2. Background

A number [1] is a mathematical object used to count, measure, and label. The different categories, called sets, of numbers are Natural numbers, Whole numbers, Integers, Rational numbers, Real numbers, and Complex numbers.

• Natural Numbers (N), also called positive integers or counting numbers, are the numbers {1, 2, 3, 4, 5, …} [2].
• Whole Numbers (W) is the set of natural numbers, plus zero, i.e., {0, 1, 2, 3, 4, 5, …} [3].
• Integer (Z) is the set of all whole numbers plus all the negatives (or opposites) of the natural numbers, i.e., {… , -2, -1, 0, 1, 2, …} [4].
• Rational Number (Q), is all the fractions where the top and bottom numbers are integers; e.g., 1/2, 3/4, 7/2, -4/3, 4/1. Note: The numerator can be 0, but not the denominator [5].
• Real Numbers (R), also called measuring numbers or measurement numbers, include all numbers that can be represented in decimal notation [6]. This includes fractions written in decimal form e.g., 0.5, 0.75 2.35, -0.073, 0.3333, or 2.142857. It also includes all the irrational numbers such as π, √2 etc. Every real number corresponds to a point on the number line.
• Complex Numbers (C) include real numbers, imaginary numbers, and sums and differences of real and imaginary numbers [7].

Generally, each number set can be expressed as a proper subset of the other (by abuse of notation), because each of these number systems is canonically isomorphic to a proper subset of the other. The resulting hierarchy, for example, allows talking, formally correctly, about real numbers that are rational numbers, and can be expressed symbolically by writing.

A number system is a system of writing for expressing numbers using digits or other symbols [8]. It is the mathematical notation for representing numbers of a given set by using digits or other symbols in a consistent manner. It provides a unique representation to every number and represents the arithmetic [9] and algebraic [10] structure of the figures. It also allows us to operate arithmetic operations like addition, subtraction, division, and exponentials. A number system representation can be positional or non-positional.

In Non-Positional number system, also known as Sign-Value Notation, each symbol represents the same value regardless of its position [11]. In sign-value notation, a corresponding number of symbols represent every natural number. If the symbol / is chosen, for example, then the number seven would be represented as ///////. Tally marks [12]represent one such system still in common use. More useful still are Sign-value notations that employ special abbreviations by introducing different symbols for repetitions of symbols, and/or for certain new values. Very commonly, these values are powers of 10; so for instance, if / stands for one, − for ten and + for 100, then the number 304 can be compactly represented as +++ //// and the number 123 as + − − /// without any need for zero. A sign-value notation, however, fails to represent all the different number categories consistently, and arithmetic operations are generally complicated.

A more elegant representation of a number system is a place-value notation [13], or the Positional Number System. A single symbol, called numerical digit, is used alone (such as "2" or "5"), or in combinations (such as "25"), to represent numbers (such as the number 25). The name "digit" comes from the fact that the ten digits (Latin digitus meaning fingers) of the hands correspond to the ten symbols of the common base 10 numeral system, i.e. the decimal digits. In a positional (sometimes-called digital) number system, a numeral is a sequence of digits, which may be of arbitrary length. Each position in the sequence has a place-value, and each digit has a value. The value of the numeral is computed by multiplying each digit in the sequence by its place-value, and summing the results. Each digit in a positional number system represents an integer. For example, in decimal (base 10) the digit "1" represents the integer one, and in the hexadecimal (base 16) system, the letter "A" represents the number ten. A positional number system has one unique digit for each integer from zero (0) up to, but not including, the radix (base) b of the number system. Thus, in the positional decimal (base 10) system, the numbers 0 to 9 can be expressed using their respective numerals "0" to "9" in the rightmost "units" position. The number 12 can be expressed with the numeral "2" in the units position, and the numeral "1" in the "tens" position, to the left of the "2", while the number 312 can be expressed by three numerals: "3" in the "hundreds" position, "1" in the "tens" position, and "2" in the "units" position.

