WARPING NORMAL DISTANCE ON THE NUMBER LINE
We usually say the number , for instance, is a distance five from on the number line because is five unit lengths away from the zero. (We usually use absolute value notation for this distance: .)
And is a distance from , , because three-and-seven-tenths of a unit fit between and on the number line. And so on.
This is a very additive way of thinking about distance: adding five s gets you from to ; adding s get you from to ; and so on. We can say that the distance of a point on the number line, in this thinking, is the number of s that go additively into .
But much of mathematics is not only concerned with the additive properties of numbers, but also the multiplicative properties of numbers. For example, many people are interested in the prime factorizations of numbers (for example, and ). There are so many unanswered questions about the prime numbers and prime factorizations still in mathematics today. These questions are, in general, very hard!
Is there are way to bring the geometry of the number line into play to possibly help with multiplicative questions? Is there a way to think about the number line itself as perhaps structured multiplicatively rather than additively?
To think about this, rather than focus on all possible factor of numbers, let’s focus on one possible factors of numbers. And to keep matters relevant to our base-ten arithmetic thinking, let’s focus on the number .
In our additive thinking for distance on the number line we use the unit of and ask how many ones (additively) go into each number for its distance from . We now want to use the unit of and ask how many times ten goes multiplicatively into each number.
What could that mean?
In the world of integers the number is the most divisible number of all: it can be divided by any integer any number of times and still give an integer result (namely ) each and every time. Focusing on our chosen factor of ten, we can divide by ten once, or twice, or thirty-seven times, and still have an integer.
The number is a little bit “zero-like” in this sense in that we can divide it by ten once and still have an integer. The number is more zero-like as it can be divided by ten twice and still give an integer result. A googol is very much more zero-like: it can be divided by ten one hundred times and still stay an integer.
The integer is not very zero-like at all: one can’t divide it by ten even once and stay an integer.
In this setting the more times ten “goes into” into a number multiplicatively, the more zero-like it is. So in this sense, a googol is much closer to zero than is.
So let’s develop a distance formula that regards numbers with large powers of ten as factors as closer to zero than numbers with less counts of powers of ten as factors. There are a number of ways one might think to do this, but let’s try to mimic the additive properties of the number line we are familiar with.
Normally we would say that is further from zero than is, and, in fact, we might even say is ten times further from zero as is. In our multiplicative thinking, is now closer to zero than is and it would be natural to have it as ten times closer.
The following formula seems a natural way to have this happen.
If can be divided by ten a maximum of times and remain an integer, then set .
For example, then, and and . Also, since . we see, indeed, that is ten times closer to zero than is.
We can also measure the distance between any two numbers in this multiplicative way. For example, the distance between and is .
With this new way to measure distance, we see that
, , , ,
is a sequence of numbers getting closer and closer to zero. We have and and and , and so on, indeed approaching a distance of zero from .
In terms of values in a machine, we see that boxes far to the left in the machine, representing high powers of ten, are representing values very close to zero. (Before, in our additive thinking, boxes to the far right for decimals represented values very closer and closer to zero.)
Mathematicians call this way of viewing distances between the non-negative integers ten-adic arithmetic. (The suffix adic means “a counting of operations” and here we are counting factors of ten.) It is fun to think how to extend this notion of distance to fractions too, and then to all real numbers.
Let’s look now at the sequence of numbers and and and so on marching off to the right on the number line. Could they possibly be marching closer and closer to the value ?
Yes, if by “closer” we mean this new multiplicative way to think of distance.
The numbers , , , , … are indeed approaching the value .
Comment: We can now justify the (very) long addition computation given below.
We first compute , and then we add to this to obtain , and then we add to obtain , and so on. The further along we go with the computation the closer our results are to the number zero.
You can intuitively see this in the machine: when you add one more dot to this loaded machine and perform the explosions, one clears away dots, pushing what remains further and further to the left where boxes have less and less significant value.
Computation of this next (very) long multiplication is justified in a similar way.
We first compute , and then add to this to get , and then add to this to get , and so on. Now the numbers , , , … are getting closer and closer to zero, so the numbers , , , … are getting closer and closer to . The further along we go with this computation, the closer our answers are to the number .
