The misunderstood zero digit

The misunderstood zero numeral

Enrico Canuto, Former Faculty, Politecnico di Torino, Torino

July 23, 2020


I will show how zero digit is still misunderstood and careless treated also by people and companies (for instance banks) deeply involved in numerical data analysis and production.

Numeral systems

Numeral systems aim to represent natural numbers with a finite set of digits (numeral symbols). Natural numbers are unlimited as given any natural number n you can increase the number to n+1, where 1 (one) is the small number (the unit). What about zero? Traditionally zero does not belong to natural numbers, and you can find the anomaly in the language of ordinal numbers, say first, second, third, … but not zeroth. You speak of first, second, … floor, but not of zeroth floor (the basement). Indeed, the Chinese basement is the first floor (第一楼)。We speak of the first year and century of the Gregorian Calendar, both BC and AD. Actually, zero numeral is claimed by the most efficient numeral system, the base-b position system, which makes use of the division with remainder (Euclidean division) of natural numbers.

Let b a natural number. We want to express another natural number  n as a multiple of b. To this end, we divide n by b, which allows to write n as

n=q_{1}b+r_{0}\: \; (1)

where quotient q_{1} and remainder r_{0}<b are natural numbers. A pair of singular cases may occur: (i) n is a multiple of b, in which case no remainder exists, (ii) n=r_{0}, in which case no quotient exists.  In both cases, non existence is indicated by the numeral zero, i.e.

\begin{matrix}n=q_{1}b, \: \; r_{0}=0 \\ n=r_{0}, \: \; b_{1}=0 \end{matrix}

This implies that natural numbers must be extended to include zero numeral and the set of digits of a base-b system is \left \{ 0,1,2, ...,b-1 \right \}. In a similar way, also q_{1}>0 can be expressed as in (1), which allows us to rewrite n as

n=\left ( q_{2} b+r_{1}\right )b+r_{0}=q_{2} b^{2}+r_{1}b+r_{0}\: \; (2)

where b^{2}=b\cdot b. The division of the  last quotient q_{\mu -1},\; \mu >1 is performed until the next quotient disappears, i.e. q_{\mu }=0. The final base-b representation of n holds

n\left ( b \right )=r_{\mu -1}b^{\mu -1}+...+r_{2} b^{2}+r_{1}b+r_{0}\: \; (3)

and, by subsuming b, it is simplified  into the right-to-left positional notation of order \mu

\begin{matrix}n=r_{\mu -1}...r_{2}r_{1}r_{0}\: \; (4) \\ r_{\mu -1}>0 \end{matrix}

Clearly, only the leftmost (highest order) remainder r_{\mu -1} must be non zero.

The first known positional numeral system is credited to be the sexagesimal Babylonian, with b=60, which appeared around 2000 BC. Only digits from 1 to 59 had their own cuneiform symbol. Zero was indicated by a space, but only amidst non zero digits, which implied ambiguity between base-60 digits and multiples of 60. For instance ||  could represent 2, 2\cdot 602\cdot 3600, … The decimal system with b=10, now in use, is credited to be invented by Indian mathematicians in the first centuries of the Cristian Era. The simplest numeral system is the base-1 system (binary system) employed by numerical computers, whose set of base digits (bits) is \left \{ 0,1 \right \}. The first binary, programmable mechanical computer, Z1, built by the German inventor K. Zuse in 1938, used a 24-bit binary representation.

Fractional numeral systems

The role of zero numeral may become clearer by introducing rational numbers, i.e. the ratio of two natural numbers f=n/m, where n\geq 0 and m>0.  The last inequality enlightens the singular role of zero, at the border of natural numbers  Let us rewrite equation (1) upon dividing both sides by the base b>0, which provides

n/b=q_{1}+r_{0}/b=q_{1}+r_{0}b^{-1}\: \; (5)

where 0\leq r_{0}/b<1. This suggests that rational numbers less than unit have a base-b representation in terms of the fractions b^{-\nu },\: \; \nu>1. Indeed, consider equation (3) and the fraction n/b^{\mu }<1.  From (3), the fraction representation becomes

n\left ( b \right )/b^{\mu }=r_{\mu -1}b^{-1}+...+r_{2} b^{2-\mu }+r_{1}b^{1-\mu }+r_{0}b^{-\mu }\: \; (6)

which can be simplified into the notation

n\left ( b \right )/b^{\mu }=\varrho\left ( b \right ) =0,r_{\mu -1}...r_{2}r_{1}r_{0}\: \; (7)

Remark. Comma vs dot.  The 22nd General Conference on Weights and Measures (2003) accepted both dot and comma as alternative fractional separators. But the ISO-8601 standard states that comma is preferable, which is our choice in (7).

