Different systems of numeration

 

A numeral is a symbol or group of symbols, or a word in a natural language that represents a number. Numerals differ from numbers just as words differ from the things they refer to. The symbols "11", "eleven" and "XI" are different numerals, all representing the same number. This article attempts to explain the various systems of numerals.

A numeral system (or system of numeration) is a framework where a set of numbers are represented by numerals in a consistent manner. It can be seen as the context that allows the numeral "11" to be interpreted as the binary numeral for three, the decimal numeral for eleven, or other numbers in different bases.

Ideally, a numeral system will:

· Represent a useful set of numbers (e.g. all whole numbers, integers, or real numbers);

· Give every number represented a unique representation (or at least a standard representation);

· Reflect the algebraic and arithmetic structure of the numbers.

For example, the usual decimal representation of whole numbers gives every whole number a unique representation as a finite sequence of digits, with the operations of arithmetic (addition, subtraction, multiplication and division) being present as the standard algorithms of arithmetic. However, when decimal representation is used for the rational or real numbers, the representation is no longer unique: many rational numbers have two numerals, a standard one that terminates, such as 2.31, and another that recurs, such as 2.309999999.... Numerals which terminate have no non-zero digits after a given position. For example, numerals like 2.31 and 2.310 are taken to be the same, except in the experimental sciences, where greater precision is denoted by the trailing zero. Numeral systems are sometimes called number systems, but that name is misleading. Let’s have a look at the types of numeral systems.

Nowadays, the most commonly used system of numerals is known as Hindu-Arabic numerals and two great Indian mathematicians could be given credit for developing them. Aryabhatta of Kusumapura who lived during the 5th century developed the place value notation and Brahmagupta a century later introduced the symbol zero.

The simplest numeral system is the unary numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol ′ is chosen, for example, then the number seven would be represented by ′′′′′′′. The unary system is normally only useful for small numbers. It has some uses in theoretical computer science. Elias gamma coding is commonly used in data compression; it includes a unary part and a binary part.

The unary notation can be abbreviated by introducing different symbols 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 number 123 as + - - ′′′ without any need for zero. This is called sign-value notation. The ancient Egyptian system is of this type, and the Roman system is a modification of this idea.

More useful still are systems which employ special abbreviations for repetitions of symbols; for example, using the first nine letters of our alphabet for these abbreviations, with A standing for "one occurrence ", B "two occurrences", and so on, we could then write C+ D′ for the number 304. The numeral system of English is of this type ("three hundred [and] four"), as are those of virtually all other spoken languages, regardless of what written systems they have adopted.

More elegant is a positional system, also known as place-value notation. Again working in base 10, we use ten different digits 0,..., 9 and use the position of a digit to signify the power of ten that the digit is to be multiplied with, as in 304 = 3×100 + 0×10 + 4×1. Note that zero, which is not needed in the other systems, is of crucial importance here, in order to be able to "skip" a power. The Hindu-Arabic numeral system, borrowed from India, is a positional base 10 system; it is used today throughout the world.

Arithmetic is much easier in positional systems than in the earlier additive ones; furthermore, additive systems have a need for a potentially infinite number of different symbols for the different powers of 10; positional systems need only 10 different symbols (assuming that it uses base 10). The numerals used when writing numbers with digits or symbols can be divided into two types that might be called the arithmetic numerals 0,1,2,3,4,5,6,7,8,9 and the geometric numerals 1,10,100,1000,10000... respectively. The sign-value systems use only the geometric numerals and the positional system use only the arithmetic numerals. The sign-value system does not need arithmetic numerals because they are made by repetition (except for the Ionic system), and the positional system does not need geometric numerals because they are made by position. However, the spoken language uses both arithmetic and geometric numerals.

In certain areas of computer science, a modified base- k positional system is used, called bijective numeration, with digits 1, 2,..., k (k ≥ 1), and zero being represented by the empty string. This establishes a bijection between the set of all such digit-strings and the set of non-negative integers, avoiding the non-uniqueness caused by leading zeros. Bijective base- k numeration is also called k -adic notation, not to be confused with p-adic numbers. Bijective base-1 is the same as unary.

Modern computers use transistors that represent two states with either high or low voltages. We have already learnt that the smallest unit of memory for this binary state is called a bit. Bits are arranged in groups to aid in processing, and to make the binary numbers shorter and more manageable for humans. More recently these groups of bits, such as bytes and words, are sized in multiples of four. Thus base 16 (hexadecimal) is commonly used as shorthand. Base 8 (octal) has also been used for this purpose. A computer does not treat all of its data as numerical. For instance, some of it may be treated as program instructions or data such as text. However, arithmetic and Boolean logic constitute most internal operations. Whole numbers are represented exactly, as integers. Real numbers, allowing fractional values, are usually approximated as floating point numbers. The computer uses different methods to do arithmetic with these two kinds of numbers. A base-8 system (octal) was devised by the Yuki of Northern California, who used the spaces between the fingers to count. The base-10 system (decimal) is the one most commonly used today. It is assumed to have originated because humans have ten fingers. These systems often use a larger superimposed base. Base-12 systems (duodecimal or dozenal) have been popular because multiplication and division are easier than in base-10, with addition just as easy. Base-12 could have originated from the number of knuckles in the four fingers of a hand excluding the thumb, which is used as a pointer in counting. Multiples of 12 have been in common use as English units of resolution in the analog and digital printing world, where 1 point equals 1/72 of an inch and 12 points equal 1 pica, and printer resolutions like 360, 600, 720, 1200 or 1440 dpi (dots per inch) are common. These are combinations of base-12 and base-10 factors: (3×12)×10, 12×(5×10), (6×12)×10, 12×(10×10) and (12×12)×10.

