dozenal
DESCRIPTION
A discussion of the merits/drawbacks of using the dozenal system.TRANSCRIPT
S H O U L D W E S W I T C H F R O M D E C I M A LT O D O Z E N A L ?
alistair o’neill
A cost benefit analysis of changing base
April 2013 – version 1.1
A good decisionis based on knowledge and not on numbers.
— Plato
A B S T R A C T
When representing magnitudes, numbers are currently repre-sented using positional decimal notation. That is to say, usingbase ten. This dissertation explores why this is the case andwhat other options are available, before assessing the impactwhich changing the de jure base would have and whether ornot society would be likely to adopt such a change. This isevaluated by using the transition to metric from imperial as ananalogue of the way in which decimal to dozenal would pro-ceed.
iii
A C K N O W L E D G E M E N T S
Thank you very much to Mr Price and Mr Budds who havesupported me in my research for the Extended Project.
iv
C O N T E N T S
1 alternatives 1
1 the possibilities 2
1.1 Binary 2
1.2 Octal 2
1.3 Decimal 3
1.4 Dozenal 3
1.5 Hexadecimal 4
2 number of digits 5
2.1 Number of Different Digits 5
2.1.1 Binary 5
2.1.2 Octal 5
2.1.3 Decimal 5
2.1.4 Dozenal 5
2.1.5 Hexadecimal 6
2.1.6 Sexagesimal 6
2.1.7 Conclusion 6
2.2 Number of Digits to Represent a Quantity 7
2.2.1 Days in a Year 7
2.2.2 Minutes in a Year 8
2.3 Standard Form 8
3 testing for divisibility 10
3.1 Divisibility Tests in Decimals 10
3.1.1 Dividing By One 10
3.1.2 Dividing By Two 10
3.1.3 Dividing By Three 11
3.1.4 Dividing By Four 11
3.1.5 Dividing By Five 12
3.1.6 Dividing by Six 12
3.1.7 Dividing By Seven 12
3.1.8 Divisibility By Eight 13
3.1.9 Divisibility By Nine 13
3.1.X Divisibility By Ten 13
3.1.E Divisibility By Eleven 14
3.2 Generalised Base 14
3.2.1 Last Digits 14
3.2.2 Summation of Digits 16
3.2.3 Alternating Summation 16
3.2.4 Composites 17
v
contents vi
3.3 Conclusion 17
4 elementary arithmetic 19
4.1 Addition 19
4.2 Subtraction 20
4.3 Multiplication 20
5 fractions 23
5.1 Definitions 23
5.2 Proving Recurring Decimals Are Rational 24
5.3 Conditions For Termination 25
6 mathematical conclusion 28
6.1 Weighing Up Pros and Cons 28
2 practicalities 30
7 infrastructure 31
7.1 Cost 31
7.2 Double Standards 31
7.3 Information Technology 31
7.4 Education 32
7.4.1 Children 33
7.4.2 Adults 33
8 parallels to metrification 34
8.1 Compatibility 35
8.1.1 Bridge Heights 35
8.1.2 Gimli Glider and NASA 35
8.2 Legislation 36
8.3 Conclusion 36
3 overall conclusion 37
9 conclusion 38
4 appendix 39
a research review 40
a.1 The Heritage Of Thales 40
a.2 The Art Of Computer Programming - VolumeOne 40
a.3 www.dozenalsociety.org.uk and www.dozenal.org 41
a.4 Conflicts in the Learning of Real Numbers andLimits 41
a.5 Code Quality: The Open Source Perspective 41
a.6 www.mail-archive.com 42
a.7 www.information-management.com 42
a.8 USMA 42
a.9 www.dft.gov.uk 43
contents vii
b activity log 44
b.1 Before October 2012 - Research 44
b.1.1 July 2012 44
b.1.2 August 2012 44
b.1.3 September 2012 44
b.2 Writing the Dissertation 45
b.2.1 October 2012 45
b.2.2 November 2012 45
b.2.3 December 2012 45
b.2.4 January 2013 46
b.2.5 February 2013 46
b.2.6 March 2013 46
c analysis of the epq process 47
d glossary 48
d.1 Notation 48
d.1.1 Summation 48
d.1.2 Logarithms 48
d.1.3 Ceiling 48
d.1.4 Floor 49
d.1.5 Product 49
d.1.6 Set Notation 49
d.1.7 Other 49
d.2 Terminology 50
d.2.1 Factor 50
d.2.2 Prime 50
d.2.3 Q.E.D. 50
d.2.4 Numerator 50
d.2.5 Denominator 50
d.2.6 Sets 50
d.2.7 Modular Arithmetic 50
I N T R O D U C T I O N
Before progressing into a mathematical discussion of the ben-efits and drawbacks of alternative notation for numbers, it isimportant to be clear what is meant by decimal notation. It refersto base-ten positional notation. That is to say, the position ofthe symbols matter, with each position being worth ten timesthat of the place to its right. This is best illustrated with an ex-ample. If we take the number 111, then the first 1 is worth 100,while the second 1 is worth ten, and the final 1 is worth justone. Hence, we can break up 111 as thus:
111 = 100+ 10+ 1 (0.0.1)
Similarly, we can break up a less trivial number.
862 = 800+ 60+ 2 = 8× 100+ 6× 10+ 2 (0.0.2)
Within this dissertation, a subscript following a number shalldenote the positional base it is written in. Therefore, the above862 would be written as 862dec. Algebraic symbols followedby subscripts act as normal mathematical arrays. Furthermore,symbols written next to each other within quotation marks rep-resent a number written out with the value of each of thosesymbols representing a digit. For example:
x = 8,y = 6, z = 2 (0.0.3)⇒ "xyz"dec = 862dec (0.0.4)
Given our definition of the base-ten positional notation system,it obviously follows that, for an n+ 1 digit integer Sigma notation,
∑is used in summingseries. Refer to theglossary for moreinformation.
"anan−1...a1a0"dec =n∑i=0
10idecai (0.0.5)
for n ∈ Z,n > 0 and 0 6 ai 6 9. In fact, (5) is a good definitionfor what is meant by the decimally notated integers.
Unfortunately, not all numbers are integers. The non-whole por-tion of the number is placed to the right of a decimal mark(usually a period "." in the United Kingdom and United States
viii
introduction ix
of America, sometimes a comma "," elsewhere) such that theinteger part of the number is separated from the fractional part.Similarly to when dealing solely with integers, to the right ofthe mark, a digit is worth ten times what it would be if it werein the position to its right. Or conversely, the position is tentimes smaller in magnitude than that on its left. So:
0.531 =5
10+
3
100+
1
1000(0.0.6)
This is an example of a terminating decimal. That is to say,a quantity which, after a finite number of decimal places, be-comes only a string of "0"s. Considering a terminating decimalto m decimal places then it is obvious that:
"anan−1...a1a0.a−1a−2...a1−ma−m"dec =n∑
i=−m
10idecai (0.0.7)
for n ∈ Z,n > 0 and 0 6 ai 6 9. To extend this to the non-terminating decimals, one need only set m = −∞.
With the mathematical definition of decimal notation complete,the structure of this dissertation is as follows. Within sectionone, there is a discussion of the features of various bases, witha conclusion describing the theoretically optimal base. In sec-tion two, the practicalities of switching are discussed.
Part 1
A LT E R N AT I V E S
In this part, the features of a generalised base arediscussed and followed by an analysis to investigatewhich base is optimal in each instance. Unless spec-ified otherwise, the use of x represents
"anan−1...a1a0.a−1a−2...a1−ma−m"b =n∑
i=−m
biai
(0.0.8)
whilst z represents
"anan−1...a1a0"b =n∑i=0
biai (0.0.9)
where b is the base in which the number was writ-ten.These stem from the fact that each digit is worth afactor of b more than that to its right, and is thedefinition of positional base system.
1T H E P O S S I B I L I T I E S
Whilst there exist an extensive variety of numeral systems, thisdissertation will focus on the following:
• Binary
• Octal
• Decimal
• Dozenal
• Hexadecimal
Whilst other bases are mentioned, there will not be extendeddiscussion of them, aside from mentioning why they are beingdisregarded as a practical alternative. This chapter gives a briefintroduction to the above systems.
