aa r X i v : . [ m a t h . HO ] M a y ON THE LEXICOGRAPHIC REPRESENTATION OF NUMBERS
VINCENZO MANCA
UNIVERSITY OF VERONA - ITALY
Abstract
It is proven that, contrarily to the common belief, the notion of zero is notnecessary for having positional representations of numbers. Namely, for any positive integer k , a positional representation with the symbols for 1 , , . . . , k is given that retains allthe essential properties of the usual positional representation of base k (over symbols for0 , , . . . , k − Keywords
Positional representation, Zero, Lexicographic ordering1.
Number Positional Representations
The positional representation of numbers was an epochal discovery that is a landmarkin the history of mathematics and science. Il is based on a notational system of Hinduorigin that, via the Arabian mathematics, was imported in Europe with the famous book
Liber Abaci published in 1202 by Leonardo Fibonacci (Leonardo from Pisa). This numberrepresentation was the seed of a new mathematical perspective, algebraic and algorithmic,that surely contributed to the important discoveries of Renaissance mathematics, fromwhich modern mathematics stemmed.An important aspect of positional representation was the notion of zero (from an arabianroot common to the noun zephyr , a gentle wind whose origin is hardly determinable). Anynatural number can be univocally represented as sum of powers of any base k > k . From this unicity it easily follows that, given an alphabet of k symbols (standing for 0 , , , , k − k ,and the symbol in that position encodes a multiplicative coefficient, in such a way that the UNIVERSITY OF VERONA - ITALY sum of these multiplied powers yields the number. When a (symbol for) zero occurs, thenthe corresponding power gives a null contribution to the overall sum providing the number.Classical texts in number theory and history of mathematics usually connect intrinsicallythe notion of zero with the positional notation [10, 3]. For example, in [10] it is writtenthat “The only complication which the positional notation involves lies in the necessity ofintroducing a zero symbol to express a void or missing class; for instance, 204 is differentfrom 24”. The same “necessity” is expressed by [3], and certainly the attitude towardthis conviction was enforced by the decimal notation and the related algorithms for basicoperations (using alignment, carry, positional shift, and decimal point), after the seminalworks of Fran¸cois Vi`ete and Simon Stevin, in sixteenth century [2]. However, in [11], in thecontext of the historical analysis of number representation systems, some remarks clearlysuggest that positional notations were present in some ancient number systems. One ofthem is the Greek one, especially in the sexagesimal notation used in Ptolemy’s Almagest,where some kind positional concepts, made possible the complex computations required inthe contexts of geometrical and astronomical problems. In [7] an illuminating synthesis ofthe historical development of number representation is reported. In passing, we recall that,in a geographically far culture, and chronologically antecedent, Maya developed a positionalnumber system, with base twenty, also having a well defined notion of zero [5].In this note we show that the connection between zero and the positional notation isnot necessary. Namely, we prove that the usual lexicographic representation of strings,in a given alphabet, is a positional number system, but without any symbol for zero.The essence of positional notations is the notion of cipher sequence, where ciphers havepositional values corresponding to the powers of the base. The main advantage of thisnotational mechanism is that arithmetical operations (firstly sum and multiplication) canbe calculated by operation tables , that is, by only knowing the values of the given op-eration when its arguments range over a finite set of numbers (whose cardinality is equalto the value of base). We will show that just this mechanism is behind the lexicographicrepresentation by strings, where the number n is represented by the string in position n according to the lexicographic enumeration. In [1] the possibility of a zeroless positionalsystem, and its connection with the string ordering, was already investigated. However,the analysis developed in [1] is very different from the one we present in this paper, and themain properties of a lexicographic representation are determined in a completely differentway. Moreover, the representation algorithms are here defined in a more direct manner(without passing through the classical positional systems as in [1]), and the correctnessproofs are formally established (in [1] they are informally motivated). Finally, here, trans-lation functions between lexicographic and classical positional representations are formallydefined. It is interesting to remark that the main idea of the present paper arose duringthe search for unconventional genome representation, where DNA sequences are viewedas numbers (in base four). In that context the following question was posed: Define afunction that assigns to any string, over a given alphabet, its order in the lexicographicenumeration . The solution to this question is given by Equation (1) of the “Lexicographic
N THE LEXICOGRAPHIC REPRESENTATION OF NUMBERS 3
A, C, G, T, AA, AC, AG, AT, CA, CC, CG, CT, GA, GC, GG, GT, TA,TC, TG, TT, AAA, AAC, AAG, AAT, ACA, ACC, ACG, ACT, AGA,AGC, AGG, AGT, ATA, ATC, ATG, ATT, CAA, CAC, CAG, CAT.
