance, even though 1_{3} and 1_{10} are quantitatively identical. I have tried to show also, in the statistical assessment of sequential radical distributions above, how this instability does not arise from random or chaotic factors, but rather determines certain patterns of affinity within a given context of variation (as exemplified by the heightened potential of *z*=7 within the context of *x*=10)^{4}. This is the case, as I understand it, because numbers, rather than behaving as stable transcendental objects, are affected by context-specific dynamic and dispositional properties, as well as by rational principles. In fact, rational principles – the ability of numerical values to express conformable ratios – do not operate as universal principles consistently governing all numerical notations, but depend locally on the terms of the particular numerical radix we happen to be employing. The rational principles governing a decimal system are incompatible with those governing an octal or a binary one, as is clearly shown in the distributions exhibited in the foregoing exercises.

Clearly, issues of relative frequency must be determined *extrinsically* – from the relations between integers as notional quantities, or between the individual members of relational groups of digits (as quantities) in series. From the conventional rational (and, it has to be said, deeply metaphysical) viewpoint of integers as self-contained transcendental objects whose values are determined intrinsically, the constitutive effect of numerical behaviour upon value, of quality upon quantity, will forever be shrouded in mystery.

These criticisms are not intended to be exhaustive, but are an initial response to the problems arising from the failure of the principle of rational proportionality, as evinced by the exercises in the foregoing sections, as well as in the extended exercises in the pdf document: Radical Affinity etc. I do not suggest here that this analysis provides a full or adequate explanation of the instances of variable proportionality revealed in these exercises. Attempts at analytical explanation tend to be predisposed towards rational solutions; and after all it was necessary to employ the distinctly rational method of logarithmic comparison in order to expose the failures in rational proportionality as the principle underpinning that method.

Having got thus far, and in the light of these results, it may be that we must accept that rationality operates only within well-defined limits – that it is ultimately undermined as an overarching principle in the analysis of quantitative systems. It begins to appear as a contingent, relational property, which fluctuates according the terms of a signifying regime (i.e., according to its numeric or indexical syntax), rather than as a necessary precondition with universal, or absolute, applicability. This suggests that a fully rational explanation of these results may ultimately be unjustifiable, or inappropriate.

Further historical and epistemological enquiry should help towards an understanding of how the selective inheritance of certain notions from Classical mathematics lent support to the overvaluation of the principle of rational proportionality, by hypostatising that which should correctly be understood as a fundamentally conceptual category (number), into something idealised as a concrete entity (integer), asserting the latter as proportionally invariant, or qualitatively indifferent. A major impulse, from the 16th Century onwards, towards the mechanical understanding of the laws of Nature, required the establishment of systems of absolute measurement and quantification; which in turn enforced the rational dichotomy between quantitative and qualitative forms of knowledge. While this may have been historically and theoretically necessary to the development of modern empirical science in its infancy, from our own developed information-based technological perspective, which mistakenly assumes a seamless correspondence between diverse numerical radices (binary, octal, decimal, hexadecimal, base-64, etc.), the continued disregard for their rational non-conformability becomes an issue of urgent critical importance.

November 2011

(revised 19 April 2024)

#### Footnotes:

- Some may find it surprising, or erroneous, that the values of
*z*in the horizontal axes in the following graphs are not aligned with the divisional markers, but between these points. I must confess to being a novice at the use of the ‘chart’ function in Microsoft Excel, and so did not override this default configuration when initially entering the data, which is the configuration most suitable when creating bar charts, for instance, rather than linear distributions of precise values. The subsequent decision not to override the default configuration for these graphs, resulting in an unconventional display, was made with regard to the fact that the precise alignment of integers with divisional markers is characteristic only of a certain limited definition of integers, as discrete*points*of value on a linear scale. With consideration to the scope of this investigation, and its limitation to the sphere of*natural numbers*– as ‘wholes’ (excluding fractions) – the resulting unaligned scale of values helps to accommodate certain roles of natural numbers in describing the (not strictly linear) apportionment of numeric value corresponding to familiar divisions in space and time, for instance. In identifying certain periods of time, or regions in space, we are accustomed to using whole numbers to represent entire periods or regions (the word ‘zone’ covers both uses). We may speak of ‘Week 1’ to cover any point in time in a particular 7-day duration, for instance; or use a numeric description for concentric zones in space, such as in the London Underground map, where ‘Zone 2’ describes*any point between*the lines of concentric division separating it from Zones 1 & 3 (stations occurring precisely on the line are understood to occupy both adjacent zones simultaneously). In physics, for instance, we might wish to consider a continuous sine wave in terms of its discrete single iterations, and then identify them consecutively as iterations 1, 2, & 3, etc., and so to dissociate the numeric notation from any particular point of amplitude of the sine wave (such a dissociation being impossible for any precise positional point on the horizontal axis). By allowing attention to such ‘periodic’ or ‘zonal’ features in certain ways of using integers we may perhaps help illuminate and further explain their apparent proportional inconsistencies. [back] - The same qualitative characteristics do not apply to instances of
*z*=7 in the series taking the decimal values of*x*from 2 to 9 as their starting point, as shown in the analysis of variations across these further series in pp.102-112 of the linked pdf document: Radical Affinity etc. (see also Note 4 below). [back] - This characteristic is of course what enables digital computer systems to employ binary code principally to convey a series of processing instructions, rather than merely as an index of quantity. [back]
- As a further example, see the distributions of variation factors given for the series where
*x*=9, on p.110 of Radical Affinity etc.. In this context there is displayed a heightened frequency of elevated values of*r*where*z*=6, occurring in 4/8 of the series (i.e., for*b*= 2, 4, 5, & 8). [back]