The numerals used when expressing numerals with digits or symbols can be categorized into two - called the Arithmetic Numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) and the Geometric Numerals (1, 10, 100, 1000, 10000 ...), respectively [14]. The sign-value systems use only the geometric numerals and the positional systems use only the arithmetic numerals. A sign-value system does not need arithmetic numerals because they are made by repetition (except for the Ionic system [15]), and a positional system does not need geometric numerals because they are made by position. However, both arithmetic and geometric numerals may be used in spoken languages.

Arithmetic operations [16] (like addition, subtraction, division, exponentials, etc…) are much easier in positional systems than in the earlier additive ones (sign-value notations); furthermore, additive systems need a large number of different symbols for the different powers of 10; a positional system needs only ten different symbols (assuming that it uses base 10). Presently, the positional decimal system is universally used in human writing. The base 1000 is also used (albeit not universally), by grouping the digits and considering a sequence of three decimal digits as a single digit. This is the meaning of the common notation 1,000,234,567 used for very large numbers. In computers and other electronic devices, the main numeral systems are based on the positional system in base 2 (binary numeral system), with 2 binary digits, 0 and 1. Positional systems obtained by grouping binary digits in three (octal numeral system) or four (hexadecimal numeral system) are also commonly used. The octal (base 8) system uses the digits from "0" through "7". The hexadecimal system uses all the "0" through "9" digits from the decimal (base 10) system, plus the letters "A" through "F", which represent the numbers 10 to 15 respectively. For very large integers, bases 32 or 64 (grouping binary digits by 32 or 64, the length of the machine word) are used.

Not all languages have numeral systems. Specifically, there is not much need for numeral systems among hunter-gatherers who do not engage in commerce. Many languages around the world have no numerals [17] above two to four, or at least did not, before contact with the colonial societies. Speakers of these languages may have no tradition of using the numerals they have for counting. Indeed, several languages from the Amazon [18] have been independently reported to have no specific words for numbers other than 'one’, and some languages of Australia [19] do not have words for quantities above two. Such languages do not have a word class of 'numeral' (called Anumeric). Most languages with both numerals and counting systems use base 8, 10, 12, or 20. It is believed that base 10 number system appears to come from counting one's fingers, base 20 from the fingers and toes, base 8 from counting the spaces between the fingers, and base 12 from counting the knuckles (3 each for the four fingers). Majorities of the traditional number systems are decimal (ancient Latin adjective decem meaning ten) [20]. However, before the positional numeral system that included the number and digit zero, traditional number systems were simply used for commerce and were all represented by the sign-value notation with no zero. With the growth of civilizations and spread of education, the need to represent very large and very small numbers, have made traditional systems superseded by the use of scientific notation, or the positional system.

The first true written positional numeral system is considered to be the Hindu–Arabic numeral system [21]. This system was first used in the 7th century in India, but was not yet in its modern form because the use of the digit zero had not yet been widely accepted. Instead of a zero (0), the digits were sometimes marked with dots to indicate their significance, or a space was used as a placeholder. Even when zero (0) was introduced to the world, it was not treated as a number at that time, but as a "vacant position". Records show that the Ancient Greeks [22] seemed unsure about the status of zero (0) as a number, and asked themselves "how can 'nothing' be something?" leading to interesting philosophical and, by the Medieval period, religious arguments about the nature and existence of zero (0). They even questioned whether 1 was a number.

