Exploding Dots

9.2 Base One-and-a-half?

Lesson materials located below the video overview.

Here is a video from Goldfish & Robin and friends “Where Young Minds Collide” demonstrating the weird 2 <– 3 machine

 

Let’s get weird!

 

What do you think of a \(1 \leftarrow 1\) machine?

What happens if you put in a single dot? Is a \(1 \leftarrow 1\) machine interesting? Helpful?

 

What do you think of a \(2 \leftarrow 1\) machine?

What happens if you put in a single dot?

What do you think of the utility of a \(2 \leftarrow 1\) machine?

 

After pondering these machines for a moment you might agree there is not much one can say about them. Both fire “off to infinity” with the placement of a single dot and there is little control to be had over the situation.

 

How about this then?

 

What do you think of a \(2 \leftarrow 3\)machine?

This machine replaces three dots in one box with two dots one place to their left.

 

Ah! Now we’re on to something. This machine seems to do interesting things.

 

For example, placing ten dots into the machine

first yields three explosions,

then another two,

followed by one more.

We see the code \(2101\) appear for the number ten in this \(2 \leftarrow 3\) machine.

 

In fact, here are the \(2 \leftarrow 3\) codes for the first fifteen numbers. (Check these!)

 

Some beginning questions:

Does it make sense that only the digits \(0\), \(1\), and \(2\) appear in these codes?

Does it make sense that the final digits of these codes cycle \(1,2,0,1,2,0,1,2,0,\ldots\)?

Can one do arithmetic in this weird system? For example, here is what \(6+5\) looks like. Is this answer indeed eleven, which has code 2102?

But the real question is: What are these codes? What are we doing representing numbers this way? Are these codes for place-value in some base?

 

Of course, the title of this section gives the answer away, but let’s reason our way through the mathematics of this machine.

 

Dots in the rightmost box, as always, are each worth \(1\). Let’s call the values of dots in the remaining boxes \(x\), \(y\), \(z\), \(w\), ….

Now three dots in the \(1\)s place are equivalent to two dots in the \(x\) place.

This tells us that \(2x=3\cdot\), giving the value of \(x=\dfrac{3}{2}\), one-and-a-half.

In the same was we see that \(2y=3 \cdot \dfrac{3}{2}\).

This gives \(y=\dfrac{3}{2}\cdot \dfrac{3}{2}=\left(\dfrac{3}{2}\right)^2\), which is \(\dfrac{9}{4}\).

 

And in the same way,

\(2z=3\left(\dfrac{3}{2}\right)^2\) giving  \(z=\left(\dfrac{3}{2}\right)^3\), which is \(\dfrac{27}{8}\),

\(2w=3\left(\dfrac{3}{2}\right)^3\) giving \(w=\left(\dfrac{3}{2}\right)^4\), which is \(\dfrac{81}{16}\),

and so on.  We are indeed working in something that looks like base one-and-a-half!

 

Comment: Members of the mathematics community might prefer not to call this base-one-a-half in a technical sense since we are using the digit “2” in our work here. This is larger than the base number. To see the language and the work currently being done along these lines, look up “beta expansions” and “non-integer representations” on the internet. In the meantime, understand that when I refer with “base one-and-a-half” in these notes I really mean “the representation of integers as sums of powers of one-and-a-half using the coefficients 0, 1, and 2.” That is, I am referring to the mathematics that arises from this particular \(2 \leftarrow 3\) machine.

 

I personally find this version of base one-and-a-half intuitively alarming! We are saying that each integer can be represented as a combination of the fractions \(1\), \(\dfrac{3}{2}\), \(\dfrac{9}{4}\), \(\dfrac{27}{8}\), \(\dfrac{81}{16}\), and so on. These are ghastly fractions!

 

For example, we saw that the number ten has the code \(2101\).

Is it true that this combination of fractions, \(2 \times \dfrac{27}{8}+1\times \dfrac{9}{4}+0\times \dfrac{3}{2}+1\times 1\), turns out to be the perfect whole number ten?  Yes! And to that, I say: whoa!

 

There are plenty of questions to be asked about numbers in this \(2 \leftarrow 3\) machine version of base one-and-a-half, and many represent unsolved research issues of today! For reference, here are the codes to the first forty numbers in a \(2 \leftarrow 3\) machine (along with zero at the beginning).

 

QUESTION 1: PATTERNS?

Are there any interesting patterns to these representations?

Why must all the representations (after the first) begin with the digit 2?

Do all the representations six and beyond begin with 21?

If you go along the list far enough do the first three digits of the numbers become “stable”?

What can you say about final digits? Last two final digits?

 

Comment: Dr. Jim Propp of UMass Lowell, who opened my eyes to the \(2 \leftarrow 3\) machine suggests these more robust questions.

