NAME: Which Base is Better?

DATE:

Part I: A Brief History

It is speculated that the first known use of numbers dates back to around 35,000 BC.

Bones and other artifacts have been discovered with marks cut into them which many

consider to be tally marks. The uses of these tally marks may have been for counting

elapsed time, such as numbers of days, or keeping records of quantities, such as of

animals.

Tallying systems have no concept of place value (such as in the currently used decimal

notation), which limit its representation of large numbers but are nonetheless considered

the first kind of abstract numeral system.

The first known system with place value was the Mesopotamian base 60 system (ca. 3400 BC) and the earliest known base 10

system dates to 3100 BC in Egypt. Since then, pretty much every civilization has adopted the base 10 system, most likely for

no better reason than because we have 10 fingers.

At this point, it would be silly to try to change to a different number system. Everyone seems to like the base 10 system just

fine. But what if we didn’t have 10 fingers? What if there were as many number systems in the world as there were languages?

Which would be considered the best or easiest to use? For some ideas on this topic, we will consider the ideas of two old white

guys:

Part II: The Ideas of Two Old White Guys

Lagrange claimed that a prime base is most advantageous. He pointed to the fact that with a prime base every systematic

fraction would be irreducible and would therefore represent the number in a unique way. In our present numeration, for

instance, the decimal fraction .36 stands really for many fractions: 36/100, 18/50, and 9/25. Such ambiguity would be lessened

if a prime base, such as eleven, were adopted.

Buffon was a proponent of the base 12, or duodecimal system. Since 2, 3, 4, 6 are factors of 12, he argued, it is a more

convenient number system for computing fractions compared to decimal system, which has only the factors 2 and 5.

Languages based on a duodecimal are uncommon. Languages in the Nigerian Middle Belt such as Janji , Kahugu , the Nimbia

dialect of Gwandara , and the Chepang language of Nepal are known to use duodecimal numerals. In fiction, the elves from J.

R. R. Tolkien's The Lord of the Rings used a duodecimal system.

Being a versatile denominator in fraction may explain why we have 12 inches in a foot, 12 ounces in a pound, 12 old British

pence in a shilling, 12 items in a dozen, 12 dozens in a gross, 12 gross in a great gross, etc.

Part III: An Introduction to Subscripts and Base 3

To keep the different number systems straight, it is helpful to use subscript. For example, the number 1210 is a number from

the commonly used base 10 (decimal) number system. 123 is a number from the base 3 number system. To see how these two

number systems compare, look at the table below:

Base 3 Base 10

0 0

1 1

2 2

10 3

11 4

12 5

20 6

21 7

22 8

100 9

101 10

102 11

110 12

a) How many symbols are used for the base 3 number system? ___________ For the base 10? ____________

b) What is the largest single digit number in base 3? __________ In base 10? __________

c) What base 10 number is equivalent to 123? ____________

d) How do you write 1310, 1410, and 1510 in the base 3 system? __________ __________ ___________

e) The first three place values of the base 10 system (from right to left) are 1’s, 10’s, and 100’s. For example, the number

23410 can be thought of and rewritten as: 2(100) + 3(10) + 4(1), or 200 + 30 + 4.

What are the first three place values of the base 3 system? __________ __________ ___________

f) Write 2013 in the base 10 system. ___________

g) Write 5310 in the base 3 system. ___________

What about fractions and decimals? The number .32110 can be thought of and rewritten as: 3(1/10) + 2(1/100) + 1(1/1000) or

.3 + .02 + .001. The first three place values after the decimal point are 10ths, 100

ths, and 1000

ths.

Base 3 fraction Base 3 decimal Base 10 fraction Base 10 decimal

1/2 .1111111… 1/2 .5

1/10 .1 1/3 .33333333….

1/11 .0202020… 1/4 .25

1/12 .0120120… 1/5 .2

1/20 .0111111… 1/6 .16666666...

h) What are the first three place values after the decimal point in base 3? __________ __________ ___________

i) Convert .23 to a base 10 decimal and a base 10 fraction. __________ __________

j) Convert the base ten fraction 7/9 to a base 3 decimal and a base 3 fraction. __________ __________

k) Why is the base 10 fraction 1/3 a repeating decimal in base 10 but not in base 3?

Part IV: Lagrange vs. Buffon (undecimal vs. duodecimal)

Now we are ready to compare Lagrange’s base 11 (undecimal) number system to Buffon’s base 12 (duodecimal).

A base 11 number system would require 11 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and T.

a) What number does the T represent in the base 11 system in base 10? __________

b) Using only the digits 0 through T, how would you write the number 1210 in base 11? __________

c) What is the base 10 value of 12311? _________

d) What are the first three place values of the base 11 number system (starting to the left of the decimal point and moving left)

__________ __________ __________

e) What are the first two places values to the right of the decimal point in a base 11 system? _________ _________

f) How many of the following base 10 decimals can be written as base 10 fractions with a denominator less than 10?

.1, .2, .3, .4, .5, .6, .7, .8, .9 __________

g) Convert 1011 to base 10. __________

h) Convert the base 11 fraction 1/10 to a base 10 fraction. __________

i. ) Rewrite the following base 11 decimals as base 11 fractions:

.1 ______ .2 ______ .3 ______ .4 ______ .5 ______ .6 ______ .7 ______ .8 ______ .9 _______ .T _______

i) How many of the following base 11 decimals can be written as base 11 fractions with denominator less than 1011?

.1, .2, .3, .4, .5, .6, .7, .8, .9, .T __________

g) The number .3610 can be rewritten many ways as a base 10 fraction: 36/100, 18/50, or 9/25 since 36 and 100 have common

factors. It is helpful to think of .3610 as: 3(1/10) + 6(1/100).

Convert .3611 to a base 10 fraction and a base 11 fraction: _________ __________

A base 12 number system requires 12 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, and B.

a) What do the A and B represent? ________ _________

b) 1/2 is equivalent to what base 10 decimal? __________? What base 12 decimal? __________

c) Convert 1/3 and 1/4 to base 10 and base 12 decimals: _________ _________ __________ __________

d) Convert 1/6 and 5/6 to base 10 and base 12 decimals: _________ _________ __________ __________

e) Convert 1/9 and 2/9 to base 10 and base 12 decimals: _________ _________ __________ __________

How to read this table

The table shows the decimal (floating point) equivalents of the first 12 unit fractions and compares the number of digits in the

mantissa (the part of the number after the point) for the base 10 and base 12 number systems. The mantissa will be either be

composed of a fixed number of digits or a series of digits that repeat (recur) endlessly. We humans typically prefer as few

digits as possible with none recurring.

f) Why is base 12 considered an improvement for writing the unit fractions 1/3 and 1/4?

g) Why are 1/5 and 1/10 easier to represent with base 10 decimals? In other words, why do 1/5 and 1/10 require a repeating

decimal when written in base 12 but can be written with a fixed decimal in base 10?

h) Which system would work better for 1/13? _________ 1/14? _________ 1/15? __________ 1/16? ___________

Part V: Conclusion

Whose system is better? Lagrange’s base 11 or Buffon’s base 12? Why? __________________________________________

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