The first known documented use of zero (0) dates to AD 628, and appeared in the Brāhmasphuṭasiddhānta [23], the main work of the Indian mathematician Brahmagupta [24]. He treated zero (0) as a number and discussed rules governing the use of it. This work considers not only zero, but also negative numbers and the algebraic rules for the elementary operations of arithmetic with such numbers. In some instances, his rules differ from the modern standard, specifically the definition of the value of zero divided by zero as zero. Indian texts used a word śūnya (Sanskrit: शून्य) [25] to refer to the concept of void. In AD 813, a Persian mathematician, Muḥammad ibn Mūsā al-Khwārizmī [26], prepared astronomical tables using Hindu numerals; and about AD 825, he published a book synthesizing Greek and Hindu knowledge of astronomy and contained his own contribution to mathematics including an explanation of the use of zero. This book was later translated into Latin in the 12th century under the title Algoritmi de numero Indorum [27]. This title means "al-Khwarizmi on the Numerals of the Indians". The word "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name, and the word "Algorithm" or "Algorism" started meaning any arithmetic based on decimals [28]. Muhammad ibn Mūsā al-Khwarizmi stated that if no number appears in the place of tens in a calculation, a little circle should be used to "keep the rows". This circle was called ṣifr. In pre-Islamic time, the word ṣifr (Arabic صفر) had the meaning "empty" [29]. The Hindu–Arabic numeral system (base 10) [21] reached Europe in the 11th century via Al-Andalus through Spanish Muslims, the Moors, together with knowledge of astronomy and instruments like the astrolabe [30]. For this reason, the Hindu–Arabic numeral system [21] came to be known in Europe as "Arabic numerals". The Italian mathematician Fibonacci or Leonardo of Pisa (1170–1250), who grew up in North Africa, was instrumental in bringing the system into European mathematics in 1202 [31]. He used the term zephyrum for the Arabic ṣifr. This became zefiro in Italian, and was then contracted to zero in Venetian. The Italian word zefiro was already in existence (meaning "west-wind" from Latin and Greek zephyrus) and may have influenced the spelling when transcribing the Arabic ṣifr. Sifr evolved to mean zero when it was used to translate śūnya from India. By the 13th century, Western Arabic numerals were accepted in European mathematical circles. They began to enter common use in the 15th century. Until the late 15th century, Hindu–Arabic numerals seem to have predominated among mathematicians, while merchants preferred to use the Roman numerals. In the 16th century, they became commonly used in Europe. The first known English use of zero was in 1598. By the end of the 20th century, virtually all non-computerized calculations in the world were done with Arabic numerals, which have replaced native numeral systems in most cultures.

Zero (0) is both a number and the numeric digit used to represent that number in positional number systems. As a digit, it is used as a placeholder in place-value systems. Below are some of its scientific applications.

• The number zero (0) fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures [22]. Zero is the integer immediately preceding 1. Zero is an even number, because it is divisible by 2 with no remainder. Zero (0) is neither positive nor negative. Many definitions include Zero (0) as a natural number, and then the only natural number not to be positive. Zero (0) is a number which quantifies a count or an amount of null size. In most cultures, Zero (0) was identified before the idea of negative things, or quantities less than zero, was accepted. The value, or number, Zero (0) is not the same as the digit zero, used in numeral systems using positional notation. Successive positions of digits have higher weights, so inside a numeral the digit zero is used to skip a position and give appropriate weights to the preceding and following digits. A Zero (0) digit is not always necessary in a positional number system, for example, in the number 02. In some instances, a leading Zero (0) may be used to distinguish a number. The number Zero (0) is the smallest non-negative integer. The natural number following Zero (0) is 1 and no natural number precedes Zero (0). The number Zero (0) may or may not be considered a natural number, but it is an integer, and hence a rational number and a real number (as well as an algebraic number and a complex number). The number Zero (0) is neither positive nor negative and is usually displayed as the central number in a number line. It is neither a prime number nor a composite number. It cannot be a prime because it has an infinite number of factors, and cannot be composite because it cannot be expressed as a product of prime numbers (0 must always be one of the factors). Zero (0) is, however, even (as well as being a multiple of any other integer, rational, or real number).

  The following are some basic (elementary) rules for dealing with the number 0 [22]. These rules apply for any real or complex number x, unless otherwise stated.

  • Addition: x + 0 = 0 + x = x. That is, 0 is an identity element (or neutral element) with respect to addition.
  • Subtraction: x − 0 = x and 0 − x = −x.
  • Multiplication: x • 0 = 0 • x = 0.
  • Division: 0/x = 0, for non-zero x. But x/0 is undefined, because 0 has no multiplicative inverse (no real number multiplied by 0 produces 1), a consequence of the previous rule.
  • Exponentiation: x⁰ = x/x = 1, except that the case x = 0 may be left undefined in some contexts. For all positive real x, 0^x = 0.