1.1 binary
Dating as far back to ancient India1 binary is a positional nota-tional system with a radix of two. Hence,
1001bin = (1× 23 + 0× 22 + 0× 21 + 1× 20)dec (1.1.1)= (8+ 1)dec (1.1.2)= 9dec (1.1.3)
It is most commonly used today in computing as a specific tran-sistor can be on (1) or off (0). This means that binary makessense for the use of logic gates as each digit only has those twopossible outcomes. Hence, a single digit of binary representsone bit of data.
1.2 octal
Octal is, as the name suggests, the use of base eight. This baseis rarely used today as its initial applications have been su-perceded by hexadecimal. In an unpublished paper from 1718
1 The Heritage Of Thales, W S Anglin and Joachim Lambek, 978-0387945446
2
1.3 decimal 3
Emanuel Swedenborg explained an octal system at the requestof Charles XII2. This didn’t gain any traction however.Base eight became useful when IBM mainframes used proces-sors which had 12, 24 or 36 bit words. As these are all multiples A word is the unit
of data which theprocessor deals with
of three, and 8 = 23, a 12-bit word, which would require twelvebinary digits, would require only 4 octal digits, with conver-sion being as simple as grouping each set of three binary digitstogether.
zbin =
n∑i=0
2iai (1.2.1)
=
bn3 c∑i=0
(8i2∑j=0
2ja3i+j)� (1.2.2)
Hence, conversions from binary to octal are as simple as look-ing at three digits at a time.
101001011101bin = 5135oct (1.2.3)
Interestingly, the Na’vi, in James Cameron’s Avatar, use octal,owing to the fact that they have eight fingers. In real life, theYuki and Pamean languages use base eight, where the spaces inbetween fingers are counted, rather than the fingers themselves.
1.3 decimal
Almost all modern societies currently use the decimal, or baseten, positional numeral system. The main reason for this is be-cause humans have ten fingers and, hence, it is easy to teachchildren basic arithmetic by getting them to count up on theirfingers and move onto the next digit when they run out. Beyondthis, there is very little practical situation where base ten is nat-urally beneficial. It’s use as the fundamental block of the metricsystem is artificially man made, as is that of our currency.
1.4 dozenal
The dozenal system is another positional numeral system. Ithas a radix of twelve. It is not an unheard of base in differentcultures; some Nigerian languages, Gwandara, Chepang andMahl are examples of where it is used. However, there are manyplaces where the number twelve is used as a fundamental value.For example, there are twelve inches in a foot; twelve ounces in A troy pound is
used to weighprecious metals andgemstones
2 The Art Of Computer Programming, Donald Knuth, 0-201-89683-4
1.5 hexadecimal 4
a troy pound; twelve pence in a shilling; and twelve months ina year. The reasons why it would make such a good base areenumerated later in this dissertation.In terms of counting, one can still use their hand, in that theyhave twelve phalanges in their fingers. Hence, they can use theirthumb to point to each phalanx in turn, while counting.
1.5 hexadecimal
Hexadecimal isoften referred to asjust "hex"
Hexadecimal, is, as the name suggests, a positional numeralsystem in base sixteen. It is most commonly used in referenceto computing as 16dec = 24 meaning that it can be used as amore condensed way of writing numbers for a computer. Asthe standard word length for processors is now 32-bit or 64-bit, these are represented by eight digit or sixteen digit hexvalues respectively. Also, the byte as a unit of storage is eightbits, which means that it can be represented by two digits inhexadecimal. The conversion process is just as simple as that ofoctal; however one has to look at groups of four binary digitsrather than just three.
zbin =
n∑i=0
2iai (1.5.1)
=
bn4 c∑i=0
(16i3∑j=0
2ja4i+j)� (1.5.2)
Values which correspond to maximum of certain length hexvalues appear very commonly in computing and gaming. Forexample, in many games, the maximum of a stat you can attainis 255dec = FFhex which is the maximum value a single byteof data can take. Hexadecimal is often used in colours wherea colour is represented by the amount of red, green and bluelight required to make it. Each two digits of the code representsthe value for each primary colour ranging from 00 to FF. For ex-ample #FFFFFF is white as all primary colours are maximised.
2N U M B E R O F D I G I T S
2.1 number of different digits
If a number is written in base b, then it means that 10b = b⇒ ∃a digit "X" such that X = b − 1. Therefore, as we are includ-ing 0, there must be b different digits allowed within base b.This doesn’t normally create any problems until a sufficientlylarge base is suggested that there are so many symbols requiredthat they become difficult to remember. Generally bases use thesame ten digits of decimal and then have new symbols for dig-its representing values > 10dec.
2.1.1 Binary
Under binary, there are only two digits: 0 and 1.
2.1.2 Octal
Under octal, there are eight digits: 0, 1, 2, 3, 4, 5, 6 and 7.
2.1.3 Decimal
The ten digits of decimal are: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
2.1.4 Dozenal
Under Dozenal there are twelve different digits. The first tenare the same as those under decimal. There is a new digit whichrepresents the value "ten" and is written as either A, T , X or X.There is also a digit representing "eleven" which is written as B,E or E. These different conventions are used by different groups,with the Dozenal Society of Great Britain preferring the X andE symbols introduced by Sir Isaac Pitman1. These are the oneswhich will be used in this dissertation and are, when read outloud, pronounced "dek" and "ell" respectively.
1 http://www.dozenalsociety.org.uk/pdfs/AboutUs.pdf
5
2.1 number of different digits 6
2.1.5 Hexadecimal
Under hexadecimal, there are sixteen digits and these are0, 1, 2, 3, 4, 5, 6, 7, 8, 9,A,B,C,D,E and F.
2.1.6 Sexagesimal
The Ancient Sumerians and Babylonians used a base sixty nu-meral system. Therefore they required 60 different symbols forthe possible different digits. They are shown below
This isn’t a pure base, however, as the digits are not distinctbecause each "digit" uses a small base ten tally system itself. Ifthere were to be sixty entirely unique symbols for the digits,then it would be very difficult to learn indeed.
2.1.7 Conclusion
Adding new digits to simple electronic displays is an issuefor a base larger than ten. This is one of the reasons that oc-tal was preferred over hex during early computer development.Programmers didn’t need to have a specialised display whichallowed letters. Indeed, installing the characters for X and E
proved tricky enough within the writing of this dissertationdespite the versatile nature of LATEX. However, as technology isprogrammed by humans, for humans, if there became widespreaddesire for a specific base, then the necessary symbols wouldsoon become standard with all fonts.
2.2 number of digits to represent a quantity 7
2.2 number of digits to represent a quantity
It is useful to consider how long a number will be when writtendown. It stands to reason that a smaller base, will result in moredigits being required to represent a quantity. This is because, ann+ 1 digit number is at most bn+1 − 1 as shown below. Note that the
largest value ai cantake is (b-1)zmaxb =
n∑i=0
bi(b− 1) (2.2.1)
=
n∑i=0
(bi+1 − bi) (2.2.2)
=
n+1∑i=1
bi −
n∑i=0
bi (2.2.3)
= bn+1 − b0 (2.2.4)= bn+1 − 1� (2.2.5)
Similarly, z is at least bn as an > 1 bydefinition andai > 0zminb =
n∑i=0
biai (2.2.6)
= bn × 1+n−1∑i=0
(bi × 0) (2.2.7)
= bn� (2.2.8)
Therefore, these combine to give us: Log is an increasingfunction so wedon’t need to worryabout the directionof the signs
bn 6 zb 6 bn+1 − 1 (2.2.9)
⇒ bn 6 zb < bn+1 (2.2.10)
⇒ n 6 logb zb < n+ 1 (2.2.11)⇒ n = blogb zbc (2.2.12)
⇒ n = b ln zblnbc (2.2.13)
⇒ n+ 1 = d ln zblnbe (2.2.14)
So we can see that the number of digits required to represent ln is the naturallogarithm of anumber
a quantity scales in proportion to the natural logarithm of itsmagnitude as lnb is a constant for a given base. Let us considerthe number of digits required to represent common numbers.