Table 1.
The first 40 strings over { A, C, G, T } in the lexicographic order.Theorem” of Section 2. From the recursive formula (1), the iterative formula of Equa-tion (2) easily follows, which directly provides a number positional representation with theproperties discussed along the paper.2. The Lexicographic Number Representation
Let S = { a , a , . . . , a k } a finite alphabet of symbols where we define an enumerationfunction ω of its symbols such that ω ( a i ) = i , for 1 ≤ i ≤ k . We denote by S ∗ the set ofstrings, that is, of finite sequences over S . In this paper we denote by | S | the cardinalityof the set S , while | α | denotes the length of string α , moreover string concatenation isexpressed by juxtaposition, or by ∗ when it avoids confusion, and α ( i ) will denote thesymbol of α in position i (the first position is 1 and the last equals the length of thestring).We can define a strict order over S ∗ by the following conditions: α < β if | α | < | β | α < β if | α | = | β | and ∃ j : ∀ i < j : α ( i ) = β ( i ) and ω ( α ( j )) < ω ( β ( j )) . The second condition is the usual criterion adopted in the enumeration of the items in alexicon. The first one is added to the second, in order to obtain a linear ordering over (theinfinite set of) all strings over the alphabet. For example, given the (ordered) alphabet { A < C < G < T } (typical of genomes), then Table 1 gives the first 40 strings over thisalphabet enumerated in the lexicographic order: Theorem 1 (The lexicographic Theorem) . Let S be an (ordered) finite alphabet. If x ∈ S ,and α ∈ S ∗ , then Equation (1) inductively defines the enumeration number ω S ( α ) of string xα , shortly indicated by ω ( α ) , according to the lexicographic ordering on S ∗ ( ω ( λ ) = 0 , if λ is the empty string): (1) ω ( xα ) = ω ( x ) | S | | α | + ω ( α ) moreover: (2) ω ( α ) = | α | X i =1 ω ( α ( i )) | S | | α |− i Proof . In fact, let us denote by ω = ( α ) the position of α when we lexicographically enu-merate the strings of length α ( a | α | is the first string in position 1, if a is the first symbol VINCENZO MANCA
UNIVERSITY OF VERONA - ITALY of S ). In order to evaluate the position of xα in the lexicographic enumeration, we observethat:i) xα follows all the strings of length ≤ | α | , which are in number: | α | X i =1 | S | i ii) and xα follows the strings of length | α | + 1 beginning with any symbol y such that ω ( y ) < ω ( x ), which are in number: [ ω ( x ) − | S | | α | iii) between the first string of length | α | + 1 that begins with x and xα (extremes included)there are ω = ( α ) strings. Therefore, the position of xα in the lexicographic enumerationis: ω ( xα ) = | α | X i =1 | S | i + [ ω ( x ) − | S | | α | + ω = ( α )that can be rewritten as: | α |− X i =1 | S | i + | S | | α | + [ ω ( x ) − | S | | α | + ω = ( α )but, summing the two terms in the middle of the sum above, we get the term ω ( x ) | S | | α | ,while summing the other two (extremal) terms we get ω ( α ). This concludes the proof ofEquation (1) (and of the inductive definition of ω ). The proof of Equation (2) followsdirectly, by iteratively applying Equation (1). If k is the cardinality of S , then Equation(2) can be rewritten as follows: ω ( α ) = | α | X i =1 ω ( α ( i )) k | α |− i Q ED.An interesting simple consequence of the theorem above is the following corollary, due tothe fact that, being the lexicographic order a total ordering over all the strings, every stringrepresents the number of its lexicographic enumeration.
Corollary 2.
Given a number k > , any non-null natural number n is univocally repre-sentable by a linear combination of powers k i , c k + c k + . . . + c j k j where < c i ≤ k for all ≤ i ≤ j . We remark that according to the corollary above, lexicographic number representation re-sults a full positional representation because all the powers of k less than a maximum valuegive a positive contribution to the sum representing the number, while classical representa-tions admit the possibility of having empty contribution (expressed by zero ciphers). The N THE LEXICOGRAPHIC REPRESENTATION OF NUMBERS 5 + A C G TA C G T AAC G T AA ACG T AA AC AGT AA AC AG AT
Table 2.
