Dozenal RPN Calculator App for Android for Android Smartphones and Numworks Beta Firmware Version 25

 Dozenal Calculator App for Android Smartphones and Numworks Beta Firmware Version 25

Dozenal Calculator App for Android Smartphones

The Dozenal RPN Calculator app is by Unum Designum for the Android smartphones. It is a classical, four-level stack RPN (Reverse Polish Notation) app that operates in both dozenal (i.e. duodecimal, Base 12) and decimal (Base 10). The calculator app features a standard set of scientific functions:

* arithmetic, square, square root

* sin, cos, tan, and inverses

* e^x, 10^x, ln, log

* DMS/Decimal conversions and HMS/Hours conversions (not sure how there are two sets, since they seem to do the same thing)

* Pol/Rect conversions

* percent function (%)

* Stack functions: roll down, roll up, swap, last x

Originally Released: July 1, 2020

Last Updated: December 17, 2025 (as of January 22, 2026)

Version I am reviewing 1.0.11

The polar/rectangular conversions follow the conventional RPN calculator format:

	Rectangular	Polar
Y Stack:	Y	Θ
X Stack:	X	R

The percent function follows the conventional RPN calculator format:

	Before	After
Y Stack:	Y	Y
X Stack:	X	Y * X ÷ 100

However, the decimal/DMS (decimal-minute-seconds) and hours/HMS (hours-minute-seconds) follow up this format:

	Decimal/Hours	DMS/HMS
Z Stack:	0 after conversion	seconds
Y Stack:	0 after conversion	minutes
X Stack:	decimal/hours as a decimal	decimal/hours

Conversion between bases is just a matter of pressing [ f ] [ DOZ/DEC ].

Symbols:

* Upside down 2 (↊) represents 10 (Unicode 218A); commonly symbolized by X

* Backwards 3 (↋) represents 11 (Unicode 218B); commonly symbolized by E

(Unicode is from Wikipedia: https://en.wikipedia.org/wiki/Duodecimal)

Two constants provided by the Dozenal RPN app:

	Base 10 (e: 10^n)	Base 12 (e: 12^n)
π	3.141592653589793	3;184809493↋9186459↊↊
Planck	6.6206070149999999e-34	1;↊79611175↊0925342846e-27

App information: https://play.google.com/store/apps/details?id=dozecal.unumdesignum.com&hl=en_NZ#/

Numworks: Beta Firmware 25

Version 25 Information: https://www.numworks.com/calculator/update/version-25/

Nuwmorks has released a firmware update 25.1. It is a beta version software where new features are tested. To try it, Numworks is inviting Numworks users to download and become a beta test or us the beta emulator on the Numworks website.

Verison 25 Beta Emulator: https://www.numworks.com/calculator/update/version-25/

Please keep in mind that this emulator is probably only available for the testing period and may become unavaiable once the official release is made.

Major updates include:

* Data in the Statistics app can either be qualitative (data points, the way the statistics mode was always used) or categorial (data points always belong to specific categories, up to 10)

* The Grapher app can shade area of intersection for a given set of inequalities.

* The Grapher app also finds intersection points of conic sections and vertical lines.

* In the Calculations app, results with five or more decimal place will have at least a fractional approximation in the Additional Results quick tab.

* The degree and radian symbol/indicators are added to the Toolbox.

* Sequences have new notations.

* Installing this version will limit roll back deinstalls to versions 24.11 or later.

No word on any additions or changes to the Python app.

I have installed version 25 on my older Numworks calculator (N0110) so I can try them out.

HP Prime and TI-84 Plus CE: Base 10 (Decimal)-Base 12 (Duodecimal/Dozenal) Conversions

HP Prime and TI-84 Plus CE:  Base 10 (Decimal)-Base 12 (Duodecimal/Dozenal) Conversions

Conventions:

X = 10

E = 11

(notation by William A. Dwiggins)

HP Prime Program DEC2DUO

Enter the integer as a number.