2.2.1 Days in a Year
The true value isapproximately365.25, but aninteger is easier todeal with at thisstage
101101101bin = 555oct = 365dec = 265doz = 16Dhex (2.2.15)
2.3 standard form 8
So there are nine digits in binary, and three for the other bases.
2.2.2 Minutes in a Year
This time days in ayear is taken as365.2510000000011010001000bin (2.2.16)
= 2003210oct (2.2.17)= 525960dec (2.2.18)= 214460doz (2.2.19)= 80688hex (2.2.20)
Here there are twenty digits in the binary representation, sevenin the octal, six in the decimal and dozenal and five in the hex-adecimal.
Clearly, when we extend this to even larger numbers, the differ-ence in the number of digits change dramatically according tothe proportionality proven earlier. Numbers which are merelyseven digits long in decimal look exceedingly complicated inbinary. Because there are only two different digits, it becomesdifficult to read through the entire number without losing yourplace. Supplementation with commas could improve the read-ability, but would make the written number even longer on thepage.For this reason, binary is not practical for everyday use by hu-mans.
2.3 standard form
Usually, when writing down numbers which are either large orsmall and are not needed to be perfectly exact, standard formis used. That is to say written of the form
α× bβ (2.3.1)
where β ∈N and 1 6 α < b
Here, β is the order of magnitude and is used to get a roughestimate of the size of the number. If the number were writtenout fully, it would mean that the number had β+ 1 digits. α isusually written to three significant figures and allows numbersof similar magnitude to be compared. If it is written to threesignificant figures then there are b3 − b2 = b2(b− 1) possible Going from 1.00 up
to "(b-1).(b-1)(b-1)"
2.3 standard form 9
values of α. Obviously the fewer possible values there are, themore difficult it is to compare values. In this instance, the largerthe base, the better.
3T E S T I N G F O R D I V I S I B I L I T Y
It is often helpful to be able to look at a number and immedi-ately decide whether or not it will be divided exactly by a spec-ified number. This is where divisibility rules come in. Manypeople are familiar with the rules for testing whether or not anumber is divisible by a specific divisor in base ten. It is im-portant to note that such rules are not solely restricted to us-ing base ten, and there are similar rules for other bases. Beforestarting to generalise, it is helpful to look at the rules as theycurrently exist so as to ascertain understand why they work.
It is important to note that a d|z, that is to say d divides z iff∃k ∈ Z : z = kd. Also, "iff" means if and
only if. It is usedwhen implicationruns in bothdirections
z = kd⇔ z ≡ 0 (mod d) (3.0.2)
"modσ" refers tomodular arithmetic.It is effectively theremainder which isleft after dividingthe number by σ
3.1 divisibility tests in decimals
3.1.1 Dividing By One
All integers can be divided by one, as doing so does not changethe value, hence remains an integer.
3.1.2 Dividing By Two
A number is divisible by two iff it is even (by definition). Usingdecimal notation, a number is even iff it ends in 0, 2, 4, 6 or 8.By exhaustion, these are all of the single digit even numbers. Sowhat the rule is really saying is that the last digit of a numberis even iff the entire number is. By the contrapositive, it is thesame saying that the parity of a number is the same as theparity of its final digit. This is proven as follows: � is equivalent to
"Q.E.D" andmeans that a proofis complete
10
3.1 divisibility tests in decimals 11
z =
n∑i=0
10idecai (3.1.1)
= a0 +
n∑i=1
10idecai (3.1.2)
= a0 + 2×n∑i=1
5× 10i−1decai (3.1.3)
≡ a0 (mod 2)� (3.1.4)
3.1.3 Dividing By Three
A decimal number is divisible by three iff the sum of its digitsis divisible by three. This can be proven if a number, modulothree, is congruent to the sum of its digits, as being 0 (mod 3)is merely a special case of this. Consider the
binomial expansion.Only the final 1term will not have afactor of 3
z =
n∑i=0
10idecai (3.1.5)
=
n∑i=0
(3× 3+ 1)iai (3.1.6)
≡n∑i=0
1iai (mod 3) (3.1.7)
≡n∑i=0
ai (mod 3)� (3.1.8)
3.1.4 Dividing By Four
A decimal number is divisible by four iff its final two digits aredivisible by four. Similar to the two situation, this is becausethe number (mod 4) is equivalent to the number formed fromits last two digits.
z =
n∑i=0
10idecai (3.1.9)
= 10× a1 + a0 +n∑i=2
10idecai (3.1.10)
= 10× a1 + a0 + 4× 25×n∑i=2
10i−2decai (3.1.11)
≡ 10× a1 + a0 (mod 4)� (3.1.12)
3.1 divisibility tests in decimals 12
3.1.5 Dividing By Five
A decimal number is divisible by five iff its final digit is a 0
or 5. Similarly to two, this is implied by the final digit beingequivalent to the entire number (mod 5).
z =
n∑i=0
10idecai (3.1.13)
= a0 +
n∑i=1
10idecai (3.1.14)
= a0 + 5× 2×n∑i=0
10i−1decai (3.1.15)
≡ a0 (mod 5)� (3.1.16)
3.1.6 Dividing by Six
Any number is divisible by six iff it is divisible by two and bythree. It is obvious that if a number is divisible by six, it mustbe divisible by three and two as 6 = 3× 2. As 2 and 3 are co- Numbers are
co-prime iff theirhighest commonfactor is one.
prime, it is also true that the reverse implication exists, that isto say, being divisible by 2 and 3 is sufficient to prove that it isdivisible by 6.
z = 2× k1 = 3× k2 (3.1.17)
By Euclid’s lemma, either 3|2= or 3|k1. Hence, Euclid’s lemmastates that n|aband n - a⇒ n|b
= is used in thisdissertation to showthat a contradictionhas been reached.
k1 = 3× k3 (3.1.18)⇒ z = 2× 3× k3 = 6× k3� (3.1.19)
3.1.7 Dividing By Seven
Whilst there exist some tests for testing divisibility by seven,none of them are particularly more efficient than simply at-tempting the long division sum and seeing if it results in aninteger answer. Those which can be done mentally are recur-sive and only shave off one digit at a time.
3.1 divisibility tests in decimals 13
3.1.8 Divisibility By Eight
A number is divisible by eight iff 8|4a2 + 2a1 + a0. This is be-cause:
z =
n∑i=0
10idecai (3.1.20)
= 100a2 + 10a1 + a0 + 8× 125×n∑i=3
10i−3decai (3.1.21)
≡ 96a2 + 4a2 + 8a1 + 2a1 + a0 (mod 8) (3.1.22)≡ 4a2 + 2a1 + a1 (mod 8)� (3.1.23)
3.1.9 Divisibility By Nine
A number is divisible by nine iff the sum of its digits is divisibleby nine. This is proven similarly to the rule for divisibility bythree.
z =
n∑i=0
10idecai (3.1.24)
=
n∑i=0
(9+ 1)iai (3.1.25)
≡n∑i=0
1iai (mod 9) (3.1.26)
≡n∑i=0
ai (mod 9)� (3.1.27)
3.1. Divisibility By Ten
A number is divisible by ten iff it ends in 0. Again, like mostof the others, this can be implied by proving that a decimalnumber is equivalent to its final digit modulo 10.
z =
n∑i=0
10idecai (3.1.28)
= a0 + 10×n∑i=1
10i−1decai (3.1.29)
≡ a0 (mod 10) (3.1.30)
3.2 generalised base 14
3.1. Divisibility By Eleven
This divisibility rule is in the same vein as that of three andnine, however this time one alternates adding and subtractingdigits. The result is the equivalent to the original number mod-ulo eleven provided the final digit is added. It is useful to notethat, under the specific case that 11|z, it does not matter whetherwe start with addition or subtraction as 0 = −0. Again, only the
final term of thebinomial expansiondoes not contain 11z =
n∑i=0
10idecai (3.1.31)
=
n∑i=0
(11− 1)iai (3.1.32)
≡n∑i=0
(−1)iai (mod 11)� (3.1.33)
3.2 generalised base
In the above examples, there are four different methods usedto tackle finding a rule to test for divisibility.
