Numbers and Numerical Systems

Tom Kando 

I was recently watching the great old musical Brigadoon. The people of that mythical fantasy land come alive once every century. So I’m thinking: Why every century? Why not some other time unit? Why did most of the world agree to go decimal? 

There are still exceptions, notably this country’s mishmash of measuring systems: Unlike much of the world, Americans measure distance and height in inches, feet, yards and miles, not centimeters, meters and kilometers. Our weights are in ounces and pounds, not grams and kilos. Volume is expressed in gallons, not liters, temperature in Fahrenheits, not centigrades, etc. But even in the US, science and medicine do most things decimally. 

There is no question that decimal is a lot better than the hodgepodge of measurement systems we use. But even decimal is an arbitrary system, hardly reflecting the nature of the world. It is often said that the choice had something to do with our ten fingers. 

A while ago I wrote a post asking whether mathematics is a discovery or an invention. In other words, is it embedded in nature, or do we humans use it to interpret nature? (See my post of Feb. 22 2024): ”Is Math a Discovery or an Invention?” or: Was God a mathematician?” 

Today, I am thinking of something related: Alternative numerical systems. The big one that immediately comes to mind is the binary system, because computers use it and computers have taken over the world. If we went binary, years would be numbered differently. We learn in history that Charlemagne was crowned Emperor in the year 800.

Converted into binary numbers, that would be the year 1100010000. This year would not be 2025. It would be called the year 11111101001. In a binary world, the number of days in a year would not be 365. It would be 101101101. Conversion of decimal numbers into binary numbers is already widespread, of course: A binary system only has 2 numbers: 1 and 0. To convert a decimal number into a binary number: Repeatedly divide the decimal number by 2, noting the remainder at each step; the binary representation is the sequence of remainders written in reverse order, where a remainder of 1 represents a "1" in binary and a remainder of 0 represents a "0" in binary.(Digital to binary conversion)

 There is something more natural  about the binary system than the digital system. After all, the difference between Nothing (zero) and Something (one ) is fundamental. Same as the difference between yes and no. 

Actually, I was first confusing this binary system with another one, namely the system which consists of the two possibilities “1" and “2." But such a system is not binary. It is called ternary and it consists of three digits not two, namely “0,” “1" and “2." I took the difference between “1" and “2" as being fundamental, the difference between singular and plural. I forgot the third possibility - “0" 

The ancient Greeks reasoned somewhat similarly when, in their language and grammar at least, they had something like a quaternary system. They not only had “1"(singular) and “3" for more (plural), but also “2" (a separate category meaning “pair.”) 

I am always intrigued by the question whether or not some numbers are inherent in nature. I said previously that in a binary world, the number of days in a year would not be 365. It would be 101101101. Obviously, the number would be the same, no matter how you count it. 

But there are some numbers that appear frequently in nature, for example in plant growth. For instance, many of the Fibonacci numbers are said to occur in nature. The Fibonacci sequence is simply this: Each number is the sum of the two preceding numbers: 0, 1,1, 2,3,5,8,13,21,34,55,89,144,233, etc. 

Related to this is the golden ratio, which is also called Phi. It is about 1.618 and it expresses the ratio of the sum of two numbers to the larger of the two numbers. For example 89/55 = 1.618 (See above Fibonacci series) 

One conclusion to be derived from my musings is this: Even mathematics, as practiced in most of the world today, is to some extent “cultural.” The decimal system is fine, but it is a choice. Other possibilities exist, and several of them have been used by other civilizations. But I am in no way arguing for upsetting the apple cart! leave comment here