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 Calculating In Twelfty, A Guide to Alternating Arithmetic
Posted: Jan 14 2015, 10:24 AM

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Introduction Post.
Posted: Jan 14 2015, 10:55 AM

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The following is how alternating arithmetic might be presented in a school text. I'm following C Pendlebury and Beard's "Shilling Arithmetic", a text suited for upper primary and secondary school levels (say aimed at 10-15 year old children.

This text does not cover this method, but we can make something that looks like what's in the text, and adapt it to the new method. There are faster ways, but i leave this to other posts.

user posted image

1. Ones and Tens, the digits.

The numerals from one to ten are

1 one, 2 two, 3 three ; 4 four, 5 five, 6 six ; 7 seven, 8 eight, 9 nine ; ten.

Numbers greater than ten are grouped into multiples of ten, and a remainder, the multiples of ten are written as the numeral, followed by a 0 for no remainder, or the symbol representing the remainder.

10 ten, 20 twenty, 30 thirty; 40 forty, 50 fifty, 60 sixty; 70 seventy, 80 eighty, 90 ninety; V0 teenty, E0 elefty

There are twelve bundles of 10 before one carries to the next unit. V0 represents ten times ten, and E0 is eleven times ten.

Numbers from 11 to 19 have an irregular form, being thus

11 eleven, 12 twelve, 13 thirteen; 14 fourteen, 15 fifteen, 16 sixteen; 17 seventeen, 18 eighteen, 19 nineteen.

The remainder of the numbers are formed by a multiple of ten and a remainder, thus 72 seventytwo, being seven bundles of ten and two.

The combination of the units and tens digits represents a place or twistaff.

2. Hundreds and Thousands. The places

The number that follows eleftynine is a hundred. A hundred is a new thing of count, and one counts the hundreds as if they were units. One has a count of hundreds counted as above, and a count of units.

The number 86 23 represents a count of eightysix piles of a hundred each, and a remainder of 23, and is said eightysix hundred and twentythree.

A Thousand is a hundred of hundreds, written as 1 00 00

A cention is a hundred of thousands, written as 1 00 00 00

A million is a hundred of centions written as 1, 00 00 00 00.

A number is then said as the residue of millions, centions, thousands, hundreds, and units. Where there is no residue, the digit is written as 0 and not said.

9 85 10 16 nine cention, eightyfive thousand, ten hundred and sixteen.
7 60 00 00 seven cention, sixty thousand.

3. Zero

One writes a zero in every digit, where no remainder is given. The zero digit is usually never spoken, but if one is saying the places by their digit-pairs only, one might say an -oh- to represent an absent tens digit, and zero to represent an empty place, so

7 60 00 00 is seven sixty zero zero,
19 08 is nineteen oh-eight
Posted: Jan 14 2015, 10:57 AM

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4. Simple Addition

There are two digits in a place, one has a carry or overflow of tens, (or dickers), while the tens digits overflow as a count of dozens. For doing calculations, it is best to leave a space or point to the left of a tens column, such to mind the dozen-carry.

The numbers are added in decimal, but the carry is found by considering whether a count of dickers or dozens are needed.

So 6+6+8 gives 20 always, but these digits are in the tens, twenty tens is writen as dozens and units of tens, ie hundreds and tens, 20 is 1 dozen and 8, so it is written as 1 8.

The student is advised to learn the dicker-dozen table, which for every number from one to 120, gives eg 86 as 8 dicker 6 is 7 dozen 2.
Posted: Jan 14 2015, 11:07 AM

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6. Simple Multiplication.

Simple multiplication is to multiply a multiple digit number by a single digit. In this case, one multiplies the two digits in decimal, and adds any carry from the previous operation.

The difference here is in the commit, when one writes the result into the answer. It the number is placed into the high or tens column, it must be converted into a tulf-form by the use of the DD table. Thus one writes twenty as antle-eight, that is one dozen and eight, and writes eight into the result, and carries the unit.

7. Simple Division

8. Multiplication and Division by factors.
Posted: Jan 14 2015, 11:09 AM

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The methods of long arithmetic is to do a series of short arithmetic calculations, one for each digit of the multiplier or divisor. Because the high digit is the tenth multiple of the low digit, this ten is transferred to another part of the calculations, so we are doing short calculations at all times.