3.2.1 Last Digits
The first method involves looking at the last few digits of anumber. The divisors this works for are 2, 4, 5, 8 and 10. Theone shared feature they have is that they all have only primefactors which are in 10. That is to say, they are of the form
d = 2µ5λ (3.2.1)
It isn’t too tricky to generalise a proof of a divisibility rule inbase b where
b =
m∏j=1
pγjj (3.2.2)
where m is the number of distinct prime factors it has and γj isthe power to which each prime factor is raised. Let the divisorbe defined as
d =
m∏j=1
pαjj (3.2.3)
3.2 generalised base 15
Where all of the pjs are the same as those of b. Also, let max will return thelargest value of thegiven setθ = dmax(
αj
γj)e (3.2.4)
We get the following Note thatiγj −αj > 0 byour definition of θ
z =
n∑i=0
biai (3.2.5)
=
n∑i=0
(ai(
m∏j=1
pγjj )i) (3.2.6)
=
n∑i=0
(ai
m∏j=1
piγjj ) (3.2.7)
=
θ−1∑i=0
(ai
m∏j=1
piγjj ) +
n∑i=θ
(ai
m∏j=1
piγjj ) (3.2.8)
=
θ−1∑i=0
(ai
m∏j=1
piγjj ) + (
m∏j=0
pαjj )
n∑i=θ
(ai
m∏j=1
piγj−αjj ) (3.2.9)
≡θ−1∑i=0
(ai
m∏j=1
piγjj ) (mod d) (3.2.10)
≡θ−1∑i=0
aibi (mod d) (3.2.11)
≡ "aθ−1aθ−2...a1a0" (mod d)� (3.2.12)
Which means we need only look at the last θ digits of the num-ber to see if it divisible by d in base b.
For example, to check if a number is divisible by 9 underdozenal then, b = 22 × 3 and d = 32.
max(αj
γj) = max(0, 2) = 2 (3.2.13)
Hence, one only need look at the last two digits and test if theyare divisible by 9. As a check, to test if
9|183E02X6doz (3.2.14)
we need only look at the last two digits.
X6doz = 9× 12doz (3.2.15)
So, this means that our original number is also divisible by nine.Indeed it is true that
183E02X6doz = 9× 231283Xdoz (3.2.16)
3.2 generalised base 16
3.2.2 Summation of Digits
The second method looks at what happens when the digits ofthe number are added. The divisors which this works for arethree and nine. These works because
10dec = (32 + 1) (3.2.17)
Again, it is simple to provide a generalisation and say that ifb = dα + 1 then providing the sum of digits is divisible by dthen so is the number itself.
z =
n∑i=0
biai (3.2.18)
=
n∑i=0
(dα + 1)iai (3.2.19)
≡n∑i=0
1iai (mod d) (3.2.20)
≡n∑i=0
ai (mod d)� (3.2.21)
As an example, to test if a hexadecimal number is divisible byFhex = 15dec then, as b = F+ 1, the sum of the digits will bedivisible by fifteen if the original number is. So, to check:
41AE42Ahex → 4+ 1+A+E+ 4+ 2+A = 2Dhex → F (3.2.22)
As F|F, then 41AE42Ahex is also divisible by F.
41AE42Ahex = F× 460F36hex (3.2.23)
3.2.3 Alternating Summation
In the decimal divisibility rule for eleven, one has to alternatelyadd or subtract the digits. This result can be generalised to anydα = b+ 1 as follows
z =
n∑i=0
biai (3.2.24)
=
n∑i=0
(dα − 1)iai (3.2.25)
≡n∑i=0
(−1)iai (mod d)� (3.2.26)
3.3 conclusion 17
As mentioned previously, if one is only interested in whetheror not the number is divisible by d, then z modulo d will bezero. Hence, it does not matter whether or not the first termis added or subtracted as multiplying the entire summation by−1 makes no difference.
3.2.4 Composites
As touched upon in the description for testing decimal divisi-bility by six, by using Euclid’s lemma, it is possible to come upwith necessary and sufficient conditions for a divisibility rulefor composites, providing there exists divisibility rules for itsconstituent powers of primes. Let
d =
m∏j=1
pαjj (3.2.27)
d|z⇒ z = k0
m∏j=1
pαjj (3.2.28)
By factorising out each pαjj in turn, it is trivial to show that theforward implication is true. The reverse implication is slightlytrickier. If we can show that pαjj |z∀0 < j < m, then the followingis true.
z = k1pα11 = k2p
α22 = ... (3.2.29)
By Euclid’s lemma, pα22 |k1 as pα22 - pα11 due to unique prime UPF is proven inthe FundementalTheorem of Alegbra
factorisation. Hence
z = k2,1pα11 p
α22 = k3p
α33 = ... (3.2.30)
By the same logic, it is seen that pα33 |k2,1 and by induction
z = k0
m∏j=1
pαjj � (3.2.31)
So it has been shown that to test for divisibility of a compos-ite number, one merely needs to test for the divisibility of thehighest power of its component primes.
3.3 conclusion
There exists a large number of fundamental tests which can beapplied to numbers of all bases in order to ascertain if they
3.3 conclusion 18
are divisible by certain numbers. It is not a feature of the dec-imal system specifically. Indeed, by looking at the conditionsfor these divisibility tests, it can be shown that there exist ahigher proportion of divisibility tests in dozenal that rely onthe simpler methods.
4E L E M E N TA RY A R I T H M E T I C
People are very comfortable with elementary arithmetic in thedecimal system and don’t generally like change. In this chapter,it is proven that switching base does not materially change anyof the processes involved.
4.1 addition
If two numbers are required to be added in decimal, then theyare written out with their digits aligned as in this example suchthat the user is adding digits in the same column, i.e. are mul-tiplied by the same power of b
Reader knowledgeof the additionalgorithm isassumed
This works in general because
x1 + x2 =
n∑i=−m
bia1,i +
n∑i=−m
bia2,i (4.1.1)
=
n∑i=−m
bi(a1,i + a2,i) (4.1.2)
If a1,i + a2,i = a3,i > b then a3,i = b+ a single digit value.Hence, by factoring the b with the digit to its left, the carryingover is accounted for. Therefore, it has just been shown that theexact same algorithm for adding works in general for all bases.Take the following example, in dozenal:
624
3E8+
——X20
——
19
4.2 subtraction 20
4.2 subtraction
The process for subtraction works in exactly the same method,by factorising out the powers of b.
x1 − x2 =
n∑i=−m
bia1,i −
n∑i=−m
bia2,i (4.2.1)
=
n∑i=−m
bi(a1,i − a2,i) (4.2.2)
If a1,i − a2,i < 0 then we "borrow" one from the bi+1 term asbi+1 = b× bi. So, once again, it has been shown that the exactsame method of subtraction works in a generalised base, b, asworks in decimal. Again, an example, in dozenal:
624
3E8−
—–228
—–
4.3 multiplication
This is where things start to get more interesting as the algo-rithm becomes more complicated. The way the algorithm worksis by multiplying the top number by each of the bottom digitsin turn and adding a further 0 to the right hand column witheach subsequent row. The user then adds up these values asthis decimal example illustrates:
4.3 multiplication 21
This works as
z1 × z2 =
n∑i=0
bia1,i ×n∑j=0
bja2,j (4.3.1)
=
n∑i=0
(bia1,i ×n∑j=0
bja2,j) (4.3.2)
=
n∑i=0
(
n∑j=0
bj+ia1,ia2,j) (4.3.3)
which is just the sum of each of the multiplicative results whenthe top number is multiplied by each digit of the second num-ber and adjusted by multiplying by b. An example, in dozenal:
000624
0003E8×———-004168
058180
167000+
———-207328
———-
This does, however, raise an interesting point. It is requiredfor the user of this algorithm to be comfortable in multiplyingany two single digit values together and returning their prod-uct near instantly for this to be efficient. That is to say, they arerequired to know their times-table off by heart.
