Dozenal: Base Twelve number system

First, let me point out that for many years there have been academics groups proposing, with many good arguments, that we should change our counting from base ten to base twelve. Twelve is mathematically a much better base than Ten is because Twelve is divisible by: 2, 3, 4, and 6; while Ten is only divisible by: 2 and 5. The "Dozenal Society" is a group advocating changing to base Twelve. To find more information just Google on the word Dozenal. Much of the best writing on the subject seems to be by a British group but unfortunately, in my opinion, too much of it is simply "anti-metric". All the world, except for the United States and a few British die-hards, realize it is dumb to not have your weights and measures consistent with your number system.

For Dozenal: We need to add two more digits to our counting system. These new digits should not be confused with letters of the alphabet, and they should be clearly representable in the familiar 7 segment displays we see on many electronic devices. Furthermore, they should be easily written with a single stroke. I would like to propose: a character that would look like a backwards capitol "L" for the first which I would call "col" and it would have the value of "nine plus one". The second which I would call "sem" whose value would be "nine plus two" would be the same with one additional up bar on the left, making it look somewhat like the capitol letter "J". The rational for these names and shapes is explained below.

As I set typing today, I am using a computer that uses the ASCII character set. Which leaves much to be desired, but what other choice do I have? It is also unfortunate that I am spelling words by a convention that is not phonetic and is FAR from optional; but what other choice do I have? If I want you to be able to read it? I decided to experiment with different ASCII options for the two new digits:

Dozenal Extra Digits Experiment Tue 2004-03-16 Examples of Dozenal Square Root of 2 to 25 (Dozenal decimal) places. Using for the additional two digits: (ASCII characters: colon, and semi-colon which follow 9), A & B (both upper and lower case) and E & T (both cases). Which do you find easier to read? 1.4;79170:07;85737704;085486853 1.4b79170a07b85737704b085486853 1.4B79170A07B85737704B085486853 1.4e79170t07e85737704e085486853 1.4E79170T07E85737704E085486853

If you print these samples. Then cover the lines up so you only see them one at a time. I think you will inadvertently find yourself using the first line to find the "new" digits. The first line is more distinct and easier to read. Agree???

Based on this and other experiments I decided to call the smaller of the new digits "col" and the larger "sem". After some thought, to avoid confusion I would call "10" in the base twelve syztem "zen", which has a value of "nine plus three". Of course!. (If you find it hard to remember which is larger, think: it takes more ink to make a semicolon, so it is the bigger.) Also, we need to be more systematic in how we "say" numbers of more than one digit.

For more practice here is a table of small prime numbers in both decimal and Dozenal:

First 100 (Dozenal) primes in decimal. Wed 03-24-2004 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829

First 100 (Dozenal) primes in "Col, Sem" Dozenal 3 5 7 ; 11 15 17 1; 25 27 31 35 37 3; 45 4; 51 57 5; 61 67 6; 75 81 85 87 8; 91 95 :7 :; ;5 ;7 105 107 111 117 11; 125 12; 131 13; 141 145 147 157 167 16; 171 175 17; 181 18; 195 19; 1:5 1:7 1;1 1;5 1;7 205 217 21; 221 225 237 241 24; 251 255 25; 267 271 277 27; 285 291 295 2:1 2:; 2;1 2;; 301 307 30; 315 321 325 327 32; 33; 347 34; 357 35; 365 375 377 391 397 3:5 3:; 3;5 3;7 401 40; 415 41; 421 427 431 435 437 447 455 457 45; 465 46; 471 481 485 48; 497 4:5 4;1 4;; 507 511 517 51; 527 531 535 541 545 557 565 575 577 585 587 58; 591

Time and Angles

Unfortunately when these units for time and angles were devised science had not advanced to where it was generally known that the earth rotated; hence, the connection between time and angles was not realized. Presently: The rotation of the earth is divided into 24 parts for time, and 360 degrees for angular measure. This is truly an unfortunate mistake of the past, and regrettably even though we think we are smart and progressive: I have not heard any arguments that we should change our units of: Time and Angles. But as any programer knows doing arithmetic on time or angles is a crazy mess. Especially if AM and PM are used! So, I will present a much better system.

If we were using Dozenal, I propose angles be measured in parts of a complete circle. Thus an angle of one complete revolution would be 1.0 rev. Instead of our crazy 360 degrees. "Rev" replacing the word "degree" for angular measures. A straight line would be an angle of 0.6 revs remember in Dozenal 1/2 is 0.6. And a right angle would be half a straight line or 0.3 revs. What we now call a 30-60-90 triangle would be a 0.1 - 0.2 - 0.3 triangle. Elegantly simple! I have used leading zeros to emphasize numbers just as we currently do, but they would not always be needed.

Time should be simply a measure of the angular rotation of the earth. Thus a day would be: 1.0 and a half day would be 0.6 and what we now call 2 hours would be 0.1 of a day. Midnight would be time 0.0 and noon would be 0.6 and half way between midnight and noon would be 0.3 Oh, yes what we now call an hour would be 0.06 and smaller time intervals would be just more digits to the right. Different but elegantly consistent.

The Longitude of where you are at would tell you your Time Zone! Think of how this would simplify the concept of "Greenwich Mean Time" or UTM.

I have in My Arithmetic Routines a calculator program to do variable precision integer arithmetic. With it you can change the input and output number bases. The above Square Roots were done with it. I also have a "tweaked" version where the defaults are Dozenal. I also converted a Prime number output routine to output in "Dozenal". If you are interested.

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