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Station  9A
Base One–and–a–Half 

Let’s get weird!

Question 1

What do you think of a 111 \leftarrow 1 machine?

What happens if you put in a single dot?
Is a 111 \leftarrow 1 machine interesting?

Question 2

What do you think of a 212 \leftarrow 1 machine?

What happens if you put in a single dot?
What do you think of the utility of a 212 \leftarrow 1 machine?

Question 3

How about this then?

What do you think of a 232 \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
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first yields three explosions,
I9S9A - Image007

then another two,
I9S9A - Image008

followed by one more.
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We see the code 21012101 appear for the number ten in this 232 \leftarrow 3 machine.

What are the 232 \leftarrow 3 codes of the first fifteen numbers?
Work them out by hand.

Some beginning questions:

Does it make sense that only the digits 00, 11, and 22 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,...1,2,0,1,2,0,1,2,0,...?

Can one do arithmetic in this weird system? For example, here is what 6+56+5 looks like. Is this answer indeed eleven, which has code 21022102?

I9S9A - Image018

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 11. Let’s call the values of dots in the remaining boxes x,y,z,w,...x, y, z, w, ....

I9S9A - Image024

Now three dots in the 11s place are equivalent to two dots in the xx place.
I9S9A - Image027

This tells us that 2x=312x=3\cdot 1, giving the value of x=32x=\dfrac {3}{2}, one-and-a-half.

In the same way we see that 2y=3322y=3\cdot{\dfrac{3}{2}}.
I9S9A - Image031

This gives y=3232=(32)2y=\dfrac{3}{2}\cdot\dfrac{3}{2}=\left(\dfrac{3}{2} \right)^{2}, which is 94\dfrac{9}{4}.

And in the same way,

2z=3(32)22z=3\left( \dfrac {3} {2}\right) ^{2} giving z=(32)3z=\left( \dfrac {3} {2}\right) ^{3}, which is 278\dfrac{27}{8},

2w=3(32)32w=3\left( \dfrac {3} {2}\right) ^{3} giving w=(32)4w=\left( \dfrac {3} {2}\right) ^{4}, which is 8116\dfrac{81}{16},

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

I9S9A - Image040

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 "22” 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 00, 11, and 22.” That is, I am referring to the mathematics that arises from this particular 232 \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 1,32,94,278,8116,1,\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 21012101.
I9S9A - Image048

Is it true that this combination of fractions, 2×278+1×94+0×32+1×12\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 232 \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 232 \leftarrow 3 machine (along with zero at the beginning).

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QUESTION 1: PATTERNS?

Are there any interesting patterns to these representations?

Why must all the representations (after the first) begin with the digit 22?
Do all the representations six and beyond begin with 2121?
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?

Is there a code that ends with 22002200?

Comment: Dr. Jim Propp of UMass Lowell, who opened my eyes to the 232 \leftarrow 3 machine suggests these more robust questions.

What sequences can appear at the beginning of infinitely many 232 \leftarrow 3 machine codes?

What sequences can appear at the end of infinitely many 232 \leftarrow 3 machine codes?

(Find a number whose code ends with five 00s. Then try to find one that ends with five 11s. This might provide some insight on this final question.)

QUESTION 2: EVEN NUMBERS

Look the list of the first forty 232 \leftarrow 3 codes of numbers. One sees the following “divisibility rule” for three.

A number written in 232 \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 232 \leftarrow 3 code? What common feature does every second code have?
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QUESTION 3: UNIQUENESS

The 232 \leftarrow 3 machine shows that each and every whole number can be written as a sum of powers of 32\dfrac{3}{2} using the coefficients 00, 11, and 22.
Now show that these representations are unique in the sense that no whole number can be written as a sum of powers of 32\dfrac{3}{2} using the coefficients 00, 11, and 22 in more than one way.

(And as an infinite number of asides:

Prove that

  • every whole number can be uniquely written as sums of non-negative powers of 75\dfrac{7}{5} using the coefficients 0,1,2,3,4,5,60, 1, 2, 3, 4, 5, 6.

  • every whole number can be uniquely written as sums of non-negative powers of 107\dfrac{10}{7} using the coefficients 0,1,2,3,4,5,6,7,8,90, 1, 2, 3, 4, 5, 6, 7, 8, 9 .

  • every whole number can be uniquely written as sums of non-negative powers of 33956\dfrac{339}{56} using the coefficients 0,12,3,...,3380,1 2, 3, ..., 338.

And so on!)

QUESTION 4: IS IT AN INTEGER?

Not every collection of 00s, 11s, and 22s will represent a whole number code in the 232 \leftarrow 3 machine. For example, looking at the list of the first forty codes we see that 201201 is skipped. This combination of powers of one-and-a-half thus is not an integer. (It’s the number 5125\dfrac {1} {2} .)

Here’s a question:
Is

21022120201200201220112011022020102210201002022122102212020120020122011201102202010221020100202212

the code for a whole number in a 232 \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 00s, 11s, and 22s 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.)

QUESTION 5: NUMBER OF DIGITS

Look again at the first forty 232 \leftarrow 3 codes.
I9S9A - Image080

Notice

00 gives the first one-digit code. (Some might prefer to say 11 here.)
33 gives the first two-digit code.
66 gives the first three-digit code.
99 gives the first four-digit code.

and so on.

This gives the sequence: 3,6,9,15,24,...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,...36, 54, 81, 123, 186, 279, 420, 630, ...

A Recursive Formula.
Let aNa_{N} represent the NNth number in this sequence, regarding 11 as the first one-digit answer. It is known that

I9S9A-image95

(If mm dots are needed in the rightmost box to get a code NN digits long, how many dots do we need to place into the 232 \leftarrow 3 machine to ensure that mm dots appear the second box? This will then give us a code N+1N+1 digits long.)

An Explicit Formula?
Is there an explicit formula for aNa_{N}? Is it possible to compute a1000a_{1000} without having to compute a999a_{999} and a998a_{998} 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 The Collatz conjecture, Littlewood-Offord theory, and powers of 2 and 3, for instance.

QUESTION 6: COUNTING EXPLOSIONS

The following table shows the total number of explosions that occur in the 232 \leftarrow 3 machine to obtain the code of each of the first forty numbers.
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Any patterns?

QUESTION 7: RATIONAL DECIMAL EXPANSIONS

We can go to “decimals” in a 232 \leftarrow 3 machine.
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Here is what 12\dfrac{1}{2} looks like as a “decimal” in this base. (Work out the division 1÷21\div 2.)
I9S9A-image102

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

Here are some more “decimal” places.
I9S9A-image103

Can 12\dfrac{1}{2} have a repeating “decimal” representation in a 232 \leftarrow 3 machine?

What’s a “decimal” representation for 13\dfrac{1}{3} in this machine?

Develop a general theory about which fractions have repeating “decimal” representations in the 232 \leftarrow 3 machine. (I don’t personally have one!)

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