Who's on first?
These are the types of thoughts that strike a scientist as he showers in the morning: "Hmm. I wonder which properties of numbers are base-independent?" It actually made me drop my soap.
Some things, like whether a number is prime, irrational, or whether it is Pi or e or some other property or constant are most likely independent of the base one is working in. We are familiar with, and tend to work mostly in base 10, but perhaps some of these numbers would look different in other bases? Is there a base in which numbers have different properties? What about irrational bases? Can you have imaginary number bases?
I pondered these questions as I lathered up what is left of my hair, and it occurred to me that certain properties of numbers are actually only psychological, and not rooted in number theory at all.
Here's an example from the news: the famous 'milestone' reached recently of 4,000 U.S. military casualties in Iraq. Four thousand is significant to us because it represents a round thousands number, like 1,000 2,000 3,000 et cetera. In this particular case, the 4,000 was being portrayed as a psychological breaking point for the will of the American public to keep fighting (more on death tolls from different events in a future post).
However, it turns out that 4,000 has only a psychological significance, rather than a truly numerical one, because just what is it about 4,000 that intrinsically makes it different from 3,999 or 4,001? After all, it's only when it is written in base 10 that it looks this way. In hexadecimal, or base 16, a system just as valid as base 10, that number looks like
FA0, which is not particularly significant next to 3,999 and 4,001, which look like
FA1in hexadecimal. In fact, in hexadecimal, the psychologically significant numbers
1000, 2000, 3000turn out to be (in base 10): 4,096, 8,192 and 12,288, which don't look very interesting at all to us base-10 biased folk.
So it turns out that certain things about numbers depend on which base you are using. Those things which are fundamental (like being prime), are independent of the base being used to represent the number itself. If the property depends on the base being used, that's probably a good sign that the property is not intrinsically part of formal number theory (but I'd love to be proved wrong).
So we could fully expect, on that auspicious day where we met an alien species with a different number of digits (fingers), that they would have a completely different set of psychologically significant numbers, because it is likely that they would count in a different base. It occurred to me that the more fingers they had, the more difficult they would be to defeat in battle, because it would take more 'kills' to reach their psychological breaking points!
Anyway, that was enough science fiction, and by this point I was towelling off anyway.
We have lots of these psychologically significant number categories floating around: centuries, decades, dozens (an example of working in a non-base 10 system), and the famous pricing strategy of leaving something at one cent below (e.g. "ON SALE FOR ONLY $2.99!!!" etc. etc.).
It was now time to adjust my bowtie, and go off and count my eggs before they hatched, and decide whether the count was significant or not.