I got so caught up last week talking about the different possible bases that I forgot to the interesting thing I had wanted to talk about those bases. I suppose that will happen as long as I write to passion rather than plan. It gives me something to speak about today, at least.
Here is one thing implied by having a consistent base for all these numbers in which position is relevant: a one in each column represents the base-number of units of whatever the next column over represents. That is, in base ten, a one in the tens column represents ten units of one; a one in the thousands column represents ten units of one hundred. I mention this obvious point because it is so familiar and simple as to pass into invisibility. (It also extends past the decimal point; a one in the hundredths column is equivalent to ten units of a thousandth. But I want to talk about divisibility, in the whole numbers, and so leave fractions for some later time.)
This is tidy, in a way that we don’t see in variable bases. It will give us one tool for neat little divisibility rules. That tool appears just by writing things in the appropriate way, which is the best sort of tool. It saves on time trying to prove it works.
Consider the number 2,038. Since this is in base ten, we can say that this number what we get from multiplying ten by 203, and then adding 8. Right away we see that it can’t be a multiple of ten. The ten times 203 part is all right, but after that, we add that pesky 8 which isn’t divisible by ten. Obviously, if we had added to the ten times 203 some number which was divisible by ten, then the whole sum would be. Of course, the only digit we could put in the ones column which is divisible by ten is itself 0, telling us that 2,030 is divisible by ten.
And this applies for any whole number, not just 2,038. We can imagine any bunch of digits, ending in some digit in the ones column. Let me call the digit in the ones column a; I may not know what it is, but, I will want to. The rest of the numbers are — well, I don’t know, but it is some number, and I will give it the name R, for Rest. Then we have some number written down as Ra, which is the ten times R plus a. The ten times R part is a multiple of ten; therefore, this sum will be a multiple of ten whenever a is. And since a has to be between zero and nine inclusive, then, any whole number ending in zero is divisible by ten; any that doesn’t, isn’t. We learned this a long time ago, so long ago we probably don’t remember ever actually learning it. This is why I spent so much time establishing a point which doesn’t otherwise seem impressive. Wonderful things can grow quickly from an obvious base, but why the obvious is true rarely will be clear.
Now let me put another obvious thing on my pile of the obvious: ten is equal to two times five.
This implies that whatever R is, ten times R is also a multiple of five. And therefore, our number Ra, ten times R plus a, will be divisible by five whenever a is. Since a is between zero and nine again, this will be divisible by five whenever a is zero or five. Sure enough we have one of those divisibility rules taught so early on that it becomes invisible. A whole number that ends in zero or five is itself divisible by five.
Of course, the argument for five works just as well for two. Whenever a divides by two, Ra divides by two. Now the only sad thing is that ten hasn’t got any more factors, so we don’t have any more numbers for which we can spot divisibility at a glance. Here those thoughts of a base twelve numbering system arise. Numbers written in base twelve can be checked at a glance for divisibility by two, three, four, six, and twelve. Bases eighteen and twenty would similarly give us an abundance of on-sight divisibility checks.
But back to base ten. It would be nice to tell on sight divisibility by four, but four doesn’t go a whole number of times into ten. It’s sad but unavoidable, dictated by a logic which transcends even the universe’s existence. There isn’t any test by which we can look at just the last digit of a whole number and say whether four goes into that integer, at least not with better than a fifty-fifty chance of being right.
Yet this does nothing to keep us from building this kind of rule.