HP Prime: Base Conversions – Joe Horn

HP Prime:  Base Conversions – Joe Horn

Special thanks to Joe Horn.  When I posted the conversion programs ( http://edspi31415.blogspot.com/2017/02/hp-prime-and-ti-84-plus-ce-convert.html ), Horn alerted me to a hidden convert command where we can convert base integers.

The HP Prime has a CONVERT (upper case letters) that converts units (i.e. length, volume, etc.).  However, for the base to work, not only convert has to be typed in lower case letters, AND it is a CAS command.

The syntax that Joe told me is:

Decimal to Other Base (D2B)

convert(number in base 10, base, desired base)

The word base is typed in lowercase.  The result is a vector of digits but the order goes from lower power to upper power.  The number will be written backwards.  To get it in proper order:

REVERSE(convert(number in base 10, base, desired base))

In Home mode or in programming, you will need a CAS command attached to the syntax.

Other Base to Decimal

convert(vector of digits, base, base that is represented)

Again, going this way the vector of digits goes from lower to upper power.  The proper order requires the following:

convert(REVERSE(vector of digits), base, base that is represented)

Once again, in Home mode or in programming, you will need a CAS command attached to the syntax.

Below are programs that help accomplish these tasks:

HP Prime Program D2B (decimal to base)

EXPORT D2B(n,b)

BEGIN

// decimal to base

// Joe Horn

CAS("REVERSE(convert(n,base,b))");

END;

Example:  D2B(17291, 12) returns [10 0 0 11]   (A00B_12)

HP Prime Program B2D (base to decimal)

EXPORT B2D(n,b)

BEGIN

// vector (decr),base to base 10

// Joe Horn

CAS("convert(REVERSE(n),base,b)");

END;

Example:  B2D([1, 0, 1, 6, 2, 5], 7) returns 17463.  (101625_7)

Thank you Joe!  As Joe Horn always says, Happy Programming!

Eddie

This blog is property of Edward Shore, 2017.

Retro Review: Texas Instruments BA II Financial Calculator (1984)

Retro Review:  Texas Instruments BA II Financial Calculator (1984)

BA II:  Finance Mode

BA II:  Statistics Mode

Company:  Texas Instruments

Mode:  BA II

Year: 1984

Batteries:  2 A76 batteries, or 2 357 batteries

Memory Registers: 1  (only 1, other variables can’t be recalled)

Where’s the Plus?

This review isn’t about the BA II Plus, this is an earlier model.  I purchased the BA II from a box in the Azusa Swap Meet last Sunday (January 28, 2017) for 1 dollar.  Thankfully the previous owner did not leave the old batteries so I didn’t have to worry about battery corrosion.  The previous owner (or one of them) has their name and address on the calculator.  Obviously for privacy issues I took of a couple of stickers containing the address.

The BA II is the second generation of the TI Business Analyst II (released in 1978).  The first generation of the Business Analysist II sold for $40-$45 new.  The Datamath page for the TI Business Analyst II:  http://www.datamath.org/Sci/Slimline/Business-Analyst-II.htm

Three Modes

The BA II has three modes which cycle by the key sequence [ 2^nd ] [ FIX ] (Mode):

‘Profit Mode (no indicator):  Profit Margin mode.  Only the [CST] (cost), [SEL] (selling price), and [MAR] (margin) keys are active.  Solve for the following variables by preceding the key by [ 2^nd ].

‘Finance Mode (FIN indicator):  Time Value of Money mode.  Only the [ N ], [ %i ], [ PMT ], [ PV ], and [ FV ] keys are active.  Solve for the following variables by preceding the key by [ 2^nd ] or if you are working with beginning of period annuity by using [DUE] instead.  Also, the P/I (Principle and Interest of a single payment) and Acc/Bal (accumulated interest and balance) are also available.

‘Statistics Mode (STAT indicator): Statistics Mode, including linear regression.  The equation is y = a + b*x where a is the intercept and b is the slope.  Mean, deviation, and predictions are also available.

Other Functions

Percent Change:  Δ% has the syntax (new number [ 2^nd ] [ % ] (Δ%) old number [ = ])

Fix Modes:  FIX 0-7 will fix the number to that decimal places.  Using FIX 8-9 engage floating mode.