In essence, 73 U is calculated as 70 U and 3 U. But we simplify the high tables, by putting 70 U = 7 10U = 7 T. In these calculations, we create T from U, and then do short multiplication against T and U, as the digits be high or low.

The table for creating T from U is the dicker-dozen table. This converts a measure in dickers (eg 86) into the matching count in dozens (7.2). Note the dozens value has a point in it, and the sense is drawing the twistaves onto the points.

The methods can be used for any combination of upper and lower column, and does not depend on the divisors of the base.

9. Long Multiplication

The long multiplication is here a sum of short multiplications, moved across to match the count of points remain constant. The product here is 12.86 5.73. The first step of the calculation is to write U = 12.86, into its tenth multiple. The place 12 is now 1.0 (one dozen zero), and 86 gives 7.2 (that is, seven dozen two). This writes eg 12.|86 into 1.0|7.2|, and there is never carry across the bars here.

The calculation then follows so to produce a term being the short multiplication of U or T, by successive digits of X, from right to left. The calculation has the unit digit of the product falling under the digit in the multiplier X.


       1.07.20  T
         12.86  U
          5.73  X
         38.18   ie 3 U
       7.50.20   ie 7 T
      63.70.--   ie 5 U
      71.38.38   the sum

10. Long Division

The method of long division is to subtract short multiples from the quotent, creating the dividend. When division is to produce a digit in the high digit, the high or T divisor is used. The low divisor produces a term which is the multiplier of the U divisor.

In the example below, we begin by subtracting a multiple of T from the quotient, and transfer the '1' to the dividend. This should leave a remainder less than T. Successive subtractions in this way give rise to successive digits of the dividend. The second try is a '2' in the units column, so 2U is subtracted from the running count.


T 56.10 ----------    
U  5.73 ) 71.38.38
        -56.10       1 T
         11.26       2 U
         ----- vv
          4.02.38    bring down next pair
          3.88.80    8 T
            33.78    6 U

11. To Find a Square Root.

The method for finding a square root, is a kind of division, with no divisors, and a T and U dividend. The divisors change each line, being twice T followed by _._0, or twice U, followed by a single _.

The theory is that suppose we find the square-root of A, and we have already subtracted a square of U from it, then the residue is then greater than 2Ud+d, since this will convert A=U+2Ud + d + r,

In the case of the units column, or low digit, U is a multiple of 10, and we seek to find some d to leave the result less than 2U+d. This is why the divisor is 2U _.

In the case of the high digit T, each time a twistaff is created in U, it is multiplied by 10 to the high position. We are then have T as a multiple of 10.00, and the divisor is 2T + 1010d, the trial digit is d, is written to the high digit of the unit dividend. The running divisor is 2T _._0, where these spaces is occupied by 10d expressed in dozens.

The square root is the Unit dividend.


    T   2  V 1  4
    U    34   16
    R   9 85.10 16
  _._0  7 60          4 3.40 is too big
   6_   2.16          divisor is twice 3_
        ----          trial digit is 4 U_
           9.10,16    bring down 4 digits,
  5.8_._0             At this point, U 34 becomes T 2.V
           5.80.V0    divisor is twice 2.V_._0.
          --------    trial digit is 1 T
  68.2_    3.49.36    divisor is 2U of 34.1_
          --------    trial digit is 6 U
      If more digits needed the 16 U becomes 1.4 in T.
Posted: Jan 18 2015, 04:08 PM

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QUOTE (wendy.krieger @ Jan 14 2015, 10:57 AM)
For doing calculations, it is best to leave a space or point to the left of a tens column, such to mind the dozen-carry.

This presumes decimal is the favored base, and dozenal is the afterthought that should be marked with a guide. As dozenalists, we might prefer assuming dozenal base for most computations, and want to look at a switch to a decimal column as being a "downgrade" that should be marked.
Posted: Jul 7 2017, 12:23 PM

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Let's hear about twelfty here, Wendy. You've explained it here and there and it ought to be gathered here so we can read about it in one place. Treisaran had done some work on it.
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