Addition table of number lexicographic representation (four symbols). ω ( CAT ) = 40 ω ( GAT T ) = 228
CAT + GAT T = GT CTω ( GT CT ) = 40 + 228 = 268
Table 3.
The additivity of the lexicographic order.lexicographic representation is the most economical way to represent the set of the first m numbers. In other words, if the cost of representing these numbers is defined (with respectto a base k ) as the sum of the lengths of strings representing them, then no representationexists that has a lower representation cost.Table 2 defines operation + for symbols A, C, G, T that is coherent with the lexicographicordering, as illustrated by Table 3. In fact, by using Table 2 we can coherently add strings,in perfect accordance with their lexicographic order.In general the additive table of a lexicographic system can be obtained in accordance toto the following proposition that follows directly from the previous discussion. In thefollowing, when we want to avoid confusion, the symbols of a k -lexicographic represen-tation (in base k ) are denoted by [1] , [2] , . . . [ k ] (with numbers in decimal notation insidebrackets), while symbols of a classical k -positional representation (with 0) are denoted by[0] , [1] , . . . [ k − Proposition 3.
Given a lexicographic system of base k , its additive table L k (+) can beobtained from the additive table P k (+) of the classical positional table: i) by removing allthe elements corresponding to sums [ j ] + [0] or [0] + [ j ] , for j < k ; ii) by replacing in P k (+) any sum result [1][0] by [ k ] ; and iii) by adding as results of the sums [ j ] + [ k ] or [ k ] + [ j ] ,for j ≤ k , the elements [1][ j ] . The following algorithm provides the string that lexicographically represents the number n over an alphabet of k symbols. It provides the inverse information with respect to thetheorem above. In fact, given a positive integer n , it solves the equation of unknown α : ω ( α ) = n. VINCENZO MANCA
UNIVERSITY OF VERONA - ITALY
Algorithm 4 (Lexicographic Number Representation) . For any positive integer h , let maxlex ( k, h ) = h X i =1 k i minlex ( k, h ) = h X i =1 k i − . The first number expresses the greatest number lexicographically representable by strings oflength h , while the second number expresses the smallest number lexicographically repre-sentable by strings of length h . Then, given a positive integer n , if minlex ( k, h ) ≤ n ≤ maxlex ( k, h ) then it is lexicographically representable by a string of length h , and we denote this value h by lex ( k, n ) , or simply by lex ( n ) when k is implicitly understood. In this case, n is univo-cally lexicographically represented by the string σ k ( n ) (over { [1] , [2] , . . . , [ k ] } ), according tothe following recursive procedure (here ∗ denotes string concatenation and · the product): σ k ( n ) = [ n ] if n ≤ kσ k ( n ) = [ m ] ∗ σ k ( n − m · k lex ( n ) − ) otherwisewhere m = max { q ≤ k | n − ( q · k lex ( n ) − ) ≥ minlex ( k, lex ( n ) − } . Q ED.
For example, let us consider 37 and 36, of course lex (4 ,
36) = lex (4 ,
37) = 3, but 37 islexicographically represented (in the base 4) by:37 = 2 · + 1 · + 1 · because 37 − · = 5 and 5 ≥ minlex (4 , · + 4 · + 4 · because 36 − · = 4 and 4 < minlex (4 , Corollary 5.
Given a positive integer n , the previous algorithm correctly provides thestring σ k ( n ) , representing n in the lexicographic system of base k , such that: ω ( σ k ( n )) = n. Proof . The asserted correctness easily follows, by induction, from the definition of lex , minlex , maxlex , and the recursive procedure defining σ k ( n ). Q ED.
N THE LEXICOGRAPHIC REPRESENTATION OF NUMBERS 7 × Table 4.
The multiplication table of the lexicographic representation inbase ten, where the first nine symbols 1 , , , , , , , , X denotes ten (zero is notpresent).The correctness of a number representation ρ implies that any (binary) operation ⊙ onnumbers, when performed on number representations, has to satisfy the morphism condi-tion (in the algebraic sense), that is, ρ has to commute with the operations (on numbersand representations, for the sake of simplicity both denoted by ⊙ ):(3) ρ ( n ) ⊙ ρ ( m ) = ρ ( n ⊙ m ) . Multiplication in lexicographic representation can be computed with the usual algorithmof classical positional systems, by using the multiplication Table 4. For example, themultiplication 37 × X = 147 X can be obtained with the usual multiplication algorithmof the positional representation (147 X corresponds to 1480 in the usual decimal system,coherently to the correspondence of (37 , X ) to the decimal pair (37 ,
40) and to the decimalmultiplication 37 ×
40 = 1480.Analogously to the additive table, the multiplicative table of a lexicographic system canbe obtained in accordance to the following proposition.