Early Installment Weirdness

1. There is no [CPT] button like it would be prevalent on later Texas Instruments financial calculators. 

2. The sign of cash flow conventions are not followed.  See the example below:

Example:  Calculate the monthly payment of a $174,000 loan.  The loan is for 30 years, 4.65% annual interest, and no balloon payment (FV = 0).  Assume that FIX 2 is in affect and Finance mode is selected.

Remember N is the number of payments and %i is the periodic interest rate (in this case monthly).

Keystrokes:

174000 [PV]

30 [ * ] 12 [ = ] [ N ]

4.65 [ ÷ ] 12 [ = ] [ %i ]

[ 2^nd ] [ PMT ]

The result of 897.21.  (There is no negative sign for outflow like in more current financial calculators, including the HP 12C)

Principal/Interest at the 60^th payment:

Keystrokes:

60 [ 2^nd ] [ x<>y ] (P/I)

Returns:  280.11 (principal) [ x<>y ] 617.10 (interest)

Accumulated Interest and Balance after 60 payments:

Keystrokes:

60 [ 2^nd ] [ K ] (Acc/Bal)

Returns: 38804.61 (accumulated interest) [ x<>y ] 158,972.13 (balance)

My recommendation is to write down your answers and key in the time.

Final Verdict

The BA II is a good entry-level financial calculator.  Being it is a 1984, the patience will be required for some calculations. 

Eddie

This is Edward Shore, 2017

The Golden Ratio: Reciprocals and Powers

The Golden Ratio: Reciprocals and Powers

Determining the Golden Ratio

Take a line segment with length of x + 1, and split it into two segments, one with width x, the other width 1. (see the black lines on the left)

The golden ratio is defined when given positive numbers a and b, where a > b, the following ratio is true:

(a + b) / a = a / b

Applying this to the above diagram:

(x + 1) / x = x / 1

Solving for x yields:

(x + 1) * 1 = x * x

x + 1 = x^2

x^2 – x – 1 = 0

By the quadratic equation:

x = (1 ± √(1^2 – 4 * 1 * -1)) / 2

x = ( 1 ± √5 ) / 2   (I)

Since x is a measure, we will consider only the positive root and define the golden ratio (ϕ) as:

ϕ = (1 + √5) / 2    (II)

ϕ ≈ 1.6180339

Reciprocals of ϕ

1 / ϕ

1 / ϕ  

= 2 / (1 + √5)

= 2 / (1 + √5) * (1 - √5) / (1 - √5)

= 2 * (1 - √5) / -4

= (2 – 2 * √5) / -4

= (2 * √5 – 2) / 4

= (√5 – 1) / 2

Here is where a trick comes in, we’re going to add and subtract 1:

= (√5 – 1 + 1 – 1) / 2

Now, note that:

= (√5 + 1 – 2) / 2

= (√5 + 1) / 2 – 2 / 2

= (√5 + 1) / 2  - 1

= ϕ – 1

Hence:  1 / ϕ = ϕ – 1

1 / ϕ^2

1 / ϕ^2

= (2 / (1 + √5) )^2

= 4 / (6 + 2 * √5)

= 4 / (6 + 2 * √5) * (6 – 2 * √5) / (6 – 2 * √5)

= (24 – 8 * √5) / 16

= (3 - √5 ) / 2

Note that 4 – 1 = 3…

= (4 – 1 - √5) / 2

= 4 / 2 – (1 + √ 5) / 2

= 2 - ϕ

Hence:  1 / ϕ^2 = 2 - ϕ

Powers of ϕ

ϕ^2

ϕ^2

= ( (1 + √5) / 2 )^2

= 1 / 4 * (1 + 2 * √5 + 5)

= 1 / 4 * (6 + 2 * √5)

= (3 + √5) / 2

= (2 + 1 + √5) / 2

= 1 + (1 + √5) / 2

= 1 + ϕ

Hence:  ϕ^2 = 1 + ϕ

Take note, this plays in determining further powers of ϕ.

