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Saturday, November 03, 2007

Integer Vitae II

Remember these?

…, 1010, 1011, 1000, 1001, 1110, 1111, 1100, 1101, 10, 11, 0, 1, 110, 111, 100, 101, 11010, 11011, 11000, 11001, 11110, ...

Here are the integers from -10 to +10 listed in the usual way--in sequence and in ascending order. So “0” is 0 and “1” is 1. If the encoding is context-free, then “0” is always 0 and “1” is always 1. In fact, this is true. So nothing represents minus sign (or plus sign). We have some sort of binary system which represents the integers without the use of signs; and judging by this modest sample, it represents all the integers uniquely, just like more familiar numerations. The fact that log2(21) = 4+ (just edging into the five-digit numerals) supports these hypotheses. Arithmetic? Looks dicey, but simple algorithms must exist, because indeed this is a familiar radix system (successive positions indicate successive powers of the so-called base). Most familiar is base 10; reformers have pushed base 12; computer geeks use base 2 (binary), base 16 (hexadecimal), and occasionally base 8 (octal). The numeration in question is radix, base -2.

I include Steadman’s interesting comment from I.V. I.

Are there negative numbers in the list? Have you dropped the signs or are they encoded too? They appear to be base four couplets, but it's 1AM, I just finished a lot of work, and my brain is dead.

Substituting as follows:A=10, B=11, C=00, D=01, your list is then

AA,AB,AC,AD,BA,BB,BC,BD,A,B,C,D,DA,DB,DC,DD,DAA,DAB,DAC,DAD,DBA,

which has enough in common with counting to make me think I'm close, but I'd be more convinced if D was followed by BA.

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