Assuming that multiplying by both zero and one are discounted Remember thatmultiplication iscommutitive
as being trivial, there are (b−2) elements to choose from. Usingcombinatorics, the total number of results is the total number ofways of taking two distinct elements from (b− 2), where orderdoesn’t matter, add on the number of possible square pairings
4.3 multiplication 22
as the "choose" function doesn’t take this into account.Therefore there are (
µ
λ
)is the number ofways of choosing λobjects from µ
((b− 2)
2
)+ (b− 2) =
(b− 2)(b− 3)
2+ (b− 2) (4.3.4)
=(b2 − 5b+ 6+ 2b− 4
2(4.3.5)
=(b2 − 3b+ 2
2(4.3.6)
=(b− 1)(b− 2)
2(4.3.7)
=
((b− 1)
2
)(4.3.8)
results to memorise. As this function is quadratic in b, it willstart to increase substantially as b increases. Under decimal,there are 36 results to memorise whilst under dozenal, thereare 55. Extended to hexadecimal, then there are suddenly 105results to learn. To take it a step further, the Babylonians wouldhave had to have known 1711 different results. Given that apupil is expected to have learnt their times-table by the end ofKey Stage 2, clearly having a base which is unnecessarily largeis not helpful.
5F R A C T I O N S
As mentioned in the introduction, not all numbers are integers.There are those which have a fractional part represented by I am using "decimal
places" to meandigits to the right ofthe separator,regardless of thebase
digits to the right of a period or dot. There are different typeson non integers:
a. Terminating decimals are those which run on for a fi-nite number of decimal places and then every subsequentdigit is zero. For example, 4.312
b. Recurring decimals are those which have a non-zero digitor set of digits which repeat over and over forever. Forexample, 6.13333333...
c. Irrational numbers are those which have infinite decimalplaces but do have any repeating pattern. An example is√2 Proving that a
number istrancendental isstill a tricky process
d. Transcendental numbers are those which have infinite dec-imal places, do not have a repeating pattern, and are notthe solutions to a polynomial equation with integer co-efficients. An example is π as proven by the LindemannWeierstrass theorem.
5.1 definitions
Given a terminating non-integer can be written
x =
n∑i=−m
biai (5.1.1)
and defining the recurring sequence as
"rαrα−1...r0" =α∑j=0
bjrj (5.1.2)
It is clear that a recurring non-integer can be written
x =
n∑i=−m
biai +
∞∑k=1
α∑j=0
b−m−k(α−1)+jrj (5.1.3)
23
5.2 proving recurring decimals are rational 24
Whereas an irrational number would be written as
x =
n∑i=−∞b
iai (5.1.4)
5.2 proving recurring decimals are rational
This dissertation will only focus on rational numbers because Rational numberscan be written as
µ
λ
where µ, λ ∈ Z
of the following property:
x =
n∑i=−m
biai +
∞∑k=1
α∑j=0
b−m−k(α+1)+jrj
⇒ bα+1x = bα+1(
n∑i=−m
biai +
∞∑k=1
α∑j=0
b−m−k(α+1)+jrj)
⇒ bα+1x =
n∑i=−m
bi+α+1ai +
∞∑k=1
α∑j=0
b−m−(k−1)(α+1)+jrj
⇒ bα+1x =
n∑i=−m
bi+α+1ai +
∞∑k=0
α∑j=0
b−m−(k)(α+1)+jrj
⇒ bα+1x− x =
n∑i=−m
bi+α+1ai −
n∑i=−m
biai +
∞∑k=0
α∑j=0
b−m−(k)(α+1)+jrj −
∞∑k=1
α∑j=0
b−m−kα−k+jrj
⇒ x(bα+1 − 1) = (bα+1 − 1)
n∑i=−m
biai +
0∑k=0
α∑j=0
b−m−k(α+1)+jrj
⇒ x(bα+1 − 1) = (bα+1 − 1)
n∑i=−m
biai +
α∑j=0
b−m+jrj
⇒ x =(bα+1 − 1)
∑ni=−m b
iai +∑αj=0 b
−m+jrj
bα+1 − 1
Therefore, it has just been shown that a recurring decimal canbe represented as the sum of two non-recurring decimals di-vided by an integer. Given the proof in the previous chapterinvolving the addition and subtraction of numbers, it is clearthat if two terminating decimals are taken, their sum and differ-ence are both also terminating. Therefore, a recurring decimalcan always be written as a terminating decimal divided by aninteger.
5.3 conditions for termination 25
Now, to show that any terminating decimal can be representedby an integer divided by an integer.
x =
n∑i=−m
biai (5.2.1)
xbm = bmn∑
i=−m
biai (5.2.2)
xbm =
n∑i=−m
bi+mai (5.2.3)
xbm =
n∑i=0
biai−m (5.2.4)
x =
∑ni=0 b
iai−mbm
� (5.2.5)
Therefore, by extension, any recurring decimal can also be rep-resented by an integer divided by an integer. The importantquestion is when a recurring decimal is formed by a quotient.In general, in everyday life, people dislike having recurring dec-imals. Consider the following proof.
ω = 0.99999999...dec (5.2.6)10decω = 9.999999...dec (5.2.7)
9ω = 9 (5.2.8)ω = 1� (5.2.9)
The majority of pupils are uncomfortable with this fact, thatthe same number can be represented with the same decimalrepresentation.1 Therefore, it is best that a base results in asfew recurring decimals as possible.
5.3 conditions for termination
So, the conditions which result in a terminating decimal arenext to be found. It has been shown above that, for x being aterminating decimal, the following is true. z ∈ Z
x =
∑ni=0 b
iai−mbm
=z
bm(5.3.1)
1 Tall, David and Schwarzenberger, R. L. E. (1978) Conflicts in the learningof real numbers and limits. Mathematics Teaching, Vol.82 . pp. 44-49. ISSN0025-5785
5.3 conditions for termination 26
The numerator of this is an integer and the denominator is apower of the base. Therefore, it follows that any integer dividedby a power of the base in question is going to result in a termi-nating decimal. This is, however, a sufficient but not necessarycondition. Consider the following:
23doz2
= 11.6doz (5.3.2)
Here, we divided by 2, which is not a power of 12dec, howeverstill attained a terminating answer. The reason for this is that 2is composed of only prime factors of 12dec.Let d be any divisor which is only made out of the prime factorsof b as thus: n is the number of
distinct primefactors of b
b =
n∏i=1
pαii (5.3.3)
d =
n∏i=1
pγii (5.3.4)
Also, let
θ = max(dγiαie) (5.3.5)
⇒ θαi > γi (5.3.6)
With these definitions in place, all that needs to be achieved isto manipulate any fraction into the form
z
bm
. x, z ∈ Z
x
d=
xn∏i=1pγii
(5.3.7)
=
xn∏i=1pθαi−γii
n∏i=1pθαii
(5.3.8)
=
xn∏i=1pθαi−γii
(n∏i=1pαii )θ
(5.3.9)
=xz
bθ� (5.3.10)
5.3 conditions for termination 27
Therefore, it has been shown that every divisor which is solelycomposed of primes which make up the base, results in a ter-minating decimal. Due to unique prime factorisation, there isno similar way to rearrange a division sum where the divisorhas a prime which is not part of the base into the desired formunless p|x, in which case there is an integer solution.In dozenal, therefore, any divisor of the form
d = 2x × 3y (5.3.11)
will result in a terminating expansion. If this is compared tounder decimal where the divisor must be in the form
d = 2x × 5y (5.3.12)
it is obvious that there are more divisors which result in a ter-minating expansion if dozenal is used. For example, if we lookat divisors under 100, there are 20 which will result in a ter-minating dozenal expansion as compared to 14 if base ten isused. Considering that in real life, only low divisors are everused without a computer taking over, it is more useful to lookat divisors under ten. Of these, 7 will give terminating dozenalexpansions and only 5 using decimal.The most obvious benefit is that dividing by three will alwaysresult in a terminating dozenal expansion. In natural life, be-ing able to divide by three is much more useful than dividingby five; it is only the current decimal system which artificiallymakes five become a common number.