ϕ^3

ϕ^3

= ϕ * ϕ^2

Using the fact that ϕ^2 = 1 + ϕ

= ϕ * (1 + ϕ)

= ϕ + ϕ^2

= ϕ + 1 + ϕ

= 1 + 2 * ϕ

Hence:  ϕ^3 = 1 + 2 * ϕ

ϕ^4

ϕ^4

= ϕ * ϕ^3

= ϕ * (1 + 2 * ϕ)

= ϕ + 2 * ϕ^2

= ϕ + 2 * (1 + ϕ)

= ϕ + 2 + 2 * ϕ

= 2 + 3 * ϕ

With ϕ^4 = 2 + 3 * ϕ.   Moving on…

ϕ^5

ϕ^5

= ϕ * ϕ^4

= ϕ * (2 + 3 * ϕ)

= 2 * ϕ + 3 * ϕ^2

= 2 * ϕ + 3 * (1 + ϕ)

= 2 * ϕ + 3 + 3 * ϕ

= 3 + 5 * ϕ

Hence:  ϕ^5 = 3 + 5 * ϕ

ϕ^6

Quickly…

ϕ^6

= ϕ * ϕ^5

= ϕ * (3 + 5 * ϕ)

= 5 + 8 * ϕ

A Connection to the Fibonacci Sequence

Sense a pattern?

ϕ^2 = 1 + ϕ

ϕ^3 = 1 + 2 * ϕ

ϕ^4 = 2 + 3 * ϕ

ϕ^5 = 3 + 5 * ϕ

ϕ^6 = 5 + 8 * ϕ

Note the coefficients come from the Fibonacci Sequence.  (1,1,2,3,5,8,13…  )  The Fibonacci sequence is defined as:

F_n = F_n-1 + F_n-2   where the first two initial terms are 1 and 1.

Can we show that ϕ^n = F_n+1 + F_n * ϕ?

Proof

Let assume that the base case of:

ϕ^n = F_n-1 + F_n * ϕ

Will this relation hold up for ϕ^(n+1)?  Then:

ϕ^(n + 1)

= ϕ * ϕ^n

= ϕ * (F_n-1 + F_n * ϕ)

= ϕ * F_n-1 + F_n * ϕ^2

= F_n-1 * ϕ + F_n (1 + ϕ)

= F_n-1 * ϕ + F_n + F_n * ϕ

= F_n + (F_n-1 + F_n) * ϕ

Using the definition of the sequence, F_n-1 + F_n = F_n+1.  Hence:

= F_n + F_n+1 * ϕ

Since ϕ^(n + 1) = F_n + F_n+1 * ϕ, the relation holds.

Next time I have a retro review of an early financial calculator from the 1980s, the TI BA-II (without any plus or other annotation).  I have couple more retro calculator reviews coming (tentatively) later this month:  Casio CM-100 and Canon FS-5. 

That’s all for now!  Take care,

Eddie

This blog is property of Edward Shore, 2017.

eCalc Scientific Calculator (Guest Post: Christine of Web Technology USA)

Today’s post is a guest post from Christine of Web Technology USA ( http://www.webtechnologyusa.com/ ), and she is going to talk about their online scientific calculator which can be downloaded, the eCalc.  I played with the eCalc and it is really nice and well organized. 

Christine, thank you for emailing me and allowing me to share.   Much appreciated.

Take it away, Christine.

Eddie

--------------------------------------------------------------------------------

eCalc Scientific Calculator

(by Christine of Web Technology USA)

Screen shot of eCalc, provided by Christine - Thanks

On the 21st century where nearly everything can be processed online, each device may now have its web-based counterpart. And calculators are no exemption. From the Abacus of 2000 BC to the Arithmometer of 18th century, the evolution of this device has been in sync with the times. Coming into the scene is eCalc, a readily available software that is offered as both online calculator and as a downloadable application.

eCalc is a free and easy to use scientific calculator that supports many advanced features including unit conversion, equation solving, and even complex-number math. This application operates in either algebraic input or Reverse Polish Notation (RPN). The algebraic input is the default mode which is widely used by most handheld calculators. RPN, on the other hand, is a stack-based notation where operators are designed to follow their operands.

Data entry on this calculator is quite straightforward. This is achieved by the use of a graphical button or through traditional keyboard keys. In fact, the user is given the option to use either input method which are both equally valid. However, due to the limited amount of space available for graphical button placements, some trigonometric functions are limited to computer keyboard entry only. Aside from certain basic functions, all hyperbolic trigonometric functions need to be entered by typing their keywords accordingly.

eCalc is divided into two sections: the scientific calculator interface on the left and the calculator pallet on the right. The pallet houses a display area for special features such as Unit Converter, Constants Library, Equation Solver, Polynomial Solver, Base Conversion, and Decimal to Fraction Conversion.

