Some Remarks on Teaching Different BasesCharles Hudson

J. S. dark High Schoolf Opelousas, Louisiana

There are several well-known reasons for teaching notation systemsin different bases. Perhaps one of the most important reasons forteaching different bases is to help pupils acquire a better understand-ing of fundamental properties of base ten. However, the usual ap-proach to teaching different bases as I have seen it presented inrecently published textbooks and current mathematical literaturedoes not give pupils a clear understanding of what they are doing.Thus, pupils see little or no relationship between notation systemsin different bases. Instead they simply memorize how to performoperations in different bases without understanding the reasons forwhat they do. It is a well established fact that teaching students toperform operations based on memory (without understanding)serves no useful purpose. In fact, Dr. Fawcett says:

No student will be guided toward an understanding of mathematical methodthrough teaching procedures which feast his memory and starve his reason.1

Thus, it will be the purpose of this article to present an effectivemethod of teaching different bases.To begin the presentation, suppose that we consider the following

set of tally marks: (arbitrarily selected)

JSL 7

FIG. i

At this point it is explained to the pupils that different basessimply mean that we count in different groups such that each groupcontains the same number of objects. Thus, using base ten first withwhich pupils are already familiar (but do not understand how itworks), the tally marks (hereafter called marks) are counted bygroups of tens. In Figure 1 there is one group of ten marks and

1 Dr. Harold P. Fawcett, "Guidelines in Mathematics Education," The Mathematics Teacher, LIII (October1960).

649

650 School Science and Mathematics

seven marks left over. The pupils can very easily verify the factthat there are seventeen marks by counting them. This means thatthe results can be written in a more concise manner as 17. Pupils cando this in base ten because they have already learned somethingabout place value but do not understand how or why it works. (Thatis one of the main reasons for teaching different bases.) Pupils havelearned something about positional notation as follows:

� � � ten thousands thousands hundreds tens ones104 103 102 10 10°

Thus, referring to Figure 1 again the pupils immediately recognizethat we have one ten and seven ones, giving 17. Next we refer toFigure 1 and count in groups of eights. The question arises now as tohow may we write the results in a concise manner? Well by using theanalogous notation for base ten we may write a similar notation forbase eight as follows:

� � � sixty-four eights ones. . . 82 8 8°

Therefore, looking at Figure 1 we have two groups of eight objectsand one left over. In a concise notation we write 21cight (meaning twogroups of eight objects and one left over). Obviously it follows thatwe have the same number of marks but the notation is different.From Figure 1 the student can immediately conclude that 21eight== 17terrAfter doing several examples as illustrated in Figure 1 we go a

step further. Suppose a number in base ten is given and we wish tochange it to base eight (without using marks). Actually what wewish to do here is to find out how many groups of eights are containedin the number given in base ten. This problem is approached by firstdoing ordinary division or reducing common fractions, which Ishall not illustrate here.

Consider the following problem: change 26ten to base eight. Thisproblem is done as follows: First the pupils write down the positionalnotation for base eight as follows: . . . eight3 eight2 eight eight0.Secondly, the pupils ask themselves these questions: Does 26 containeight0? Yes. Does 26 contain eight? Yes. Does 26 contain eight2? No.Therefore eight is the largest divisor. After pupils have answeredthese questions they proceed to solve the problem as illustrated inexample 1.

Teaching Different Bases 651

Eight0 ==1

8

1

2624

22

3

2

EXAMPLE 1

Thus we started with the largest possible divisor (eight) and con-tinued "on down the line" to eight0. Therefore 26ten=32eight. Theresults can be checked by simply converting 32cight to base ten asfollows:

32 means 3(eight) + 2(eight°)=^32eight ^ 3(8) + 2(8°)==24+2- 26t.n.

For a second example consider the problem 786 containing threedigits to be changed to base eight. First the pupils write . . . eight3eight2 eight eight0. Second the pupils determine the largest possibledivisor (by asking same questions asked in example 1). Then thepupils proceed to solve the problem as illustrated in example 2.

Eight3 =

Eight2 =

Eight1 ==

Eight0 =

512

64

8

1

786512

274256

1816

22

1

4

2

2

EXAMPLE 2Thus 786ton = 1422eight

The method which has been illustrated in examples one and twoeliminates the process of "repeated division" and "keeping the re-mainders" as I have seen it done in a very large sampling of recentlypublished textbooks and current literature on the teaching of mathe-matics. The fault that I have found in the ’repeated division^ processis that pupils on the junior high school level as well as other levelsfind it rather difficult to UNDERSTAND precisely what they aredoing.Beyond this point of understanding, we begin to use the polynomial

form for changing any base to another base. However, this method

652 School Science and Mathematics

requires that pupils know how to add and multiply in the base underdiscussion (which has already been taught before we reach thispoint). The polynomial method eliminates the division process asillustrated in examples 1 and 2. For example, consider the problemin example 2. (change 786ten to base eight)

a. 786ten = 7(102) + 8(10) + 6(10°)= [7(122)+10(12)+6(12°)]e;ght (convert notation in base ten to

notation in base eight)== [7(144) + 10(12) + 6(l)]c;ght (perform operations in base eight)= [1274 + 120 + 6]ei.ht= 1422eight

b. Change 1422eight to base ten1422oi.ht = 1 (eighty + 4(eight2) + 2(eight) + 2(eight°)

= 1(512)+4(64)+2(8)+2(1)= 512 + 256 + 16+2= 786ten

Finally, pupils are able to discover that in a place value system ofnumeration a numeral like 6,305, is simply an abbreviation for apolynomial in base 6, like d^+S^+OS+SS0. Once pupils have dis-covered this fact, then they can change any number in any basedirectly to the corresponding number in another base. That is, forexample, they can change a number in base five to base twelve with-out any difficulty. The only small matter involved in going to a basebeyond base ten is that we simply must "invent" new symbols.

Thus, once pupils reach this point of understanding, they can per-form all of the operations we do in base ten, and above all have abetter understanding of how base ten works and what makes baseten work as it does. Furthermore, pupils discover that any numberA7, base bj may be written symbolically as follows:

Nb = Z ^.1=0

which implies that

Nb == Z ^- == ^kn + ^-^n-1 + ^-^-2 + ’ ’ ’ + bk + koi=0

COMPOUND OF XENONThe January 1963 issue of SCHOOL SCIENCE AND MATHEMATICS earned a

"filler" item concerning the discovery of a stable compound of Xenon by scien-tists at Argonne Laboratory. Later in the year, the Editor received a letter,together with a newspaper clipping from a publication of the University ofBritish Columbia indicating that the discovery was first made by Dr. NeilBartlett of the U.B.C., and later confirmed by the Argonne chemists.