  The sum of 0 numbers (the empty sum) is 0, and the product of 0 numbers (the empty product) is 1. The factorial of Zero (0!) is evaluated to 1, as a special case of the empty product.

  In set theory [22], 0 is the cardinality of the empty set: if one does not have any apples, then one has 0 apples. In fact, in certain axiomatic developments of mathematics from set theory, 0 is defined to be the empty set. When this is done, the empty set is the von Neumann cardinal assignment [32] for a set with no elements, which is the empty set. The cardinality function, applied to the empty set, returns the empty set as a value, thereby assigning it 0 elements. Also in set theory, 0 is the lowest ordinal number, corresponding to the empty set viewed as a well-ordered set. In propositional logic [33], 0 may be used to denote the truth value false. In abstract algebra [34], 0 is commonly used to denote a zero element, which is a neutral element for addition (if defined on the structure under consideration) and an absorbing element for multiplication (if defined). In lattice theory [35], 0 may denote the bottom element of a bounded lattice. In category theory [36], 0 is sometimes used to denote an initial object of a category. In recursion theory [37], 0 can be used to denote the Turing degree of the partial computable functions.

  A zero of a function [22] f is a point x in the domain of the function such that f(x) = 0. When there are finitely many zeros these are called the roots of the function. This is related to zeros of a holomorphic function. The zero function (or zero map) on a domain D is the constant function with 0 as its only possible output value, i.e., the function f defined by f(x) = 0 for all x in D. The zero function is the only function that is both even and odd. A particular zero function is a zero morphism in category theory; e.g., a zero map is the identity in the additive group of functions. The determinant on non-invertible square matrices is a zero map. Several branches of mathematics have zero elements, which generalize either the property 0 + x = x, or the property 0 • x = 0, or both.

• The value zero plays a special role for many physical quantities [38]. For some quantities, the zero level is naturally distinguished from all other levels, whereas for others it is more or less arbitrarily chosen. For example, for an absolute temperature [39] (as measured in Kelvin) zero is the lowest possible value (negative temperatures are defined, but negative-temperature systems are not actually colder). This is in contrast to—for example, temperatures on the Celsius scale, where zero is arbitrarily defined to be at the freezing point of water. Measuring sound intensity in decibels or phons, the zero level is arbitrarily set at a reference value—for example, at a value for the threshold of hearing. In physics, the zero-point energy [40] is the lowest possible energy that a quantum mechanical physical system may possess and is the energy of the ground state of the system.

• Zero has been proposed as the atomic number of the theoretical element tetra-neutron [41]. It has been shown that a cluster of four neutrons may be stable enough to be considered an atom in its own right. This would create an element with no protons and no charge on its nucleus. As early as 1926, Andreas von Antropoff [42] coined the term neutronium for a conjectured form of matter made up of neutrons with no protons, which he placed as the chemical element of atomic number zero at the head of his new version of the periodic table [43]. It was subsequently placed as a noble gas in the middle of several spiral representations of the periodic system for classifying the chemical elements.