6M AT H E M AT I C A L C O N C L U S I O N
6.1 weighing up pros and cons
Here, a judgement must be made, based on the theory outlinedso far as to what the optimal base is.In terms of the number of different digits required, everybodyhas already proven they can easily learn 36 symbols just bylearning their alphabet and decimal numerals. As a result, thequantity of different digits should not be an issue unless it be-comes a ridiculously high number. At any rate, there is no issuewith going all the way up to hexadecimal.
There are however issues with the number of digits being ableto represent a number. As discussed previously, having too lowa radix means that representing large quantities requires ex-tremely large numbers of digits. For this reason, I exclude bi-nary or quaternary (base four) from being a viable base for usein everyday life.
In order to both have many simple divisibility tests and havemany divisors resulting in a terminating expansion, the base isrequired to have many different prime factors. Furthermore, itshould be an expectation that one is able to halve numbers andalways get a terminating expansion. Therefore the base musthave two as a factor. However, only having two as a prime factormeans that not many other numbers divide into values nicely.As a result I exclude octal and hexadecimal from being viablebases in real life.
This only really leaves dozenal and decimal as viable bases.They are both similar, having the same number of prime factorsand being similar in magnitude, meaning that they neither havetoo many results to remember for the times-tables nor requiretoo many digits to express large values. However, the benefit oftwelve containing the first two primes, makes it far more use-ful as a radix as there are so many more values which can bedivided by giving us a terminating expansion. It also resultsin marginally shorter numbers when writing large quantities.
28
6.1 weighing up pros and cons 29
Whilst the differences may appear subtle, the ability to divideby the first four natural numbers seems to make it a much moreintuitive base than decimal.Dozenal strikes the optimal balance between magnitude anddistinct prime factors. Six would be too small of a base, andthirty (2× 3× 5) would be too large; learning the times-tableswould require knowing 406dec unique results.
For these reasons, according to the theory, the optimal radixfor general use is twelve.
Part 2
P R A C T I C A L I T I E S
This section evaluates the practicalities of changingto dozenal and concludes whether or not it is ulti-mately worth changing
7I N F R A S T R U C T U R E
7.1 cost
Change is not cheap.
Regardless which base makes the most sense, it has to be con-sidered how much it would cost to implement the change. Ifone looks around a room, there will be countless objects whichmake use of numbers. Every single product which contains anumber larger than 9 would be required to be redesigned, newmoulds purchased, reprinted. Consumers often undervalue howmuch these production processes cost as they are happy bene-fiters of mass production. For example, from discussion withengineers, it can easily cost upward of $50,000 for a single in-jection mould. Obviously, this value varies with size, materialand intricacy. So, to have to change every single product whichhas numbers on it would cost an astronomical sum. As a result,there would not be universal adoption of the new base untilthey were going to have to buy new moulds anyway.
7.2 double standards
If there are two different numeral systems in use, there mustbe a way to distinguish if a number is written in decimal ordozenal. This dissertation used a subscript; however, it is likelythat a new symbol would have to be devised; not unlike sym-bols for different currencies are used today. This would increasethe amount of space that numbers would take to representand introduce another level of thought before one can quicklygauge the magnitude of a written number.
7.3 information technology
It isn’t only physical costs which need to be considered. It costsmoney to change pre-existing IT systems. Given that all num-bers are converted into binary for usage by the processors, thisisn’t as big an issue as physical objects; however the sheerbreadth of content and usage of IT means that it would be
31
7.4 education 32
no easy feat even to simply reconfigure input and output pro-cesses. There is also the potential for unforeseen errors to occurwith the change. In particular, overflow glitches may occur with When the
maximum value of avariable is exceededand it wraps backaround to 0
standard default values which do not cause issues in decimal.Take for example, the value
1000000dec < 220 < 1000000doz (7.3.1)
If a datum with a bit limit of 21 was applied to the value then The first digit beingused to representsign
inputting 1000000 would not cause an issue if it were decimal,but would if it were dozenal. While this example seems con-trived; issues like this do actually arise. For example, the Year2038 problem1 reared its head for AOLserver2 in 2006. Here, adefault time-out of one billion seconds was used for a process The length of time a
computer willattempt a problemfor before giving upand reporting anerror.
which shouldn’t time-out.
This was fine until 2006, when this meant that systems wouldtime out after 03:14:07 UTC on Tuesday, 19 January 2038. Thisis the time when a unix time stamp overflows its limit as a32-bit signed integer. As a result, it wrapped around to 1901.This caused the process to instantly throw up a time-out er-ror. This situation occurred when the initial programmer knewwhat quantity one billion typed out would be. The prevalenceof issues like this if every typed number were to be assumed tobe written in dozenal in the source code would be massive; andpotentially cost a lot of money in down time. A single hour ofdowntime can cost as much as $6.48 million for an online bro-ker3.
7.4 education
By far, however, the largest obstacles to widespread adoptionof a new numeral system are human beings. People are re-markably stubborn when it comes to change, as is illustratedin the next chapter. However, even if people are totally willingto adopt a new system, they still need to learn how to use itefficiently. Society has segregated people into two groups whoboth have different needs when it comes to learning dozenal.
1 Code Quality: The Open Source Perspective; 607820146
2 www.mail-archive.com/[email protected]/3 www.information-management.com/infodirect/2009_133/downtime_cost-
10015855-1.html
7.4 education 33
7.4.1 Children
Children who are still at school have the advantage of not be-ing entirely set into the ways of using decimal. As can be seenin mathematics students who learn radians after having beentaught degrees for most of their education, children are able torapidly adapt to a new system. Children have the opportunityto start entirely from scratch and learn dozenal from the groundup; in fact, things like times-tables will be easier in dozenal be-cause there are nicer patterns formed by the digits than thereare in decimal.
7.4.2 Adults
There are, however, problems for adults who want to learn anew system. Given how ingrained the old decimal system is, itwould be difficult to truly adapt. Firstly, there is very little in-centive for an adult to put effort into practising the new radixto the same extent of a child; there is no detention if they don’tfinish the worksheet. This means that, whilst they may under-stand the theory behind the new system, they are not nearly asfluent with it as they are with decimals. Furthermore, their abil-ity to glance at a number and ascertain size will be off as thesame digits can now represent a substantially different quan-tity.Also, when first using dozenal, people will convert each num-ber into decimal in their head; do the calculation; and thenconvert back into dozenal. This is massively inefficient. How-ever, until one is entirely fluent with the new system, this iswhat they are forced to do, rather than doing the calculationsin dozenal.When doing things like multiplication, someone who used touse decimal may confuse their decimal times-table with theirdozenal and thus make calculation errors. They need to remem-ber things like:
5× 3 = 13doz = 15dec (7.4.1)
8PA R A L L E L S T O M E T R I F I C AT I O N
The previous chapter outlined theoretical issues with a gener-alised conversion from decimal to dozenal. However, there ex-ists no data to confirm if the concepts would actually occurin practice. There has never been such a massive shift in nu-merical notation within recent history. Historically, mathemat-ics was the reserve of those few people who had an education.Ordinary people would simply be counting, using tallies etc.
Therefore, it makes sense to compare the transition from deci-mal to dozenal with the transition between imperial and metricunits. Whilst most countries have mainly converted to metric,there still remain countries who do not use metric for everydayuse.This is a map showing when countries adopted the metric sys-tem.
1
The first thing to notice about the image is the timespan whichhas passed since the metric system was first introduced in Francein 1795. To this day, there is little to no support of metric in theUSA with the exception of the fields of science. This illustratesthe societal inertia to new concepts where the general publicare not fully convinced of the benefits.
1 Data from USMA (U.S. Metric Association)
34
8.1 compatibility 35
8.1 compatibility
Whilst it could be said that there is no issue with there being anoverlap of the two systems whilst dozenal is being learned, wecan see from history the problems caused by both metric andimperial standards being in play at the same time.
