The Paraná Numerical System
Raymond Cheng
*
Introduction.
The Paraná Numerical System is a graphical number system which expresses
3
-adic place values via colour and shape, and digit values via relative placement of symbols. The
System admits a direct translation to and from the balanced ternary numbers. Currently, the
System supports the
9
place values
3
i
for
0 ≤ i ≤ 8
and is therefore capable of expressing positive
integers between
1
and
9841 = 3
0
+ 3
1
+ · · · + 3
8
; the place value
3
i
is represented by
i + 1
vertical
monochromatic bands, the colours with warmth increasing with i.
Balanced ternary numbers.
Every integer
n
admits a unique expression as a finite sum and
difference of powers of 3: there exists a unique expression
n =
X
∞
i=0
a
i
3
i
in which
a
i
∈ {−1, 0, +1}
and only finitely many
a
i
are nonzero. Indeed, take the usual
3
-adic
expansion of
n
and iteratively replace every occurrence of the digit
2
by
2 = 3 − 1
; since the lowest
power of
3
with a digit
2
increases upon each replacement, this process terminates after a finite
number of steps. The balanced ternary number system is this representation of the integer
n
by the
sequence of digits (a
0
, a
1
, a
2
, . . .).
Construction of a Paraná number.
The Paraná representation of an integer
1 ≤ n ≤ 9841
is
constructed as follows: First, compute its balanced ternary expansion
n = a
0
· 3
0
+ a
1
· 3
1
+ · · · + a
8
· 3
8
where a
i
∈ {−1, 0, +1} as above. Second, let
I
+
·
·
= { i | a
i
= +1 } and I
−
·
·
= { i | a
i
= −1 }
be the indices for which
a
i
is either
+1
or
−1
. Third, let
L
be the ordered list obtained by taking
I
+
in increasing order followed by
I
−
in decreasing order. Finally, the Paraná representation of
n
is
obtained by vertically concatenating, top-down, the graphical place values in the order given by
L
.
Reading a Paraná number.
To convert a Paraná number into an integer
n
, let
L
be the ordered
list of place values obtained by reading the Paraná number top-down. Then
L
is unimodal with a
unique apex. Let
L = I
+
t I
−
be the unique decomposition in which
I
+
is the set of place values up
to the apex, and I
−
is the set of place values after the apex. Then
n =
X
i∈I
+
3
i
−
X
i∈I
−
3
i
.
*