It can be proved that, given a base b, any (nonnegative) rational number (i.e. any nonnegative fraction) less than unit can be represented as in (6) by the sum of a finite number \mu of fractions r_{k}b^{-\mu +k}<1, k=0,...,\mu -1. The result applies to periodic rational numbers as well, since a rightmost portion of the fractional digits (in the limit all digits after the comma) repeat themselves, as in 7/12=0,5833333...=0,58\bar{3}.

Factorization. Notation (7) shows that any natural number can be factorized as the product of a natural number b^{\mu } (the scale or characteristics) and a rational number \varrho \left ( n \right )<1 possessing  \mu digits (the mantissa or significand), that is

\begin{matrix}n\left ( b \right )=\varrho \left ( b \right )\cdot b^{ \mu } \\ \varrho \left ( b \right )=0,r_{\mu -1}...r_{2}r_{1}r_{0}\: \; \end{matrix},\, \; (8)

The \mu fractional digits (after the comma) are known as the significant digits  of n, r_{\mu -1} being the most significant and r_{0} the least significant. The factorization in (8) separates the scale of n expressed by the characteristics and the numerical accuracy expressed by the mantissa. The factorization concept in (8) dates back to Archimedes (287-212 BC) who used the myriad M=10000 as a scale factor for counting the grains of sand that fit into the universe as required by the Syracusan king Gelo II. The modern scientific notation was eased by the exponential notation b^{\mu } adopted by the French philosopher and mathematician R. Descartes (1596-1650) in the Discours de la Méthode (Discourse on the Method, 1637) and by the decimal point (or comma) adopted by European mathematicians during the same epoch. 

Numerical accuracy. Per se, a natural number is numerically accurate since the cardinality \mu of significant digits is finite and equal to the exponent of the characteristics b^{\mu }. In practice, some of the rightmost digits (the less significant ones) have to be truncated to fit available memory space in computers or accounting tables. In that case, equation (8) is rewritten as

\begin{matrix}n\left ( b,\mu \right )\cong \hat{n}\left ( b,\nu \right )=\varrho \left ( b,\nu \right )\cdot b^{ \mu },\; \nu \leq \mu \\ \varrho \left ( b ,\nu \right )=0,r_{\mu -1}...r_{\mu -\nu +1}r_{\mu -\nu }\: \; \end{matrix}\: \; (9)

and the truncated representation is numerically accurate for \nu =\mu and inaccurate for \nu <\mu. To clearly distinguish between accurate and inaccurate representations, the rightmost zero digits  r_{0}=...=r_{\mu -\nu -1}=0 should not be dropped as they cannot be dropped form original notation (4). For instance, given n\left ( 10 \right )=10140, the truncated notation \hat{n}\left ( 10,4 \right )=10^{5}\cdot 0,1014 should be kept as inaccurate in the least significant digit, although n\left ( 10,5 \right )=\hat{n}\left ( 10,4 \right ). The issue looks subtle and at first sight idle, but becomes prominent when treating irrational numbers. Before passing to them, we define the truncation (numerical) error as follows

\begin{matrix}\tilde{n}\left ( b,\mu \right) = n \left ( b,\mu \right )-\hat{n}\left ( b,\nu \right )=\tilde{\varrho }\left ( b,\nu \right )\cdot b^{ \mu },\; \nu \leq \mu \\ \tilde{\varrho} \left ( b ,\nu \right )=0,00...0r_{\mu -\nu -1}...r_{1}r_{0 }\cdot b^{ \mu }=0,r_{\mu -\nu -1}...r_{1}r_{0 }\cdot b^{ \mu-\nu }\\ \left |\tilde{\varrho} \left ( b ,\nu \right ) \right |<b^{\nu-\mu} \end{matrix}\: \; (9)