EXPORT DEC2DUO(n)

BEGIN

// integer

// EWS 2017-01-30

// base 10 to base 12

LOCAL s,d,k,q,str1;

s:=IP(LN(n)/LN(12))+1;

d:=n;

str1:=":";

FOR k FROM 1 TO s DO

q:=IP(d/12^(s-k));

IF q<10 THEN

str1:=str1+STRING(q);

END;

IF q==10 THEN

str1:=str1+"X";

END;

IF q==11 THEN

str1:=str1+"E";

END;

d:=d-q*12^(s-k);

END;

RETURN MID(str1,2,s+1);

END;

Example:  Convert 12501 from Base 10 to Base 12

DEC2DUO(12501) returns “7299”

HP Prime DUO2DEC

Enter the integer as a string, use X for 10 and E for 11

EXPORT DUO2DEC(str1)

BEGIN

// as a string, X=10, E=11

// 2017-01-30 EWS

// base 12 to base 10

LOCAL s,n,k,str2,q;

s:=DIM(str1);

n:=0;

FOR k FROM 1 TO s DO

str2:=MID(str1,k,1);

q:=EXPR(str2);

IF str2=="X" THEN

q:=10;

END;

IF str2=="E" THEN

q:=11;

END;

n:=n+q*12^(s-k);

END;

RETURN n;

END;

Example:  Convert X2X3 from base 12 go base 10

DUO2DEC(“X2X3”) returns 17691

Note that the latest version (as of 1/30/2017) of the TI-84 Plus CE operating system is used.

TI-84 Plus CE Program DEC2DUO

Input "DEC: ",N

iPart(ln(N)/ln(12))+1→S

DelVar Str1

":"→Str1

N→D

For(K,1,S)

int(D/12^(S-K))→Q

If Q<10

Str1+eval(Q)→Str1

If Q=10

Str1+"X"→Str1

If Q=11

Str1+"E"→Str1

D-Q*12^(S-K)→D

End

sub(Str1,2,length(Str1)-1)→Str1

Disp ">DUO"

Pause Str1

Example:  5825 in base 10, result is 3455 in base 12

TI-84 Plus CE Program DUO2DEC

Disp "X=10, E=11"

Disp "DON'T INCLUDE QUOTES"

Input "DUO: ",Str1

length(Str1)→S

0→N

For(K,1,S)

sub(Str1,K,1)→Str2

expr(Str2)→Q

If Str2="X":10→Q

If Str2="E":11→Q

N+Q*12^(S-K)→N

End

Disp ">DEC"

Pause N

Example:  3EE1 in base 12, result is 6901 in base 10

This blog is property of Edward Shore, 2017

Base 12 Arithmetic: The Dozenal Society

Base 12 Arithmetic: The Dozenal Society

Introduction

Base 12 is a numerical base system that is based on the powers of 12 instead of 10.  The number line would like:

…

12^3 (1728)

12^2 (144)

12^2 (12) Dozens

12^1 (1)

;

12^-1 (1/12)

12^-2 (1/144)

12^-3 (1/1728)

…

For example, the number 49_12 in base 12 would read 4 dozen, 9 singles.  Its decimal equivalent is 4*12 + 9 = 57.

The number 24;53_12 in base 12 would be equivalent to 2 * 12 + 4 + 5/12 + 3/144 = 455/16 = 28.4375 in decimal.  Note that the semicolon is used as the fractional point.

Converting from base 10 to base 12 involves dividing the number by the powers of 12.  For example, 188 in decimal,

188/144 = 1 R11 

188 – 1 * 144 = 44

44/12 = 3 R8

The dozenal equivalent is 138_12  (1 * 144 + 3 * 12 + 8). 