 

What sequences can appear at the beginning of infinitely many \(2 \leftarrow 3\)  machine codes?

What sequences can appear at the end of infinitely many \(2 \leftarrow 3\) machine codes?

What sequences can appear somewhere in the middle of infinitely many \(2 \leftarrow 3\) machine codes?

 

QUESTION 2:  EVEN NUMBERS?

Look the list of the first forty \(2 \leftarrow 3\) codes of numbers. One sees the following “divisibility rule” for three.

 

A number written in \(2 \leftarrow 3\) code is divisible by three precisely when its final digit is zero.

 

What is a divisibility rule for the number two for numbers written in \(2 \leftarrow 3\) code? What common feature does every second code have?

(I am personally very intrigued by this question. I don’t have an answer!)

 

QUESTION 3: UNIQUENESS?

Prove that these \(2 \leftarrow 3\) machine representations of whole numbers using the digits \(0\), \(1\), and \(2\) are unique.

 

[And as an infinite number of asides: Prove that every whole number can be uniquely written as sums of non-negative powers of \(\dfrac{7}{5}\) using the coefficients \(0,1,2,3,4,5,6\). And that every whole number can be uniquely written as sums of non-negative powers of \(\dfrac{10}{7}\) using the coefficients \(0,1,2,3,4,5,6,7,8,9\). And that every whole number can be uniquely written as sums of non-negative powers of \(\dfrac{339}{56}\) using the coefficients \(0,1,\ldots,338\). And so on!]

 

QUESTION 4:  IS IT AN INTEGER?

Not every collection of \(0\)s, \(1\)s, and \(2\)s will represent a whole number code in the \(2 \leftarrow 3\) machine. For example, looking at the list of the first forty codes we see that \(201\) is skipped. This combination of powers of one-and-a-half thus is not an integer. (It’s the number \(5\frac{1}{2}\).)

 

Here’s a question: Is

 

 the code for a whole number in a \(2 \leftarrow 3\) machine?

 

Of course, we can just work out the sum of powers this represents and see whether or not the result is a whole number. But that doesn’t seem fun!

Is there some quick and efficient means to look as a sequence of \(0\)s, \(1\)s, and \(2\)s and determine whether or not it corresponds to a code of a whole number? (Of course, how one defines “quick” and “efficient” is up for debate.)

 

BY THE WAY … Have you noticed that if you delete one or more end digits of a code for a number you are sure to have a valid code for an even integer? Can you explain why this must be the case?

 

QUESTION 5:  NUMBER OF DIGITS

Looks again at the first forty \(2 \leftarrow 3\) codes.

Notice

0 gives the first one-digit code. (Some might prefer to say  here.)

3 gives the first two-digit code.

6 gives the first three-digit code.

9 gives the first four-digit code.

and so on.

This gives the sequence: 3, 6, 9, 15, 24, …. (Let’s skip the questionable start.)

 

Are there any patterns to this sequence?

 

If you are thinking Fibonacci, then, sadly, you will be disappointed with the few numbers of the sequence.

\(36, 54, 81, 123, 186, 279, 420, 630, \ldots\) 

 

A Recursive Formula.

Let \(a_{N}\) represent the \(N\)th number in this sequence, regarding \(1\) as the first one-digit answer.  It is known that

(If \(m\) dots are needed in the rightmost box to get a code \(N\) digits long, how many dots do we need to place into the \(2 \leftarrow 3\) machine to ensure that \(m\) dots appear the second box? This will then give us a code \(N+1\) digits long.)

 

An Explicit Formula?

Is there an explicit formula for \(a_{N}\)? Is it possible to compute \(a_{1000}\) without having to compute \(a_{99}\) and \(a_{98}\) and so on before it? (This question was posed by Dr. Jim Propp.)

 

FURTHER: There is a lot of interest about problems involving the power of two, and three, and of three-halves. See Terry Tao’s 2011 piece , for instance.

 

 

QUESTION 6:  COUNTING EXPLOSIONS

The following table shows the total number of explosions that occur in the \(2 \leftarrow 3\) machine to obtain the code of each of the first forty numbers.

Any patterns?

 

 

QUESTION 7:  RATIONAL DECIMAL EXPANSIONS

We can go to “decimals” in a \(2 \leftarrow 3\) machine.

Here is what \(\dfrac{1}{2}\) looks like as a “decimal” in this base. (Work out the division \(1\div 2\).)

(Do you have choices to make along the way? Is this representation unique?)

 

Here are some more “decimal” places.

Can \(\dfrac{1}{2}\) have a repeating “decimal” representation in a \(2 \leftarrow 3\) machine?

What’s a “decimal” representation for \(\dfrac{1}{3}\) in this machine?

 

Develop a general theory about which fractions have repeating “decimal” representations in the \(2 \leftarrow 3\) machine. (I don’t personally have one!)

 

 

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