The Maya civilization and other civilizations of Pre-Columbian Mesoamerica used base-20 (vigesimal), possibly originating from the number of a person's fingers and toes. Evidence of base-20 counting systems is also found in the languages of central and western Africa. Danish numerals display a similar base-20 structure. Base 60 (sexagesimal) was used by the Sumerians and their successors in Mesopotamia and survives today in our system of time (hence the division of an hour into 60 minutes and a minute into 60 seconds) and in our system of angular measure (a degree is divided into 60 minutes and a minute is divided into 60 seconds). Base-60 systems are believed to have originated through the merging of base-10 and base-12 systems. Many ancient counting systems use 5 as a primary base, almost surely coming from the number of fingers on a person's hand. Often these systems are supplemented with a secondary base, sometimes ten, sometimes twenty. In some African languages the word for 5 is the same as "hand" or "fist" (Dyola language of Guinea-Bissau, Banda language of Central Africa). Counting continues by adding 1, 2, 3, or 4 to combinations of 5, until the secondary base is reached. In the case of twenty, this word often means "man complete". This system is referred to as quinquavi gesimal system. It is found in many languages of the Sudan region. (see APPENDIX and REFERENCES)

 

5. Find and learn Russian equivalents for the following words and expressions:

 

1) real number	a)
2) Elias gamma code	b)
3) floating point	c)
4) decimal representation	d)
5) data compression	e)
6) rational number	f)
7) pica	g)
8) trailing zero	h)
9) bijective numeration	i)

6. Find and learn English equivalents for the following words and expressions:

 

1) целое число	a)
2) позиционная система (счисления)	b)
3) система знакового обозначения	c)
4) натуральное число	d)
5) пустая строка	e)
6) арифметическое число	f)
7) цифровая строка	g)
8) натуральное число	h)

Bijective numeration

It is any numeral system that establishes a bijection between the set of non-negative integers and the set of finite strings over a finite set of digits. In particular, bijective base- k numeration represents a non-negative integer by using a string of digits from the set {1, 2,..., k } (k ≥ 1) to encode the integer's expansion in powers of k.

Gesimal system

The system which is based on five and twenty.

Integer

Any of the natural numbers (positive or negative) or zero; "an integer is a number that is not a fraction".

Elias gamma code

is a universal code encoding positive integers developed by Peter Elias. It is used most commonly when coding integers whose upper-bound cannot be determined beforehand.

T railing zeros

A sequence of 0s in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow.

Floating point

Describes a system for representing numbers

that would be too large or too small to be represented as integers. Numbers are in general represented approximately to a fixed number of significant digits and scaled using an exponent. The base for the scaling is normally 2, 10 or 16. The typical number that can be represented exactly is of the form: significant digits × base^exponent

The term floating point refers to the fact that the radix point (decimal point, or, more commonly in computers, binary point) can "float"; that is, it can be placed anywhere relative to the significant digits of the number. This position is indicated separately in the internal representation, and floating-point representation can thus be thought of as a computer realization of scientific notation. Over the years, several different floating-point representations have been used in computers; however, for the last ten years the most commonly encountered representation is that defined by the IEEE 754 Standard. The speed of floating-point operations is an important measure of performance for computers in many application domains. It is measured in FLOPS.

FLOPS (or flops or flop/s)

An acronym meaning FL oating point OP erations per S econd. The FLOPS is a measure of a computer's performance, especially in fields of scientific calculations that make heavy use of floating point calculations, similar to the older, simpler, instructions per second. Since the final S stands for "second", conservative speakers consider "FLOPS" as both the singular and plural of the term, although the singular "FLOP" is frequently encountered.

! Study the following: The papers were late and the post was too.

If we want to say that people, things, actions or events are similar, we can use as or like; so, neither do I and similar structures; or adverbs such as too, also, and as well.

To say that they are identical, we can use  the same (as).

To say that people, things etc are equal in a particular way, we often use the structure as (much/many)…as.

To say that people, things etc are unequal in a particular way, we can use comparative adjectives and verbs, or  more (…than) with adjectives, adverbs, verbs or nouns.

To say which one of a group is outstanding in a particular way, we can use  most.

We can use double comparatives to say that something is changing. …er and …er/more and more….

! Study the following: The more illustrated it is, the more I like it.

We can use comparatives with  the +comparative expression + subject + verb, the… to say that things change or vary together, or that two variable quantities are systematically related.

! Study the following: This experiment made it all the more important to continue the research.

Another use of the meaning ‘by that much’ is in  all/any/none the +comparative. This structure can be used only to express abstract ideas when we say something is or should be ‘more…’

! Study the following: It was  ten times more difficult  than we expected.

Instead of  three/four etc  times as much, we can use three/four etc times + comparative.

10. Find, underline and translate the other examples in the text.