Exploding Dots

3.4 Negative Bases? Irrational Bases? Going Wild!



A \(-1 \leftarrow 4\) machine operates by converting any four dots in one box into an anti-dot one place to the left (and converts four antidots in one box to an actual dot one place to the left).


a)    This machine is a base machine:


Explain why \(x\) equals \(-4\).

b)    What is the representation of the number one hundred in this machine? What is the representation of the number negative one hundred in this machine?

c)     Verify that \(2|-3|-1|2\)  is a representation of some number in this machine. Which number? Write down another representation for this same number.

d)    Write the fraction \(\dfrac{1}{3}\) as a “decimal” in base \(-4\) by performing long division in a \(-1 \leftarrow 4\)  machine. Is your answer the only way to represent \(\frac{1}{3}\)  in this base?




BASE PHI  Consider the very strange machine \(1|0|0 \leftrightarrow 0|1|1\). Here two dots in consecutive boxes can be replaced with a single dot one place to the left of the pair and, conversely, any single dot can be replaced with a pair of consecutive dots to its right.


Since this machine can move both to the left and to the right, let’s give it its full range of “decimals” as well.



a)    Show that, in this machine, the number \(1\) can be represented as \(0.10101010101….\)  (It can also be represented just as \(1\) !!)


b)    Show that the number \(2\) can be represented as \(10.01\).


c)     Show that the number \(3\) can be represented as \(100.01\).


d)    Explain why each number can be represented in terms of \(0\)s and \(1\)s with no two ones consecutive. (TOUGH: Are such representations unique?)


Let’s now address the question: What base is this machine?


e)    Show that in this machine we need \(x^{n+2} = x^{n+1} + x^{n}\)  for all \(n\).


f)     Dividing through by \(x^{n}\) this tells us that \(x\) must be a number satisfying \(x^{2} = x + 1\). There are two numbers that work. What is the positive number that works?


g)    Represent the numbers \(4\) through \(20\) in this machine with no consecutive \(1\)s. Any patterns?





The Fibonacci numbers are given by:


1, 1, 2, 3, 5, 8, 13, 21, 34, ….


They have the property that each number is the sum of the previous two terms.


In 1939 Edouard Zeckendorf proved (and then published in 1972!) that every positive integer can be written as a sum of Fibonacci numbers with no two consecutive Fibonacci numbers appearing in the sum.

For example:




\(46 = 34 + 8 + 3 + 1\).


(Note: \(17\) also equals \(8 + 5 +3 + 1\) but this involves consecutive Fibonacci numbers.)


Moreover, Zeckendorf proved that the representations are unique:


Each positive integer can be written as a sum of non-consecutive Fibonacci numbers in precisely one way.


This result has the “feel” of a base machine at its base.


Is there a way to construct a base machine related to the Fibonacci numbers in some way and use it to establish Zeckendorf’s result?


Comment: Of course, one can prove Zeckendorf’s result without the aid of a base machine. (To prove that a number \(N\) has a Zeckendorf representation adopt a “greedy” approach: subtract the largest Fibonacci number smaller than \(N\) from it, and repeat. To prove uniqueness, set two supposed different representations of the same number equal to each other and cancel matching Fibonacci numbers. Use the relation \(F(n+2) = F(n+1) + F(n)\) to keep canceling.) It would be lovely, however, to see a visual proof of the result via the mechanics of a machine.





Invent other crazy machines …


Invent \(a|b|c \leftrightarrow d|e|f\)  machines for some wild numbers \(a,b,c,d,e,f\).


Invent a base half machine.


Invent a base negative two-thirds machine.


Invent a machine that has one rule for boxes in even positions and a different rule for boxes in odd positions.


Invent a base \(i\) machine or some other complex number machine.


How does long division work in your crazy machine?


What is the fraction \(\frac{1}{3}\) in your crazy machine?


TOUGH QUESTION … Do numbers have unique representations in your machines or multiple representations?


Go wild and see what crazy mathematics you can discover!


Comment: Dr. Jim Propp invites folk to consider a machine that converts any three dots in a box to one dot to its left and two dots to its right: \(0|3|0 \leftrightarrow 1|0|2\). He call’s this machine “unary-binary” because it gives correct representations of numbers in both base 1 and base 2. For example, the number three in this machine is represented as \(10.2\). In base one this is \(1 \times 1^1 + 0 \times 1^0 + 2 \times 1^{-1}\), which does equal three, and in base two it is \(1 \times 2^1 + 0 \times 2^0 + 2 \times 2^{-1}\), which is also three!

(The machine is unary because an explosion does not change the number of dots and so the sum of dots in any representation is the same as the initial number of dots, and the machine is binary because \(3x = x^2 + 1\), the fundamental relationship for the base number, holds for \(x = 2\).)


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