Base 12 Advocates

There are a number of advocates for replacing Base 10 arithmetic with Base 12 arithmetic.  Two of the prominent groups are:

The Dozenal Society of America (Website: http://www.dozenal.org/   )

The Dozenal Society of Great Britain (Website:  http://www.dozenalsociety.org.uk/index.html )

I also found a number of videos on YouTube who extols the virtues of Base 12, including:

Numberphile:  https://www.youtube.com/watch?v=U6xJfP7-HCc

ParchitaFM:  https://www.youtube.com/watch?v=snz-81omkFQ

So why base 12 instead of base 10?  The Dozenal Society of America argues that the dozenal way of counting was arrived by societies worldwide:

*  Bakers sold baked goods in twelves

*  Rulers (non-metric) are usually a foot long divided into 12 sections (inches) – used by carpenters

* Historically, pharmacists and jewelers divided a pound into 12 ounces (the pound in this sense is called an Apothecary pound)

I will add that we have 12 months in a year, non-military clocks have work in 12 hours (with an AM/PM indicator), and in most astrological practices where Ophiuchus the Serpent Bearer isn’t included, there are 12 signs of the zodiac.

The factors of 10 are 1, 2, 5, and 10.  The factors of 12 are 1, 2, 3, 4, 6, and 12.  Base 12 advocates argue that the increase in number of factors would make more fractions easier to represent in dozenal than in decimal.  Examples include 0;2 for 1/6, 0;3 for 1/4, and 0;4 for 1/3. 

Base 12 would also facilitate counting, instead of using two hands and their fingers, people would count using the twelve phalanges of a single hand, where the thumb is used as a counter.   One hand can represent singles while the other represents dozens. 

Symbols

Thankfully, the symbols for digits 0 through 9 are retained. Unlike hexadecimal representation (Base 16), there is no uniform consensus of how ten and eleven are represented in Base 12.  Symbols include:

*  A for 10 and B for 11.  These symbols would align with the hexadecimal representation.

*  An upside down 2 for 10 and upside down 3 (Ɛ ) for 11.  In Microsoft Alt codes, the   Ɛ is associated hexcode 0190.  These symbols are popular in Britain.  No offense, but personally, these are not my favorite as look confusing and too close to the 2 and 3 we have. 

*  One suggestion is to use T for 10 and E for 11.  In this video Bon Crowder explains the digits of Base 12:   https://www.youtube.com/watch?v=BJRYCwl5Rgw

* Another set of symbols are called dek (χ ) for 10 and el ( Ɛ except the bottom line is flat) for 11.  These two symbols have been suggested by William A. Dwiggins, and is used in The Dozenal Society of America’s publications.  For typing on the computer, the capital letter X stands for dek and capital letter E for el. 

For the purposes of this blog and the tables presented, I will use X for 10 and E for 11. 

Let’s compare how the basic operations addition and multiplication work in both decimal and dozenal.

Adding:  Base 10 vs. Base 12

Below is an adding table (jpeg image) for numbers 1 through 12 represented in both Base 10 and Base 12.  

Multiplying:  Base 10 vs. Base 12

Below is a multiplication table (jpeg image) for numbers 1 through 12 represented in both Base 10 and Base 12. 

Fractions: Base 10 versus Base 12

Here are some common fractions listed below in both decimal and dozenal systems.  Note the interesting patterns for 0.1 to 0.9 (the table to the right).

Dozenal Representation of Pi

The numerical constant π to 30 places in base 12 is:

π = 3.18480 9493E 91866 4573X 6211E E1515 51X05…

You can find representations of π to 100 places here:  http://turner.faculty.swau.edu/mathematics/materialslibrary/pi/pibases.html

Conclusion

What do you think, would a dozenal, base 12 system work for you?  Should it replace base 10 in everyday mathematics?  Personally I find value in both systems and the ability to quickly double and halve is partially base 10 has survived as the dominant base system for centuries.

In the upcoming week, I work on a program to convert to and from base 10 to 12 integers.  I am thinking about either keep the X (dek) and E (el) or using A for 10 and B for 11.

Eddie

Sources

Schiffman, Jay.  “Fundamental Operations in the Duodecimal System”.  The Dozenal Society of America.  1982 (1192_12)  http://www.dozenal.org/drupal/sites_bck/default/files/db31315_0.pdf

Retrieved January 27, 2017

Zirkel, Gene. “A Brief Introduction to Dozenal Counting”.  The Dozenal Society of America.  1995 (11X3_12)  http://www.dozenal.org/drupal/sites_bck/default/files/db38206_0.pdf

Retrieved January 27, 2017

This blog is property of Edward Shore, 2017  (1201_12)