Again, we can intuitively see this reasoning at play by tripling all the values in this loaded machine.
And did you discover that behaves like the fraction ?
Question: Doubling gives which is one more than , which is . Is this consistent?
Challenge: Show that in a machine that is negative one! Show that when multiplied by gives , and so represents . (What measure of distance might we be using on the number line this time for these “numbers” to make sense?)
CONSTRUCTING NEGATIVE INTEGERS
In our base-ten thinking with our multiplicative notion of distance on the number line, we set
where is the largest count of times can be divided by ten and remain an integer.
And we have made sense of as a meaningful number with value .
So what’s in this unusual system of arithmetic?
Let’s think in terms of a machine. Since , and is double , we should have
With explosions we get
And one can check that this long addition does give zero.
We can see now how to readily construct any negative integer. For example, we can see that adding to will give zero and so this latter quantity must be , and that adding to gives zero and so this quantity must be .
Challenge: What is in a machine? What is ?
We saw that is the fraction : multiply this quantity by three and you get .
The machine provides a natural way to compute such fractions. For example, let’s find the ten-adic representation of . That is, let’s find a number such that . Start by writing
as for a machine. Then
We want ,after explosions, to leave a . So we need a multiple of four greater than a multiple of . We see that is good. So let’s set .
Now we want to be a multiple of so that all dots in that box explode to leave zero behind. This suggests .
Now we need a multiple of . Choose .
Now choose .
And then .
And now I am doing the same work as I did for a value , making a multiple of . We are in a cycle and so is represented as
Challenge: This process felt reminiscent of the task of writing as a decimal in ordinary arithmetic using a machine with decimals. We argued there too that the decimal represent had to fall into a cycle.
Can you argue that the fraction will also have a repeating ten-adic expansion?
Challenge: What is the ten-adic expansion of ?
Write as and add some dots and antidot pairs to make all the terms positive.
Let’s try to compute the ten-adic representation of the fraction . Here we seek a number
This means we have a number so that, after explosions, leaves a single dot. That is, we need to be one more than a multiple of ten. This is not possible!
Challenge: Contemplate the ten-adic expansions for and and .
In general, which fractions seem to be problematic?
Challenge: Develop a general theory that if is a reduced fraction with sharing no factor in common with ten (other than ), then it is for certain possible to express as a ten-adic number . Show further that its expression is sure to fall into a repeating cycle.
BROADENING OUR DEFINITION A TAD
It seems we have defined a ten-adic value to be an expression of the form with each digit one of the standard digits through , allowing for non-zero digits to appear infinitely far to the left.
In this system we have the ordinary positive integers,
eg is ,
the negative numbers
and some fractions
eg is .
But not all fractions. It turns out that the troublesome fractions are the ones which, when written in reduced form, have a denominator a multiple of or or both.
We can obviate this problem if we allow a ten-adic number to extend finitely far into the decimal places on the right. That is, set a ten-adic expression to be one of the form with each digit one of the standard digits through , allowing for non-zero digits to appear infinitely far to the left of the decimal point, and only finitely far to its right. (After all, we do the analogous thing in ordinary arithmetic by writing , for example, for thirty-three and a third.)
Now we have
We can also handle by thinking of this as . Since is we must have .
Challenge: Show that and hence find the ten-adic expression for .
What is the ten-adic expression for ?
Challenge: Explain why every fraction is now sure to have a ten-adic representation.
Challenge: Show that is the number in ten-adic arithmetic. (Hint: Multiply the quantity by and subtract.)
One can use the technique of this question to show that every ten-adic number that eventually falls into a cycle going leftwards is a rational number.
Challenge: In ordinary arithmetic, the quantity is the fraction . We see this by setting and noticing that . Subtracting then yields .
Show that the same algebra applied to the ten-adic number shows that it this number has value .
In fact, prove the following general result. Suppose , , …, are single digits.
If is the fraction in ordinary arithmetic,
then is the fraction in ten-adic arithmetic, and vice versa.
Challenge: Explore a theory of “3/2-adic” representations of fractions using a machine.