Something fascinates me about number systems. When I was younger I even created my own system and taught myself the rules of how it worked (at the time I thought I was “discovering” these rules, but alas, I was no forerunner in the field).
My system was Base-Two system, not a Base-Ten like the one the world uses today (except for computers). My system worked like this…
I had many different symbols for different values (a flaw I see now), so my system, though a Base-Two system could not truthfully be called a “binary system”… I contend that “binary” used in this context means “two names” or “two terms.” Computereeze I call a “binary” system.
Each of my symbols (hand-drawn in those days) represented different powers of two. Actually, at first, as I remember it, I did not intend to start with numbers which were ONLY powers of two… What I had started doing was allowing each new symbol to equal the previous symbol DOUBLED. I just thought this was cool.
It worked like this (making do here with typed symbols to replace my hand-drawn ones)…
a = 1, b = 2, c = 4, d = 8, e = 16, f = 32, g = 64, etc…
It was only after I started doing this doubling that I realized that each of my symbols was a power of two. By simply adding to my list of symbols one standing for 2^0 (or 1), my symbols then became a true Base-Two system. I could then also represent ANY Counting-Number. It now looked something like this…
z = 2^0, a = 2^1, b = 2^2, and so on…
I had no special symbol for zero. I just used a “0” if there was no value at all. But I found that when you’re not using a place-value system, the whole zero thing becomes far less important. No wonder the Romans never bothered with it (theirs was not a Place-Value system either).
The idea of using Place-Value did not even occur to me… and this turns out to be a rather inefficient way to represent a large number. For example, to represent the number 63, I would had to write…
…. for edcbaz = 2^5 + 2^4 + 2^3 + 2^2 + 2^1 + 2^0 = 63
In this way, it took me six symbols to communicate what we could write with only two symbols using today’s “normal” way. Realizing that this was a fairly long string of numbers to represent “63”, I artistically decided that any string of consecutive Base-Two symbols could be apostrophized so that, for instance…
edcbaz = e’z
What I found most exciting about my invented number system [that may just be the nerdiest introductory phrase I’ve ever written] was that I began to see certain patterns or rules emerging… For instance, the above number and those like it clued me in that…
2^(x) + 2^ (x-1) + 2^(x-2) + … + 2^(0) = 2^(x+1) – 2^0
It also dawned on me that I could multiply two numbers of my system by merely adding exponents. For example…
(c * d) = (2^3 * 2^4) = (8 * 16) = 128 = 2^(3 + 4) = 2^7
In other words…
2^x + 2^y = 2^(x+y)
I thought this was groovy.
My system also highlighted commonsensical relationships like…
2^x + 2^x = 2^(x+1)
2^y – 2^(y-1) = half of 2^y
Playing around with subtraction in my system, I also learned that I could re-state a large number into smaller chunks which would facilitate subtraction. For example, if I was subtracting 4 from 128, I could restate my symbol representing “128” so that it contained a value for “4” which I could then take away…
Example problem: g – b = ?
given that g = fedcbb, then we can restate the problem as…
g – b = fedcbb – b
[I’m wondering now looking at this if this is a rule, that x – y always equals (x-1)’y, but I don’t want to take the time to look into at the moment]
In the realm of Base-One or Base-Two numbering/counting systems, I later learned I was preceded by other mathematicians by a few hundred years or so…
According to Michael J. Bradley’s Age Of Genius: 1300 to 1800, John Napier [NAY pee yur] (1550 – 1617) created a binary number system– only he was smart enough to utilize a positional system. Although Bradley tells us next-to-nothing about this system, he does say that Napier was able to do all the basic calculations (add, subtract, multiply, divide) using his notation.
Gottfried Leibniz [LIEP nitz] (1646 – 1716) also experimented with a place-value binary system, and he used the numbers 0 and 1 to boot (no computer pun intended). Again, Bradley is unforthcoming with the details, but apparently– like yours truly– the numbers used by Leibniz were expressed as sums of different powers of 2… which power of two was determined by place-value (why didn’t I think of that?!).
For example, the term “1101” represents: 2^3 + 2^2 +  + 2^0 . This means that “1101” equals 8 + 4 + 0 + 1, or 13, in Leibniz’s binary system. He described his ideas in a 1701 essay entitled Essay On A New Science Of Numbers.
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Also, since I don’t have anything worth a full post to say about it at the moment, I’ll just note here that Leibniz also attempted to bequeath to human speech a logical precision similar to that used in mathematics. He attempted this by inventing a system of universally recognizable symbols to represent, as Bradley explains, “a small number of fundamental concepts,” together with logical operations to relate them. His hope was that his new “Universal Language” could express all human thought as combinations of universally recognized symbols.
The list of the fundamental concepts Leibniz used included: the null set, identity, negation, and class inclusion. Many years later, George Boole would use similar ideas when creating his Boolean Algebra (which I also need to eventually take a look at).
As I understand it, Leibniz never got very far with his idea. Nevertheless, as a writer always in search of clarity (both of in giving and in receiving communications), I stand intrigued and impressed by his attempt. I, myself, have dabbled in this area, and still have hopes of playing more with the idea. My approach thus far has been to mix Venn Diagrams with the verbal equivalent of a “color-wheel,” with prefixes and suffixes near-central to the whole plan. If I ever get any farther with the idea, maybe I’ll post about it.