Least Significant Digits

Copyright 2003 Sophisticated:The Magazine. All rights reserved.

Decimal numbers... hmm, it seems we are stuck with them.  For the reason why, "look no further than just beyond your own wrists."
by Paul Niquette

 
eing least significant does not mean a digit has no significance.  Ironically, being most significant can mean exactly that for some digits -- utter insignificance.  Remember the Y2K bug? 
A misguided form of sophistication spent a whole century throwing away most significant digits in dates, treating ciphers that represented hundreds and thousands of years as utterly insignificant.  Plenty of shiny people built whole computer systems based on Least Significant Digits for governmental institutions and business enterprises as if there were no tomorrow -- or at least no tomorrow that would call for subtracting 99, say, from 00.

In and of itself, a most significant digit lacks the authority to assert whether a number is whole or rational, real or imaginary.  Oddly enough, a most significant digit prancing around over there on the left cannot even tell you if the number is odd or even.  Only Least Significant Digits can do that.  A mindless machine programmed to make a fifty-fifty split of an integer won't be able to ascertain if it can be done until after hacking a pathway through a forest of most significant digits to that quiet meadow where Least Significant Digits make their revelations.

A most significant digit has no significance in determining whether an integer is prime either.  On the other hand, Least Significant Digits, when they're even, can at least assure you that an integer is not prime (see Prime Numbers are Odd); moreover, when one particular odd cipher, Private Five, is seen patrolling the ramparts of Least Significant Digits, you know every single time that another aspiring integer has been disqualified as a prime.  Well, almost every single time.

In Benford's Law sophisticated solvers discovered that most significant digits, as they appear in natural settings,  suffer from maldistribution of their decimal ciphers -- that frequencies range from over 30% for cypher 1 to under 5% for cypher 9.  Similar patterns have been confirmed for the sum of the first n Integers, along with other integers, which are not statistically determined but generated algorithmically, such as Fibonacci numbers and factorials (see Factorial Factoids).

Oh sure, if all you care about is the size of a number, then for you the most significant digit will be most significant.  Even so, the real significance of all your digits depends on how many, whatever their sizes, are placed between them and the decimal point. 

Sophisticated solvers, however, often find significance in a number beyond its mere size. We have speculated elsewhere that Least Significant Digits are necessarily biased...

An integer operand can be either even or odd, and it takes two operands to make a sum or a product, in four combinations -- even|even, even|odd, odd|even, odd|odd

The respective sums will be even, odd, odd, even; the respective products will be even, even, even, odd.

Accordingly, when setting about to tabulate sums of consecutive integers ("totorials"?), the sophisticated solver might well expect Least Significant Digits to come out evenly distributed even|odd and would have no reason to suspect that the ciphers would be anything but equally represented in the right-most position.

The expression S(n) = n(n + 1)/2 reveals that the sum of consecutive integers is determined by a product, n(n +1), which is always even, making all consequent values of S(n) fully capable of selecting their Least Significant Digits from all 10 decimal ciphers.  

Let L{x} symbolize the least significant digit of a positive integer x, such that, in Decimal Land, L{x} = 0, 1, 2,... 9.  The following table systematically summarizes the calculation of all possible Least Significant Digits for S(n):
 

L{n}
L{n + 1}
L{n}L{n + 1}
L{n}L{n + 1}/2
L{S(n)}
0
1
0
0
0
1
2
2
1
1
2
3
6
3
3
3
4
12
6
6
4
5
20
10
0
5
6
30
15
5
6
7
42
21
1
7
8
56
28
8
8
9
72
36
6
9
0
0
0
0

WatchdogNota bene, only two of the following relationships are valid:

L{L{n} + L{n + 1}} = L{2n + 1},
L{L{n}L{n + 1}} = L{n(n + 1)}
L{L{n}L{n + 1}}/2 = L{n}L{n + 1}/2



Almost forgot to mention: Quietly missing from the table above are ciphers 2, 4, 7, 9.

Which reminds me of the sound of a dog not barking.



 

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