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 $|-|_{10}$. Two numbers will be “close to” each other when the size of their difference is small.

For now we will define $|-|_{10}$ on the integers. First, every nonzero integer $n$ can be factored as $10^k \cdot m$, where $m$ is not divisible by $10$. In other words, $k$ is the largest power of $10$ which evenly divides $n$. For example, $23000 = 10^3 \cdot 23$; $-50 = 10^1 \cdot (-5)$; $297 = 10^0 \cdot 297$. Now, for $m$ not divisible by $10$, we simply define

$|10^k \cdot m|_{10} = \frac{1}{10^k}$.

So, $|23000|_{10} = 1/1000$; $|-50|_{10} = 1/10$; $|297|_{10} = 1$. This definition doesn’t cover zero, however, because $0 = 10^k \cdot 0$ for any $k$. But notice that the distance from any number to itself will be $|x - x|_{10} = |0|_{10}$, so we define $|0|_{10} = 0$, 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 $10587$ and $10588$? We compute $|10587 - 10588|_{10} = |1|_{10} = 1$. 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 $1/10^k$ for some nonnegative $k$, and it’s easy to see that $1$ is actually the largest possible value for this metric! So $10587$ and $10588$ 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, $2 + 2$ still equals $4$.) $10587$ and $10588$ 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 $10587$ and $10588$? How about $34976$ and $20734346$? “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:

$|20734346 - 34976|_{10} = |20699370|_{10} = |10^1 \cdot 2069937|_{10} = 1/10$

Aha, so they really are a bit closer together. What about $39645$ and $99945$?

$|99945 - 39645|_{10} = |60300|_{10} = 1/100$

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?

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### 5 Responses to The decadic metric

1. Jonathan says:

I guess your last question means two unique integers? It looks like things are closest when their difference is a number ending in the longest string of zeroes. To recall junior high, the closest decadic numbers are the ones whose differences are the least annoying to write in “scientific notation”

I can’t think of a neat trend yet. At first I thought numbers would have to be of the same order of magnitude to be close, but 300000000001 and 1 are pretty close. So are 300009 and -1, or 300000 and -300000.

It seems to me that closeness is defined by two things:
• The “complementarity” of the numbers: whether they differ to a factor of 10
• The largest magnitude of the two numbers: we can get a smaller distance when at least one number is a big factor of ten

I am trying to think of how I’d map this. It sure isn’t a line, since given any number and any distance I can pick a lot of integers that are 1 away. I can also pick an arbitrarily large number of much larger-order numbers that are closer and closer. What I mean is (and sorry, since there’s no preview button I can’t really try out this LateX thing)

For 987:
• Most numbers are 1 away
• -3, 7, -23, and 17 are 1/10 away
• -113, 87, and 187 are 1/100 away
• -1013, -13, and 1987 are 1/1000 away

The most special number here seems to me to be -13. I am getting the biggest bang for my buck in the low-order number -13, order being the power of 10 it is closest to (ignoring sign).

So I propose: if O(x) is the order of x, an N and J are integers, then for every number N there is just one number J where

O(J) ≤ O(N)
and
O(|N-J|_10) > O(N)

and you can find numbers closer and closer to N by choosing high-order powers of ten and adding J.

2. Mark James says:

In a sense, you can think about writing the numbers backwards to get an intuition for the difference with this metric.

For example, 3000001 and 1 can be rewritten as 0.10000003 and 0.1 which are quite close together.