• The most common practice throughout human history has been to start counting at one, and this is the practice in the early classic computer science programming languages such as FORTRAN [44] and COBOL [45]. However, in the late 1950s, LISP introduced zero-based numbering for arrays while ALGOL, introduced completely flexible basing for array subscripts (allowing any positive, negative, or zero integer as base for array subscripts) [46], and most subsequent programming languages adopted one or other of these positions. For example, the elements of an array are numbered starting from 0 in C [47], so that for an array of n items the sequence of array indices run from 0 to n−1. This permits an array element's location to be calculated by adding the index directly to address of the array, whereas 1-based languages pre-calculate the array's base address to be the position one element before the first. There can be confusion between 0- and 1-based indexing, for example Java's JDBC [48] indexes parameters from 1 although Java itself uses 0-based indexing. In databases, it is possible for a field not to have a value. It is then said to have a null value. For numeric fields it is not the value zero. For text fields this is not blank nor the empty string. The presence of null values leads to three-valued logic. No longer is a condition either true or false, but it can be undetermined. Any computation including a null value delivers a null result. A null pointer is a pointer in a computer program that does not point to any object or function. In C [47], the integer constant 0 is converted into the null pointer at compile time when it appears in a pointer context, and so 0 is a standard way to refer to the null pointer in code. However, the internal representation of the null pointer may be any bit pattern (possibly different values for different data types). In mathematics −0 = +0 = 0; both −0 and +0 represent exactly the same number, i.e., there is no "positive zero" or "negative zero" distinct from zero. However, in some computer hardware signed number representations, zero has two distinct representations, a positive one grouped with the positive numbers and a negative one grouped with the negatives; this kind of dual representation is known as signed zero, with the latter form sometimes called negative zero. These representations include the signed magnitude and one's complement binary integer representations (but not the two's complement binary form used in most modern computers), and most floating point number representations (such as IEEE 754 [49] and IBM S/390 [50] floating point formats). In binary, 0 represents the value for "off", which means no electricity flow. Zero is the value of false in many programming languages. Many APIs and operating systems that require applications to return an integer value as an exit status [51] typically use zero to indicate success and non-zero values to indicate specific error or warning conditions.

Ge’ez [52], which is one of the world's most ancient alphabets and languages, is used in Ethiopia and Eritrea to writing Ge’ez, Tigrigna, Amharic, etc..., languages. It has its own scripting characters, which consist of letters, punctuations and numeric characters. Ge’ez numeric symbols consist of the arithmetic numerals (1, 2, 3, 4, 5, 6, 7, 8, 9) and the geometric numerals (10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 10000), represented as (፩, ፪, ፫, ፬, ፭, ፮, ፯, ፰, ፱) and (፲, ፳, ፴, ፵, ፶, ፷, ፸, ፹, ፺, ፻, ፼), respectively. The ancient civilization, which might date back to Axum [53], only needed a number system for the counting numbers, or a method of counting, and used a base 10 (or decimal) number system, believed to have come from counting with one's fingers, represented by a sign-value notation. Although the concept of "nothing" or "empty" was obvious to the religious people of this civilization, the number and digit zero has no representation in Ge’ez numeric symbols. Representation of counting numbers in Ge’ez is additive using its arithmetic and geometric numerals. However, it gets inconsistent for very large numbers and cannot have all the representations for them. The Ge’ez number system, base 10 sign-value notation with no zero, cannot represent all the different number types and consistently, and even the simplest arithmetic operations are generally complex in sign-value notation. Moreover, there are no Ge’ez representations for decimal numbers. Therefore, the Ge’ez number system cannot represent very large and very small numbers. For this and other reasons, we cannot use Ge’ez numerals in scientific applications, such as business, education and technology. We witnessed the "Millennium Bug" [54] that hit the world, and waited until it was our Millennium 7 years, 8 months, and 11 days later only to realize that we are unable to correctly represent it using the Ge’ez numerals, to even celebrate it! To make the matter even more complicated, we just represented it with a Ge’ez numeral and a Ge’ez word, ፪ሺህ, and posted it in bold for the public to learn from it!!!

Positional number system is believed to have come to Ethiopia not with the advancement of commerce but with the spread of modern education [55], which accounts mainly to the time of Emperor Haile Selassie-I [56]. Zero, the symbol for both the digit and the number, was also introduced along with the positional number system. The modern Ethiopia constitutionally grants its citizens the right to learn in their native languages and transcript using their native symbols (Article 39, section 2 [57]). However, the country uses the standard base 10 positional number system, or the Hindu-Arabic number system, along with the 0 through 9 Arabic numerals. In other words, Arabization [58] of the existing 1-through-9 Ge’ez numeric symbols (፩, ፪, ፫, ፬, ፭, ፮, ፯, ፰, ፱) on an attempt to standardize the national number system. The Hindu-Arabic number system, along with its numeric symbols, is now the official national number system used in education, business and government communications in the country. This is against the constitution, and has negative social, cultural, educational and economic impacts.