Clearly, the truncation error is equal to zero (again, the difference ask for the zero numeral) without truncation, i.e. \nu=\mu.  The fractional (or relative) error

\eta \left ( b,\nu \right )=\tilde{\varrho}\left (b,\nu \right ) /n\left ( b,\mu \right )=b^{-\nu }\eta \left ( b,\nu \right )=\tilde{\varrho}\left (b,\nu \right ) /n\left ( b,\mu \right )=b^{-\nu }\frac{0,r_{\mu -\nu -1}...r_{1}r_{0 }}{0,r_{\mu -\nu -1}...r_{1}r_{0 } }

becomes independent of the scale b^{\mu }. Since n\left ( b,\mu \right )<b^{\mu }, an upper

Irrational number representation

To speak of irrational numbers, we have to extend natural numbers to integers, by considering negative natural numbers  -n, as they are the result of the subtraction n-m, when m>n. Zero numeral separates positive and negative integers.

Irrational numbers cannot be expressed as the ratio of two integer numbers, and they are the result of non arithmetic operations, like the square root. Addition, subtraction, multiplication and division of rational numbers produce other rational numbers.

The first proof of the existence of irrational numbers, or of incommensurable (άλογος) line segments, is credited to the ancient Greek mathematicians, and specifically to the Pythagorean school in the 5th century BC. The proof that \sqrt{2}  is irrational was ascribed to Hippasus of Metapontum (Magna Graecia, see the Appendix) and mentioned by Aristotle in his Prior Analytics (Άναλυικά Πρότερα) work on deductive reasoning (syllogism). The concept of the magnitude of line segments (the precursor of the modern length or norm concept) which was developed in the 4th century BC by Eudoxus of Cnidus (South-west Asia Minor) enabled him to work with abstract geometrical entities, thus avoiding the need of irrational numbers and reversing the emphasis of Pythagoreans on numbers and arithmetic. Eudoxus’ theory laid down the foundations of the abstract geometry, later formulated in an axiomatic way by the influential Euclid’s Elements (Στοιχεϊα, 3rd century BC) .

The set which combines rational and irrational numbers is the set of real numbers, formulated in the XIX century. It was demonstrated that real numbers cannot be counted, in other terms a real number cannot be associated to an integer number, unlike rational numbers that are countable. Thus in the sequel we will refer to real numbers.

Coming back to base-b representations,  a real number r, positive or negative, can be represented as in (8) as the product of sign s=\pm 1, characteristics and mantissa, but the mantissa has an unlimited number of digits

\begin{matrix}r\left ( b \right )=s\cdot \varrho \left ( b \right )\cdot b^{ \mu } \\ \varrho \left ( b \right )=0,r_{\mu -1}...r_{2}r_{1}r_{0}r_{-1}...r_{-k}...\: \; \end{matrix}\, \; (10)

Real number approximation by rational numbers. A careful examination of (10) shows that the fractional factor \varrho \left ( b \right ) has an unlimited but countable set of digits as they are denoted by the integer subscript k. Is there a contradiction with the uncountability of real numbers?

The significant property of rational numbers coherent with the countable digits of \varrho \left ( b \right ) is that any real number can be arbitrarily approximated by rational numbers. In other terms, the absolute value of their difference can be made arbitrarily small. In other terms, given areal r and a rational

Appendix – The proof of Hippasus

Hippasus proved that the ratio  h/l between the length h of the hypotenuse and the length of the legs of an isosceles right triangle cannot be rational. Let us assume that h/l is rational and that both integers have no common factors. by Pythagorean theorem we get h^{2}=2l^{2} and that h must be even. If h is even, we can write h=2y where y is an integer, and in turn l^{2}=2y^{2}. The last identity would imply that l also is even, and that h and l have the factor 2 in common, which contradicts the assumption. Thus, h and l cannot both be integers and their ratio cannot be rational. Indeed, from  Pythagorean theorem we get h/l=\sqrt{2} which is an irrational number. If their ratio is not rational, no small segment exists such that both lengths are multiple of it, in other terms, both hypotenuse and leg are incommensurable.