Continuing my series of posts exploring the decadic numbers… in my previous post, I explained that we will define a new “size function”, or metric, different from the usual “absolute value”, and written . Two numbers will be “close to” each other when the size of their difference is small.
For now we will define on the integers. First, every nonzero integer can be factored as , where is not divisible by . In other words, is the largest power of which evenly divides . For example, ; ; . Now, for not divisible by , we simply define
So, ; ; . This definition doesn’t cover zero, however, because for any . But notice that the distance from any number to itself will be , so we define , since the distance from anything to itself ought to be zero.
So, what pairs of integers are close to each other, under this new metric? How about and ? We compute . At first sight that might seem pretty close. After all, under the usual absolute value metric, that’s the closest two natural numbers can possibly be to each other without being equal. But the decadic metric is always of the form for some nonnegative , and it’s easy to see that is actually the largest possible value for this metric! So and are actually rather far apart. Weird, right? (Before reading on, can you come up with a pair of integers that are closer?)
As an aside, note that the “number line” is no longer helpful: it gives us a visual sense of how far apart any two integers are, but only for the usual absolute value metric!
For our current purposes it’s more useful to think of the integers as living in a big “soup”: each integer still exists and means the same thing it usually means, but the physical relationship of the integers to one another is quite different than what we are used to. (Their arithmetic relationships are unchanged: for example, still equals .) and are right next to each other on the number line, but they are on completely opposite sides of the soup! Part of our job will be to figure out more about what that “soup” actually looks like.
So, what about a pair of integers that are closer together than and ? How about and ? “Those don’t seem very close at all!” You might protest. Aha, but you are still thinking in terms of the old absolute value metric! Let’s compute their decadic distance:
Aha, so they really are a bit closer together. What about and ?
Those are closer still!
Do you see what’s going on? Can you come up with a concise description of when two integers will be close? How close can two integers be?