## Notes on Representation – Coexistence

In my earlier post on tally marks, I look at the way the tally mark representation system coexists with the decimal number system – with the tally marks thriving in a small but stable niche.  In some ways, they coexist like lions and butterflies, not really competing in any obvious way, and not in any obvious way a threat to each other.

Where representations of quantity differ from species is that there pretty much needs to be a way to convert quantities back and forth from one representation system to another.  If we were to picture a hypothetical situation in which one country uses the decimal number system exclusively, and another uses tally marks exclusively, it would be hard to picture any trade between them unless tally marks could be converted to decimals and vice versa.  The same principle applies within a country.  Two representational systems can be used side by side, but there needs to be a way to cross over to the other side.  If you’ve heard that computers, internally, use a representational system for numbers and quantities different from the one we use, you would also assume that these same computers are able to convert back and forth whenever a number “crosses the line” from you on its way in, or on the way out back to you.  If a computer does such conversion well, its doing so becomes transparent to you, and you might not care.  If you give a computer two numbers to add, a computer therefore must do much more than addition: it converts each number into its internal representation (the binary number system), adds the two binary numbers producing a binary sum, and then converting this binary sum back into a decimal number to show you.  However, if you ask it to compute 103 + 37 + 57 for you, the result of its first addition, 103 + 37 is not shown to you and need not be converted into decimal.  This highlights a general principle, that conversions are only needed at the boundaries between two systems of representation.  If a large number of calculations is done on one side of the boundary, the relative cost or difficulty of the conversion matters less.

The metric system is an example of a system of representing quantities that can coexist with the traditional system of measurements used in the USA (also called the English system of measures, though England has moved to the metric system).  The USA is the major holdout in the world, the rest of the world has pretty much standardized on the metric system.  Yet metric countries who trade with the USA are willing to do the conversion to the US system, and so the systems can coexist.  Even in the USA there are niches where the metric system has taken hold – in science, primarily.  Many rulers in the USA show inches on one side, and centimeters on the other.

When you first work with a system of representation of quantity, you are naturally tempted to constantly convert numbers back to the system you’re already familiar with.  As you gain more traction in the other system, you feel safer in staying in the other system for longer amounts of time without converting back.  And the less you need to convert, the more you notice differences in convenience, efficiency, and ease that may be available in the other system.  Personally, I’m not interested in trying to talk you into or out of using the metric system or the USA system of measurements.  I am interested in looking at different ways to represent numbers and looking at what those representations reveal about the underlying mathematics.  There is rich mathematics to be revealed in looking at various representational systems, even in a simple one like tally marks!