Proposition 6.
Given a lexicographic system of base k , its multiplicative table L k ( × ) can be obtained from the multiplicative table P k ( × ) of the classical positional table i) byremoving all the elements corresponding to multiplications [ j ] × [0] or [0] × [ j ] , for j < k ; ii)by replacing in P k ( × ) any multiplication result [1][0] by [ k ] , and any multiplication result [ j ][0] by [ j − k ] ; iii) by adding as results of the multiplications [ j ] × [ k ] or [ k ] × [ j ] , for j ≤ k , the elements [ j − k ] . Corollary 7.
Given a positive integer n , in the lexicographic system of base k (symbols [1] , [2] , . . . [ k ] ) the result of multiplication σ k ( n ) × [ k ] is represented by σ k ( n − k ] ( σ k (0) is the empty string). VINCENZO MANCA
UNIVERSITY OF VERONA - ITALY × X =4 2 2 X X X Table 5.
A multiplication in the lexicographic representation of baseten. In the corresponding zero positional system the same multiplica-tion becomes 423 ×
90, which usually is performed by adding a final 0 to423 × X X (and viceversa ). It seems that the usual multiplication is much sim-pler than the lexicographic one. However, also in the lexicographic systemit is possible to get the result in a shorter way. In fact, the multiplication423 × X can be obtained, according to Corollary 7, by appending X to thepredecessor of 423, getting 422 X , then 423 × Lexicographic and ZeroPositional representations
Rules can be established that transform a lexicographic representation, in a given base,into a classical (with zero) positional representation in the same base and vice versa . Letus denote by δ k ( n ) the k -positional representation with zero of the number n in base k ,while σ k ( n ) is the k -lexicographic representation of n . Algorithm 8 (Conversion between Lexicographic and ZeroPositional representations) . The k -positional (with zero) and k -lexicographic representations are mutually translated,in a 1-to-1 way, by the functions θ → k and θ k → computed with the following algorithm.Given a string α that is a positional representation with zero in base k , and that does notstart with the symbol [0] (apart the string [0] that is intended to have order number zero),then factorise α as a concatenation of substrings (some of α i , γ i can be empty strings): α γ α γ . . . α h − γ h α h where α factors are 0-free and γ factors have the form [ j ][0] i with j > and i > . Now,let: N THE LEXICOGRAPHIC REPRESENTATION OF NUMBERS 9 γ ∗ i = ([ j ][0] i − − [1])[ k ] where minus operation is computed in the k -positional representation with zero. Then: θ → k ( α ) = α γ ∗ α γ ∗ . . . α h − γ ∗ h α h . Analogously, given a string β that is a lexicographic representation in base k , factorise β as a concatenation of substrings (some of β i , η i can be empty strings): β η α η . . . β g − η g β g where β factors are [k]-free, and η factors have the form [ j ][ k ] i with j < k and i > . Now,let: η oi = [ j + 1][0] i where in the case that j + 1 = k the cipher [ k ] has to be replaced by [1][0] . Then: θ k → ( β ) = β η o β η o . . . β g − η og β g . Q ED.The following proposition states the correctness of the translation functions θ → k and θ k → defined by the algorithm above. Proposition 9. θ → k ( δ k ( n )) = σ k ( n ) θ k → ( σ k ( n )) = δ k ( n ) UNIVERSITY OF VERONA - ITALY
Proof . Firstly, we observe that if we order the alphabet { [0] , [1] , . . . , [ k − } with thenatural order (where [0] is the minimum symbol), then the number represented by a string α is the enumeration order of this string, according to the lexicographic ordering (restrictedto the strings that do not start with the symbol [0]). Let us denote by ω ( α ) the ordernumber of α in this ordering. Then, the asserted equations are equivalent to the followingones. In other words, the translations between the two representations preserve theirinterpretations as numbers: ω ( θ → k ( δ k ( n ))) = nω ( θ k → ( σ k ( n ))) = n. In fact, the translation rules of the algorithm above guarantee that translations preserve thenumerical value expressed by the strings, because they essentially express [ k ] as [1][0] and vice versa , but at same time, when this substitution is applied, a carry cipher [1] has to bepropagated (in addition or in subtraction, respectively) to the ciphers on the left of the po-sition where substitution was performed. Q ED.For example (here base 10 is denoted by X ): θ X → (2 X X
5) = 301005 .θ → X (301005) = 2 X X θ X → (2 XXX XX
5) = 300010005 θ → X (300010005) = 2 XXX XX . The proof of the previous proposition puts in evidence that classical positional representa-tions can be seen as a restricted form of lexicographic representation (limited to a subsetof strings over the k -base alphabet). Corollary 10.
