A Symmetric Numeration SystemDonald D. Paige

University Junior Hi^h School, Bloomin^tony Ind.

There has been an increased emphasis on the teaching of numera-tion systems in the past decade. The justification for this area of thejunior high mathematics program is many fold. Interest is one factorwhich supports this topic and the fact that students better under-stand the base ten numeration system is another. In fact an entirearticle could be written in support of this topic for junior high mathe-matics. This writer assumes that the inclusion of numeration systemsin the junior high program is needed and will present a special"symmetric numeration system7^ which could be used for this area.

A SYMMETRIC NUMERATION SYSTEMFor convenience a base five numeration system will be used for

this article but it should be noted that any odd numbered base systemcan be chosen. Base eleven is of special interest because the placevalues follow a nice pattern, i.e. 1, 11, 121, 1331, 14641, etc.

In this symmetric base five numeration system we need five sym-bols to represent the various face-values in question. The symbols2, I, 0, 1, 2, will be used as the digits for this system. Again for con-venience these digits can be called negative two, negative one, zero,one, and two respectively. The face-values of the digits 0, 1, and 2 willbe_fche same as they are in any numeration system. The face-valueof I and 2 will be one and two except they will indicate the operationof subtraction instead of addition. The use of both addition and sub-traction in this numeration system makes it unique from numerationsystems currently being taught.The place-values in this symmetric base-five numeration system are

the same as the place-values in a regular base-five system. That is, thevalues going left from the decimal (or pentimal) point are 5°, 51, 52, 53,or 1, 5, 25, 125, etc. If one were to represent numbers between zeroand one the place-values would follow the same reciprocal pattern asa regular numeration system.The first use of a numeration system is that of counting. As in all

systems the zero is used to count the objects in the null set so webegin our counting like every numeration system. We also have adigit to represent a single object and a digit to represent a pair of ob-jects. They are 1 and 2 respectively. We now are faced with theproblem of representing (***) (three) objects. To do this we shall usethe numeral 12s which places a one in the second place-value columnand a negative two in the first place-value column. Since the place-

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402 School Science and Mathematics

value of the second column is five, in base ten, the one in the numeral125 represents 5 objects (base ten). The place-value of the first col-umn is one therefore the 2 in the numeral 12s represents 2 objects(base ten) but the operation is subtraction. The entire numeral 12sis then evaluated by taking a group of five and subtacting two. Theresult of 5�2 of course is three which is the number we were repre-senting by the numeral 125.

Counting is now an easy process. Four is represented by ^one fivesubtract one55 or 115 and so forth. Whole numbers through seventeenare listed in figure one with their base ten equivalents. The numeralsin figure one are arranged in the most convenient pattern as thenumbers 1-100 are generally arranged in textbooks.

Base 10

0=01=12=2

Base 10

3=124=115=106==117=12

Base 10

8=229=2110=2011=2112=22

Base 10

13=12214=12115=12016=12117=122

FIG. 1

Larger numbers can be placed in this notation with very fewproblems. For instance the numer 97io can be written by thinking 125,subtract 25, subtract 5, then add 2 which becomes the numeral1 1 1 25. Numbers already written in this system can easily bechanged to base ten numerals. The numeral 1221 means add 125,subtract 50, add 10, subtract 1 and is therefore the base ten numeral84.A symmetric numeration system can be used to form a number

system. For this we will need the mathematical operations of addi-tion and multiplication. The inverse operations of subtaction and di-vision will not be discussed except to note they can be handled byusing the addition and multiplication tables in reverse.

For the binary operation of addition a table should be formulatedwhich shows the result of adding any pair of the original five digits.This table is represented by figure two.