8.1.1 Bridge Heights
In the UK, a larger proportion of crashes which involved lor-ries attempting to go underneath bridges which were too lowfor them was recorded for European lorry drivers comparedto British drivers than should have been expected.2 This wasput down to the fact that European drivers were used to see-ing bridge heights in metres rather than feet, which resulted inthem glancing at UK signs and assuming the bridge to be largerthan it actually was. It was the use of two different standardsin two different locations which caused these incidents to occur.It was proposed that bridge signs should have heights markedin both units.
8.1.2 Gimli Glider and NASA
It isn’t just laypeople who make mistakes involving units andconversions, however. In 1983, a Canadian plane narrowly avoidedcrashing after it ran out of fuel mid-flight. Upon investigation, itwas discovered that the plane had been provided with 22, 300lbsof fuel rather than 22, 300kg of fuel. This mistake could easilyhave resulted in a massive loss of life if one of the pilots hadnot also happened to be very adept in flying gliders and wasable to guide the plane to safety.
Even NASA has had problems coping with two different sys-tems. When it contracted out production of the Mars ClimateOrbiter, it stipulated that units were to be SI (metric). The sub-contractor in charge of creating the Earth side control systemsrecorded output commands in pound force seconds rather thannewton seconds. This meant that the Orbiter was not in theplace it was expected and communication was lost when it at-tempted to get into orbit of Mars. It is assumed to have been
2 www.dft.gov.uk/consultations/open/trafficsignsamendmentregs/annexd.pdf
8.2 legislation 36
destroyed due to the higher pressure as it got closer to Marsthan initially anticipated.
8.2 legislation
There were two main approaches taken with respect to con-verting into metric. The first involved, in a short space of time,making metric legal and outlawing the old units. This occurredin India between 1 April 1960 and 1 April 1962. This methodresults in all units being converted into metric with a two yeargrace period for companies to relabel packaging and for peopleto get accustomed to the change. This was massively successfuland countries such as Australia and New Zealand employedsimilar tactics.
The second involved phasing the new units in over time andallowing a long period of time between legalising metric andbanning imperial. For example, the [then] British Empire, mademetric legal in 1873 and the UK still hasn’t fully completed theconversion and exists in a limbo between metric and imperial.With the shift in UK education towards only teaching metric,however, it is likely that the conversion process will be com-pleted the children of today’s lifetime. However, legally, milesare still used in road signage and speed limits, with this un-likely to change in the near future.
8.3 conclusion
By analysing the similarities that would occur in the conver-sion process, it is seen that going from decimal to dozenal isnot something that would happen overnight. Indeed, it maytake multiple centuries until the world has fully transferred.Furthermore, people resist change and if even just one majorcountry decides not to adopt the new system, then they willforce the old base to remain in use for compatibility. Also, youwill still get older people who simply refuse to try the newsystem in an attempt to keep with tradition.
Part 3
O V E R A L L C O N C L U S I O N
9C O N C L U S I O N
A balance has to be reached between the theoretical benefitsof switching to dozenal and the practical limitations when itcomes to changing. Whilst dozenal makes elementary arith-metic easier and has more convenient answers, these benefitsdo not outweigh the extremely large cost which adoption wouldincur. Even if, somehow, all global governments decided to sup-port the change; the financial cost and potential disarray anddamage easily offset the advantages of dozenal.
Furthermore, the advantages of dozenal do not transfer to highlevel mathematics. It only changes the notation of a quantity,not any of the features of the value. As a result, it does not helpdifficult problems nor particularly assist research.
Therefore, in conclusion, while dozenal has been shown to offersubstantial benefits from a mathematical perspective comparedwith the current decimal system, the practicalities of implemen-tation as a replacement to the current system render the changenon-viable for general use.
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Part 4
A P P E N D I X
AR E S E A R C H R E V I E W
Whilst it may seem that there are few sources, this is becauseof the nature of mathematics. After getting a prompt from asource, I set about investigating it on my own, with all of theequations in the dissertation being of my own creation.
a.1 the heritage of thales
W S Anglin and Joachim Lambek, 978-0387945446
This was a fascinating book which, rather than simply present-ing the mathematical concepts, placed them in historical con-text. As it was in chronological order, it ensured that I under-stood why certain types of proof were used. For example, if onewere working before Newton / Leibnitz, they would be unableto use calculus in their proofs as it hadn’t yet been invented. Asa result, it opened me up to different methods of solving prob-lems rather than the prescribed methods used for A-level. Itis a book aimed at undergraduate maths students which usedmathematical proof which means that I trusted what it saidwith regard to the mathematics and also the history of mathe-matics, despite being less certain.
a.2 the art of computer programming - volume one
Donald Knuth, 0-201-89683-4
Given the majority of the first volume was written using As-sembly Language in all of the examples, I struggled to under-stand the majority of the concepts. However, whilst not know-ing exactly what the algorithms were for, I was able to do themaths side of them in terms of calculating their efficiency andrun time order. I was also able to do the exercises which wereto do with processing in order to learn about the usage of bi-nary, octal and hexadecimal. Donald Knuth is one of the mostrespected computer scientist in the world and, as such, I foundhis work extremely reliable.
40
A.3 www.dozenalsociety.org .uk and www.dozenal .org 41
a.3 www.dozenalsociety.org .uk and www.dozenal .org
These are the Dozenal Society of Great Britain and of Americarespectively. I found these sites useful as a starting point interms of researching the benefits of dozenal. While the contentin them is mostly anecdotal, the concepts they suggest wereenough to springboard me to actually doing original researchon the topics they mention. I also found them interesting as acommentary on societies in general. These are two groups whoboth have a shared goal, however can’t even agree on whichdigits they should be using. Therefore, whilst I shouldn’t takeanything they had written directly as fact, I was able to workwith their assertions and prove them myself.
a.4 conflicts in the learning of real numbers and
limits
Tall, David and Schwarzenberger, R. L. E. (1978) MathematicsTeaching, Vol.82 . pp. 44-49. ISSN 0025-5785
This paper looked at difficulties in teaching concepts of recur-ring decimals. I learned about the education system as it wasin 1978, specifically with regard to series and limits in the syl-labus. Whilst a lot has changed, there are still some glaringgaps which have not been patched. So, whilst it must be takenwith a pinch of salt, I think that the point which I used in thedissertation is still valid despite the age of the source.
a.5 code quality : the open source perspective
Diomidis Spinellis - 978-0321166074
Similar to the Knuth book, I couldn’t make much of the actualprogramming content of the book, however I was able to learnabout the Year 2038 problem and understand how it workedin theory. Quite ironically, the book was published a month be-fore the issues I later mentioned. Spinellis is a well respectedacademic in the field of computer science and so I trusted thisas a source.
A.6 www.mail-archive .com 42
a.6 www.mail-archive .com
www.mail-archive.com/[email protected]
It was very hard for me to find academic papers on the 2006
AOLserver issues. However, I was able to find an archive ofemails sent to and from sysadmins about the server issues.These, as primary sources, were extremely interesting to readdespite having to learn a little about server commands in orderto understand the diagnostic questions being asked. While asingle instance of the issue would not have been sufficient tosuggest it was caused by the Unix 2038 problem, the sheer vol-ume of issues seemingly caused by the time-out overflow ledme to conclude that, indeed it was Unix 2038 causing the servercrashes. This taught me the significance and danger which hav-ing arbitrary values could lead to.
a.7 www.information-management.com
www.information-management.com/infodirect/2009_133/downtime_cost-10015855-1.html Last retrieved 23 February 2013
This was probably the least reliable source I used within myresearch. I found it hard to actually find data about how muchdowntime cost companies as very few companies wanted to di-rectly report on it. Whilst this website did not cite its sources,it is, as far as a website can be trusted, fairly reputable in itsfield. As the specific value was not particularly relevant to thepoint in hand beyond being a "big number", I did not considerit worthwhile searching deeper to see where the value camefrom.
a.8 usma
United States’ Metric Association
Data from the USMA was used in generating the map showingwhen the metric system had been adopted in various countries.Their data had been gathered by looking at the legislation ineach of the countries to see when they had legally enforced themetric system. As such, it is an extremely reliable source.
