Revisiting There Is No Zero

Here’s an imitation of a pseudomathematician’s rant about the use of the symbol 0 (zero).

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I will hereby prove that all of Modern Mathematics is based on a FALSE foundation: ZERO.

Q: Since ZERO is NOTHING, how can we have a SYMBOL to represent it?
A: We can’t. NOTHING is NOTHING, the symbol we call zero (‘0’) is a FALSEHOOD!

Q: But HOW can we represent numbers without a ZERO!?!?
A: Like this! I’ll use ‘I’ to represent a unit, which is the SMALLEST possible quantity. I’ll use ‘D’ to represent ten units.

COUNT	UNITS
1	I
2	II
3	III
4	IIII
5	IIIII
6	IIIIII
7	IIIIIII
8	IIIIIIII
9	IIIIIIIII
#	D
11	DI
12	DII
13	DIII
14	DIIII
15	DIIIII
16	DIIIIII
17	DIIIIIII
18	DIIIIIIII
19	DIIIIIIIII
1#	DD
21	DDI

You get the picture. No symbol for NOTHING need exist, and in fact CAN exist. It stands to reason that you can’t SYMBOLIZE NOTHING. With the introduction of ‘#’, we can quit using zero for NOTHING or for anything. Now, a way exists to represent EVERY number without a symbol for NO NUMBER.

QED, Mathematics must be rebuilt upon this Firm, LOGICAL foundation.

Suggested pronunciation for the ‘#’ sign: “futplex”.

Example: “We humans all have futplex toes,” Tom said tensely.
Example: “The act of killing one of every futplex prisoners is called ‘decimating’,” Tom concluded.

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I dredged this rant up from a backup of my old web pages. I made a Usenet post in September of 1998 with the text of this rant, and then some time in 1999, turned it into an XHTML standalone web page. The only feedback I ever got on this rant was from my retired father. He discovered Google search maybe 2005, and looked me up. This rant was the first thing he found about me. Luckily, he misinterpreted it, or at least pretended to misinterpret it.

Apparently lots of people have the same view of zero that I developed in my pseudomathematics. You can search for there is no zero to find articles like: Is Zero Really A Number. I’m surprised by this.

A John D. Cook Consulting blog post about how you could determine whether some ancient culture used this kind of numeral system or not reminded me of my old web page. This situation isn’t out of the question, Robert Forslund proposed that ancient cultures may have used just such a numeral system in 1995, in A logical alternative to the existing positional number system, Southwest Journal of Pure and Applied Mathematics, 1: 27–29. Apparently, Forslund’s “logical alternative” is usually called “bijective base 10”, and instead of my ‘#’, the digit with value 10 is usually given as ‘A’.

Returning to John Cook’s blog post about numeral systems of ancient cultures, Cook looks at hypothetical strings of symbols that an archaeologist might believe represent numbers in some positional numeral system. How should the archaeologist decide which symbol stands for 1, which for 2 and so forth? One of the methods John Cook suggests is using Benford’s Law on leading symbol frequency.

I thought about this assertion for a bit, and I couldn’t make it work out intuitively. For example, 1000₁₀ == 99A_A. That is, some fraction of leading 1-digits disappear when converted to bijective base 10. Counting the number of leading ‘1’ digits that turn into ‘A’ digits is not as easy as counting all the numbers that begin with ‘10’, because 110₁₀ == AA_A, and 200₁₀ == 19A_A. Best to try it out.

I came up with an algorithm to write base 10 numbers in bijective base 10 that’s O(n²) where n is the count of digits in the number’s base 10 representation.

I still have data I used to look at whether or not Linux file sizes satisfied Benford’s Law. I ran the file size data from Machine E of that post through my algorithm to get some real Bijective Base 10 numbers. I counted leading digits of the string representations of file sizes in both Base 10 and Bijective Base 10.

Leading digit	Base 10	Bijective Base 10
1	2600301	3171027
2	1737165	1130817
3	233097	217177
4	156836	167141
5	129463	111592
6	101804	101921
7	86039	83615
8	80806	81089
9	260176	257780
0	21711
A		257780

That’s the numerical data. Leading digits of file sizes represented in two numeral systems. Since There is No Zero, I have no files without a length in the Bijective Base 10 leading digits. My intuition was correct in being confused: the counts of leading digits changed a bit when converted to Bijective Base 10.

There’s the numerical data in a more visual form. Clearly, data that fits Benford’s Law in Base 10 also fits it when the data is rendered in Bijective Base 10.