Given a base k > , any number has, in this base, a unique positionalrepresentation with zero. Proof . The unicity of lexicographic representation is obvious (see Corollary 2), therefore,the unicity of positional representation with zero follows immediately from it, via the trans-lation θ k → . Q ED.A final remark will be useful to the discussion of the conclusive section. Even when thebase of a positional system is a number greater than 10 (for example, 20 or 60), multi-plication algorithms based on positional representation can be very efficient. In fact, byusing some strategy of distributed carry, as the lattice (or sieve) multiplication method,originally pioneered by Islamic mathematicians, and described by Fibonacci in his
LiberAbaci , and using only a subset of the multiplication table (and/or other tricks), even com-plex multiplications can be easily developed. Tables 6 and 7 give two examples, where a
N THE LEXICOGRAPHIC REPRESENTATION OF NUMBERS 11 ×
35 =(5 · · · · · · ′′′′ ′′ ′ (3 ·
2) 6 (3 · ′′′ (3 ·
2) 1 ′′ (3 · ′′′′ ′′′ ′
56 6 56 1 62 1 31 4 9 4 5
Table 6.
A multiplication in decimal representation, developed by amethod of distributed carry , where only multiplications by 2 and 5 are used.In this method, when a multiplication provides a carry, then it is appendedto the column on the left. Multiplications by ciphers different from 2 or 5are reduced to sums of them (3 · · · UNIVERSITY OF VERONA - ITALY [7][7] × [35] =([35] · [7])([35] · [7])([10] · [7])([10] · [7])([10] · [7])([10] · [7])([10] · [7])([10] · [7])([5] · [7])([5] · [7])[10][10][10][10][10][10][35][35][1] [1][1] [1][1] [1][4] [9] [5] Table 7.
The same multiplication of Table 6 realized in sexagesimal rep-resentation, with the method of distributed carry , and where only mul-tiplication by 10 and 5 are used (if [ j ] is the j -sexagesimal cipher, then427 = [7][7] ,
35 = [35] , Almagest ) is only matter of “syntactic sugar”.4.
Conclusions
The French word for computer is ordinateur . Certainly, it relates to the fundamentalarithmetic ordering generated from zero by means of successor operation: 0 , , , . . . asbasis of any numerical calculus. From successor all the operations come via iteration: sumsare successor iterations, multiplications are sum iterations, exponentials are multiplicationiterations, differences iterate the predecessor operation (the inverse of successor ordering), N THE LEXICOGRAPHIC REPRESENTATION OF NUMBERS 13 divisions iterate difference, and so on. What we discover from the previous discussion is afurther intriguing connection between calculus and ordering. In fact, string orderings areintrinsically related to number positional representations, which provide efficient methodsfor computing the arithmetical operations.In [6] the fundamental role of positional representation in the process of understandingnumbers is explained by many specific examples and analyses. For instance, the existenceof irrational numbers, a masterpiece of Greek mathematics, becomes a simple corollaryof sequence representations of numbers. In fact, it is easy to prove that the division oftwo positive integers, in positional representation, provides a finite sequence of ciphers oran infinite sequence of ciphers that is periodic (it is a consequence of simple combina-torial arguments). Then, any rule generating an infinite sequence of ciphers that is notperiodic has to denote a number that cannot be rational. More sophisticated examplesinvolve results about real and transcendental numbers [6]. It is also worthwhile to recallhere that the famous Turing’s paper of 1936 [12], which was the starting point of the the-ory of computability, was aimed at providing a definition of computable real number , asalgorithmically generated infinite sequence of ciphers.The kind of positional representation that is inherent to the lexicographic notation couldbe the basis for explaining some intriguing aspects of ancient mathematics. In fact, weknow that, even before Greek mathematics, in ancient civilisations, complex calculationswere developed (Babylonians were able to manage, in base 60, numerical algorithms relatedto complex astronomical problems). This ability is surely impossible without an efficientnumber representation.In [11] the solution is reported that was given by Fibonacci to the cubic equation x +2 x + 10 x = 20 in sexagesimal notation (with an error only to the sixth fractional sex-agesimal cipher (60 − ). This reveals an amazing mathematical mastery of computationswith sexagesimal fractions, additions, multiplication, extraction of square roots, and so on.However, sexagesimal