A.9 www.dft.gov.uk 43
a.9 www.dft.gov.uk
www.dft.gov.uk/consultations/open/trafficsignsamendmentregs/annexd.pdf
I did not read the entirety of this source as the majority of itwas of no consequence to my dissertation. It would, however,be very reliable as it is from the Department For Transport.
BA C T I V I T Y L O G
b.1 before october 2012 - research
Before October, my dissertation was going to be based on usingmaths to "solve" games, that is to say find a strategy which isunbeatable using the rule set.
b.1.1 July 2012
I generated a flowchart of the solution to noughts and crosses and,from this, programmed an unbeatable AI using Game MakerStudio. I probably learned more about programming from thisexercise than solving games as it is a rather simplistic game.I researched the solution to four in a row and was disheart-ened to discover that it had required a computer to "solve". Iattempted to learn how the algorithm worked. Aside from afew tricks for simplifying the algorithms it used, the programhad simply brute forced the game and worked backwards frompossible final positions.The same was true for Chequers.I also discovered that chess, despite top computers having beenunbeatable since 2005, has not yet been solved. The stage hasmerely been reached that the computer can think far moremoves ahead than a human and can play endgames perfectly.
b.1.2 August 2012
I was in Ecuador for World Challenge so didn’t get any researchdone.
b.1.3 September 2012
According to my research most games in the normal sense ofthe word are requiring computers to brute force them afterenough simplifications had been applied.I read the first volume of The Art Of Computer Programming toget an idea of how these algorithms to solve games were con-
44
B.2 writing the dissertation 45
structed. It was heavy going so I didn’t understand nearly asmuch as I would have liked. The only games which were trulybeing solved mathematically were those which were mathemat-ical in nature. For example, Conway’s Game Of Life and nim.Reading the solution to nim, made me research binary additionwhich led me to a Numberphile Youtube video.Their video on the dozenal system intrigued me and so I startedresearch into it.
b.2 writing the dissertation
b.2.1 October 2012
Upon reading the arguments put forward on the various DozenalSocieties’ websites, I decided that switching to dozenal was agood idea. Changed the EPQ to looking at Dozenal as my pre-vious title would have been about computer advances insteadof mathsI spent most of the month trying to prove their assertions ratherthan just take them at their word.I wrote my abstract.
b.2.2 November 2012
I read The Heritage of ThalesI wrote the introduction.I realised that using Microsoft Word for a maths based disserta-tion would look horrible.I started to learn how to code in LATEX
b.2.3 December 2012
I realised that my report is massively one sided and I only con-sider the maths side of the issue.I noticed (from an internet argument no less) that in the USA,the metric system still has not been adopted.I started to research metrification and decided it made a goodmodel for how a numeral change would occur.I typed up my proofs in LATEX
B.2 writing the dissertation 46
b.2.4 January 2013
I didn’t do much work for my EPQ as it was the exam season.I was looking up computer viruses our of interest and cameacross the Unix 2038 problem.I tried reading through Code Quality: The Open Source Perspective.I understood enough for the points I needed to make from it.
b.2.5 February 2013
I decided that my initial conclusion was wrong as I had onlyconsidered the theoretical arguments. When looking at the prac-ticalities, however, it seemed that keeping the status quo wasthe only solution.I rewrote my conclusionI redrafted everything I had writtenI started preparing for the presentation
b.2.6 March 2013
I made final preparations for the presentation.I gave the presentation.I put the EPQ together and created a glossary.I wrote my analysis of the process.I handed the EPQ in.
CA N A LY S I S O F T H E E P Q P R O C E S S
The process of writing an EPQ is a long one, which requiredsubstantially more work than I had originally realised. Themost significant thing I have learned from the exercise is theimportance of time management as it is something I have al-ways struggled with.
I have also acquired new skills with respect to analysing andusing sources. Given my A Level subject choices, looking at thereliability of sources is not something I have had much experi-ence in. Within mathematics, something has either been provento be true, or it has not: There is no grey area, it is very muchbinary. In writing this EPQ, I have had to use my grey matterto explore this grey area for the first time and decide whetheror not certain sources are to be trusted.
One of the most beneficial things about the EPQ is probablythe motivation it has given me to actually see research throughto the end. Normally, when I am learning about new extra-curricular concepts, I will find out as much as I can and workthrough problems for a day or so. After this point, I get boredand move on to something else entirely. The EPQ process hasforced me to keep working at a specific topic for a longer pe-riod of time, which is very useful.
The presentation also forced me to work on my speaking skills.I decided to go for a "chalk and talk" method rather than usingpresentation software because I find that, with mathematics, itcan be very difficult to see how one line gets to the next if it iswritten down instantly, rather than a number at a time. The per-formance aspect of the process has increased my confidence inpublic speaking and my ability to answer questions on the spot.
Overall, I would say that the EPQ has been an incredibly fulfill-ing endeavour and I very much enjoyed putting together thisdissertation.
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DG L O S S A RY
d.1 notation
d.1.1 Summation
n∑k=0
f(k) = f(0) + f(1) + f(2) + ... + f(n) (D.1.1)
k,n ∈ Z,n > k (D.1.2)
The sigma notation indicates that a series is being summed.Underneath the sigma refers to the variable which is being usedas a counter (convention dictates k,i or j) and which number itstarts on. The value above sigma is the maximum value forthe counter. Therefore, add together all of the values of f(k)between these limits.
d.1.2 Logarithms
x = by ⇔ y = logb(x) (D.1.3)
Put simply, the logarithm of x, base b, is the power to whichb must be raised to equal x. There exist a few rules which oneshould be aware of.
logb(x) + logb(y) = logb(xy) (D.1.4)y logb(x) = logb(x
y) (D.1.5)
logb x =logc xlogc b
(D.1.6)
Finally, ln(x), the natural logarithm, is defined to be loge(x)where e is Euler’s number
d.1.3 Ceiling
The ceiling function is used to find the smallest integer greaterthan or equal to a value. For example
d5.42e = 6 (D.1.7)
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D.1 notation 49
d.1.4 Floor
The floor function is used to find the largest integer less thanor equal to a value. For example
b5.42c = 5 (D.1.8)
d.1.5 Product
n∏k=0
f(k) = f(0)× f(1)× f(2)× ...× f(n) (D.1.9)
k,n ∈ Z,n > k (D.1.10)
The pi notation indicates that a series is being multiplied. Un-derneath the pi refers to the variable which is being used as acounter (convention dictates k,i or j) and which number it startson. The value above pi is the maximum value for the counter.Therefore, multiply together all of the values of f(k) betweenthese limits.
d.1.6 Set Notation
∈ means that a value belongs to a set.
N is the set of natural numbers.Z is the set of integers.Q is the set of quotients.R is the set of real numbers.
d.1.7 Other
� means Q.E.D.∀ means for all values.≡ means equivalent.∃ means "there exists".: means "such that".⇒ means "implies"⇔ means "implies and is implied by"
D.2 terminology 50
d.2 terminology
d.2.1 Factor
x is a factor of y iff ∃k ∈ : xk = y
d.2.2 Prime
A number is prime iff its only factors are 1 and itself.
d.2.3 Q.E.D.
Quod erat demonstrandum is latin for "which had to be demon-strated" and is used to signify when a proof has been com-pleted.
d.2.4 Numerator
The numerator is the top number in a fraction.
d.2.5 Denominator
The denominator is the bottom number in a fraction.
d.2.6 Sets
Integers are whole numbers.Natural numbers are positive whole numbers.Quotients are numbers which can be written as a fraction withan integer numerator and denominator.Real numbers are those which have no imaginary component.
d.2.7 Modular Arithmetic
Also known as "clock arithmetic", modular arithmetic is wherenumbers are equivalent to each other if you are able to add orsubtract a multiple of the base to get to one from the other. Thatis to say:
x ≡ y (mod n)⇔ x = y+ kn (D.2.1)
where k ∈ Z
D E C L A R AT I O N
I declare the dissertation to be entirely my own work, withother’s ideas and theorems referenced when used
April 2013
Alistair O’Neill