Another notable feature of eCalc is “stack”. This particular attribute is inherent providing the user a way to view a history of results. Although the stack displays only 4 items at once, one can scroll up and down the stack by clicking the up and down arrow above it. A user even has the option to “pop” down a value currently existing in the stack into the input field of the calculator by clicking on the down arrows to the left of the line in the stack.

Simple yet robust, eCalc can solve other forms of mathematical equations be it basic or complex placing it a notch above battery-powered handheld calculators. And due to its nature being a standalone software, it is capable of being updated with future possible functionalities.

 eCalc Scientific Calculator is available for Windows Desktop and Mac OS X Dashboard.

--------------------------------------------------------------------------------

Link:  https://www.eeweb.com/toolbox/calculator

This blog is property of Edward Shore, 2017.

HP Prime and TI-84 Plus CE: Convert Integers: Decimal to Any Base

HP Prime and TI-84 Plus CE:  Convert Integers: Decimal to Any Base

Below are two programs that allow conversion of integers from decimal to any designated base.  Integers used in base other than 10 are presented in list format.  Each digit is separate and listed in order of descending place value.

For example: To input 1282_9 (base 9), enter {1, 2, 8, 2} and designate 9 as the base.

Digits are shown in numerical form and not in alphabetic.  Hence, if you were working for hexadecimal, enter 10 for A, 11 for B, and so on. 

HP Prime Program DEC2BASE

EXPORT DEC2BASE(N,B)

BEGIN

// decimal, base

// 2017-01-31 EWS

LOCAL S,L1,Q,D,X;

S:=IP(LN(N)/LN(B))+1;

L1:=MAKELIST(0,X,1,S);

D:=N;

FOR K FROM 1 TO S DO

Q:=IP(D/B^(S-K));

L1(K):=Q;

D:=D-Q*B^(S-K);

END;

RETURN L1;

END;

See the TI-84 Plus CE section for examples.

HP Prime Program BASE2DEC

EXPORT BASE2DEC(L1,B)

BEGIN

// list of digits, base

// 2017-01-31 EWS

LOCAL S,N,K,Q;

S:=SIZE(L1);

N:=0;

FOR K FROM 1 TO S DO

N:=N+L1(K)*B^(S-K);

END;

RETURN N;

END;

See the TI-84 Plus CE section for examples.

TI-84 Plus CE Program DEC2BASE

(This should be work with the classic TI-84 Plus)

Input "DEC :",N

Input "TO BASE: ",B

int(ln(N)/ln(B))+1→S

S→dim(L₁)

N→D

For(K,1,S)

int(D/B^(S-K))→Q

Q→ L1 (K)

D-Q*B^(S-K)→D

End

Disp ">BASE "+eval(B)

Pause L1

** For classic TI-84 or any operating system not working with OS 5.2 or later, you can leave out the “+eval(B)” for the second to the last line.

Examples:

Convert 1254 to base 6.

Result:  {5, 4, 5, 0}   (5450_6).  

(5 * 6^3 + 4 * 6^2 + 5*6)

Convert 17291 to base 13.

Result:  {7, 11, 4, 1}   (7B41_13) 

(7 * 13^3 + 11 * 13^2 + 4 * 13 + 1)

(Here I use A = 10, B = 11, C = 12, and so on, like calculator notation)

TI-84 Plus CE Program BASE2DEC

(This should be work with the classic TI-84 Plus)

List:  {q_n, q_n-1, … , q_1, q_0} where

q_n * b^(n)

q_n-1 * b^(n-1)

…

q_1 * b

q_0  (constant)

Treat this like if you were entering a polynomial of coefficients of descending order from power n to 0.

Disp "LIST OF DIGITS"

Input L1

Input "BASE: ",B

dim(L1)→S

0→N

For(K,1,S)

N+ L1 (K)*B^(S-K)→N

End

Disp ">DEC"

Pause N

Examples:

Convert 4281_9 to decimal (4 * 8^3 + 2 * 9^2 + 8 * 9 + 1)

Input:  {4, 8, 2, 1}, Base: 9

Result: 3583

Convert C107_16 to decimal (12 * 16^3 + 1 * 16^2 + 7)

Input:  {12, 1, 0, 7}, Base: 16

Result:  49415

I hope you find this helpful. 

Coming up, I am reading about the golden ratio, I will have another retro calculator review (Texas Instruments BA II from 1984) coming shortly, and I may have a guest post as well.  (10 posts away from post #700!)

Thank you everyone.  Welcome to February!

Eddie

This blog is property of Edward Shore, 2017

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)