tradition goes back to the Almagest of Ptolemy, composed aboutA.D. 150, within Greek culture, where tables of chords (essentially sines) are given forevery (1 / o from 0 o to 180 o , with sexagesimal primes and seconds, which is equivalentto an approximation greater than three decimals fractions (for instance, sin 1 = 1 ′ ′′ ′′′ ).Ptolemy never showed numerical computations, therefore we do not know to what extenthe used positional principles in computing, but according to our previous analysis, we knowthat they are independent of zero. Moreover, as it is remarked in [11], Ptolemy uses sym-bol o in the first position as indication of angle degree, but in the following positions (forprimes, seconds, and so on) symbol o is used to denote the absence of any value for thatsexagesimal position, that is, what now we call zero. In conclusion, also this confirms thatsexagesimal notation is a positional system, within which arithmetic efficient algorithmscan be available, with the full computational power of positional notation, even withoutany explicit notion of zero. In passing, with Ptolemy, Greek mathematics incorporatedconcepts coming from oriental cultures (especially Babylonian astronomy), and, at thattime, they were something well known and established. Probably, the positions of the UNIVERSITY OF VERONA - ITALY sexagesimal system are related to the notion of time periods, as suggested by astronomi-cal observations (already Archimedes, investigating on the representation of big numbers,elaborated a method of number representation based on periods).More details on the historical aspect are beyond the aims of our investigation (see [7, 5, 4]for deeper analyses), however, surely positional systems were related to the abacus whereciphers are expressed, in some way, by the number of some items in each compartments(compartments play the role of positions of symbols). If we consider a sort of “altimetric”variant of the abacus, we can suggest a strict relationship between positional representationwith zero and lexicographic positional representation. In fact, let us encode ciphers with theheight of some object, say a rod, in a vertical compartment. In this way, zero is representedby the flat position of the rod, while 9 is the highest position, at some distance from the top,which is a forbidden position (cipher positions are located at equally increasing levels). Thisis a “zero altimetric abacus”. If otherwise, flat position is forbidden, but at same time, thetop position is the highest possible one, we pass from the zero abacus to a zeroless abacus,and all the properties that we investigated in the paper easily translate in the different,but related, ways of positioning rods in the two abaci.In the Middle Ages, positional (decimal) notation reached maturity and a decimal mark(a point or a comma) appeared in order to extend the decimal notion to fractions. Inthis way, following a different track, a denotation for zero naturally could arise. In fact, ifthe decimal mark is a point separating the integer from the proper fractional part, then adecimal point alone (without preceding or following ciphers) is a denotation of zero.So far we discussed some historical implications of our analysis about positional systems(with and without zero). However, this analysis is not only relevant for the past math-ematics. In fact, some aspects of number representation structures and algorithms arerelated to emergent aspects in discrete mathematics. Firstly, the algorithm outlined inTable 6 can be easily expressed as a particular membrane system (or P system) in thesense of [9], where columns are membranes and the manipulation rules are transformationand movement of objects between membranes. In [8] (Chapter 2) an example is givenof an abacus as a membrane system following the same idea, the multiplication methodof Table 6 provides an improvement of the system in [8], because computation can fullybenefit from the distributed mechanism of membrane computing. Moreover, the interestin string enumeration, as suggested by the examples of Section 1, is related to the mathe-matical analysis of genomes. They can be viewed as strings over the four letters
A, C, G, T .When we fix a representation system of numbers, by means of sequences over the genomicalphabet, then genomes become numbers of gigantic sizes, therefore the methods of string(and number) representations have a direct impact for the investigations that aim at find-ing unconventional ways of considering genomes, and new perspectives to their study andunderstanding.
N THE LEXICOGRAPHIC REPRESENTATION OF NUMBERS 15