+[2101 2

2 11 12 2 I 0

1 12 2 1 0 1

0 21012

1 1 0 1 2 12

2 0 1 2 12 11

FIG. 2

A Symmetric Numeration System 403

A very close examination of the addition table is necessary to dis-cover all the patterns present in this system. The table, of courseshows the commutative law and the diagonal of zeros represents theinverse property. Another very interesting property is noted whenyou compare the addition of 2+2 and 2+2. The answers are 7 1 andll respectively, that is, the additive inverse is always found bymerely changing the prefix signs on all the digits. By the use of theaddition table in figure two a person should be able to work all addi-tion problems in this numeration system by using the same formatfor addition which is used in other systems, i.e. add the first column,then the second column, etc.

If one were to do many additions in this symmetric notation hewould be struck by one outstanding property. This would be the factthat most of the digits in a column addition would be paired withtheir additive inverse and therefore add to zero. This would eliminatemuch of the carrying which is found in the addition problems inregular notation systems.The binary operation of multiplication can be handled in about

the same manner as addition. Again a table should be formulatedwhich shows the results of multiplying any pair of the original fivedigits. This table is represented by figure three.

� I 2 I 0 l 2

2 11 2 0 2 ll

1 21012

0 00000

1 21012

2 11 2 0 2 11

FIG. 3

The multiplication table like the addition table contains many pat-terns. The cummutative law is again demonstrated as are the proper-ties of one and zero under multiplication. Both tables use the proper-ties of signed numbers which should be introduced during the juniorhigh years.By use of the multiplication table and the regular algorism for

multiplication students should be able to perform all problems. Aneasier form of multiplication could be used if one were to teach the^Napier^s bones^ system.One last property should be noted in a symmetric numeration

system. The problem of rounding in a base ten system is many timesmore difficult than it should be, but in a symmetric numerationsystem it is no problem. Since the base is odd we never are exactly

404 School Science and Mathematics

between numbers to which we should round. Also a quick check offigure one will show that the zero is always the center of each pat-tern of numbers. Therefore to round you simply replace any numbersby zero and you are to the nearest base involved.One could now show many computations in this numeration sys-

tem but that is not the purpose of this article. I hope its presentationhere is enough to create your interest and that you will consider itsuse in your mathematics class.

URANUS, NEPTUNE, PLUTO FORMED FROM SNOW STORMThe three most distant planets in the solar system�Uranus, Neptune, Pluto

�as well as hundreds of comets, all were formed from a giant snow storm 50 to60 times as wide as the distance from the earth to the sun.This is the theory of Dr. Fred L. Whipple, director of the Smithsonian Astro-

physical Observatory, Cambridge, Mass.As the huge hot cloud of gas surrounding the already-formed inner planets

condensed, it began to cool, freezing first one element and then another intosolids. Iron and similar elements formed dust, after which hydrogen, oxygenand other gases turned into frozen vapors or snow.The resulting snow storm has 200 times the mass of the earth and may have

measured more than 5.5 billion miles across.The snow and dust condensed into solid lumps of ^dirty ice," the three largest

becoming planets and the rest going on their way as comets.

IMITATION HUMAN BONE MADE FROM CLAY-LIKE MATERIALOut of a clay-like material made from aluminum, science has artificially

fashioned an imitation human bone. The new material promises to serve the liv-ing body almost better than the natural skeleton.

Produced by the cooperative research of a prominent orthopedic surgeon, thenation’s largest pottery giftware manufacturer, and a major pharmaceuticalfirm, this ceramic may also be used to replace eyeballs and develop non-spark-ing, non-reflective surgical tools.

Initial experiments on rabbit knee bone transplants at Baxter Laboratories,Inc., Morion Grove, 111., have shown that animal tissue will stick to the materialas natural bone does.However, three to four years of further evaluation is necessary before its

safety for human use can be determined.The material, known as Cerosium, is made from a porous ceramic composed

mainly of an oxide of aluminum that is impregnated with epoxy resin, an inertplastic compound.

This combination provides a tough, nonreactive, flexible material, very similarto bone.Haeger Potteries produced the ceramic material at the suggestion of Dr.

Lyman Smith, Elgin, III. surgeon and assistant professor at Northwestern Uni-versity.