Number representation

Numbers in the Unary system on a 25-year anniversary card.

Number representation (also number representation ) describes a format for representing a number . The number font for written representation, number words and number names for oral representation are known, but an (analog) number representation using a physical variable such as length or angle is also possible. Numerical representations are not limited by the natural numbers that can be represented in number systems . B. a computer can approximate a real number using a floating point number .

introduction

Depending on the area of application, there are a variety of numerical representations, therefore only a rough overview can be offered in the following. Some figures serve as temporary interim results and therefore have representations optimized for further calculations , other representations have a special format due to their processing history. In addition, other representations are common depending on the number range .

Number as a collection of numerals

It is easy to represent a number by the corresponding number of concrete objects:

Stones, shells, pearls, coins (see Calculus )
Notches on a notch wood
Ticks in a tally sheet
Finger (see also finger arithmetic )

These objects then fulfill the function of a number sign . The numerals of a spoken language can also derive from terms that contained a number, e.g. B. “Sun” for one, “eyes” for two, “animal paws” for four, “hand” for five. By arranging the calculi on the abacus or in the abacus , similar calculi can be assigned different numerical values depending on their position - in this way, arithmetic is supported in simple number systems. Larger numbers can also be represented by distinguishable types of objects if a different value is assigned to each type. Cash allows an amount of money to be represented as a mere collection of coins and banknotes .

Written representations

Natural, whole and rational numbers

The number font is probably the most important means of number representation. In order to represent the natural numbers , written numerals are put together according to the rules of a number system . For the purpose of compact notation and suitability for written arithmetic procedures (e.g. written addition , written multiplication ), various numerals have been superseded or further developed in the course of human history. Today the decimal system with Arabic numerals is predominant .

A preceding sign can also be used to represent negative numbers, i.e. any whole number in total . Rational numbers can be written as fractions , i.e. as pairs of whole numbers. This representation makes it possible to calculate exactly with the four basic arithmetic operations , and the comparison of numbers can also be decided . With the completely shortened fraction there is even a clear representation. Floating numbers enable the notation in the decimal system for place values less than one .

Representations using the example of 2.875
${\ displaystyle {\ begin {aligned} 2 {,} 875 & = 2 \ cdot 10 ^ {0} +8 \ cdot 10 ^ {- 1} +7 \ cdot 10 ^ {- 2} +5 \ cdot 10 ^ { -3} = {\ frac {2875} {1000}} \\ & = {\ frac {5 \ cdot 5 \ cdot 5 \ cdot 23} {2 \ cdot 2 \ cdot 2 \ cdot 5 \ cdot 5 \ cdot 5 }} = {\ frac {23} {8}} \ end {aligned}}}$

Irrational and other numbers

Not every real number can be written down, since there are only countably infinite numbers of finite representations over a finite alphabet (the uncountability of real numbers applies according to the diagonal argument ). Nevertheless, every real number can be represented in the place value system with an infinite number of decimal places (in the decimal system this is an infinite decimal fraction expansion ) - if the written notation of an infinite representation were not naturally impossible. For this reason, reference can only be made to the sections on purely mathematical representations and geometric representations using straight lines.

The fact that no universal written notation is known may be astonishing, in view of the fairly standardized notations today:

By default, some numbers are represented as complex expressions using arithmetic operators (ex. ). ${\ displaystyle {\ sqrt {2}}}$
Some constants are represented by a special symbol (e.g. , ). ${\ displaystyle \ pi}$ ${\ displaystyle e}$
Complex numbers can be represented
- using the imaginary unit (e.g. ), ${\ displaystyle \ mathrm {i}}$ ${\ displaystyle 3 + 2 \ mathrm {i}}$
- in polar form with polar coordinates as . ${\ displaystyle a \ cdot e ^ {\ mathrm {i} \ phi}}$
Physical constants are also represented by symbols (example ). ${\ displaystyle c}$
The scientific notation is a floating point representation to the approximate representation of all real numbers.
Many numbers can also be represented by infinite series , e.g. B.
- ${\ displaystyle \ pi = {\ frac {1} {4}} \ sum _ {k = 0} ^ {\ infty} {\ frac {(-1) ^ {k}} {2k + 1}}}$ ( Leibniz series ).

Linguistic representations

Numerals are as part of speech solid components of the respective language. It is not just a matter of pronouncing the number, nor is it a “copying” of the language. The Chinese numerals are an exception , where the writing corresponds exactly to the pronunciation. For example, 四千八百七十九 stands for 4 1000 + 8 100 + 7 10 + 9 = 4879:

character	四	千	八	百	七	十	九
value	4th	1000	8th	100	7th	10	9
translation	four	thousand	eight	hundred	sieve	umpteen	*nine

This representation strictly follows the principle referred to by Georg Cantor as the law of magnitude order (GGF), so the powers of ten 1000, 100, 10 and 1 are arranged in descending order. However, the translation is not "* seventieth | zig | nine", but "nine | and | seventieth | zig", so the GGF in German is violated by the interchanged naming of units and tens . Similar inconsistencies can be found in almost all languages. Such an inconsistency can also be found in the words " eleven " and " twelve ", not "* one" and "* two thirteen". This is probably a remnant of a duodecimal system .

The powers of ten and thus the orders of magnitude are explicitly mentioned, unlike in the usual number font, where a glance at the number of digits is sufficient (“4879” has four digits, i.e. in the order of 1000). The word formation is much more powerful, however. B. between cardinal (one, two, three) and ordinal numbers (first, second, third) can be distinguished directly. Fractional numbers and thus the representation of rational numbers are made possible by the suffix "-tel", e.g. B. "fourteen thirty-seventh". Decimal other hand, are very true to the script, for example, is 24.193 to "twenty-four point one nine three".

Geometric representations

An analog clock can visually display 12 hours and 60 minutes and seconds each.

Construction of the root 2 on the number line

A “basic concept of real numbers” assumes that every real number corresponds to exactly one point on the seamless number line . Accordingly, all real numbers can be represented on the number line. Furthermore, since the basic arithmetic operations can be constructed geometrically, the construction of the real numbers can be illustrated and justified very well using interval nesting .

Other number ranges, the remainder classes , could be represented according to the principle of number circles . The value of a number then corresponds to the angle. The analog clock known from everyday life is such a circular display, 12 hours and 60 minutes can be read off with clock hands in relation to the 12-hour count .

To represent the complex numbers, the number line is extended to the complex number level .

Representations in computers

Computer arithmetic has always been an integral part of a computer . Due to the finite memory size, however, the number representation options are limited. Another relevant criterion is the speed of the arithmetic operations with regard to the respective number representation in order to achieve a good execution speed of a computer program . For the purpose of optimization, these operations are often calculated in the arithmetic-logic unit , which only accepts binary-coded numbers of a fixed word size .

Due to memory limitation and speed optimization, programmers prefer certain number representations: Integers with a limited range of values stores whole numbers . Rational and real numbers are often replaced by floating point numbers . The limited display options of these data types can lead to arithmetic overflow , rounding errors or similar calculation errors. In order to be able to calculate with larger natural numbers without overflow, modern programming languages support long number arithmetic in addition to the integer data type , with which theoretically any large natural numbers can be represented.

There is also a remedy for the real numbers, although the representation of all real numbers is not possible, since only countable numbers can be coded. The representation of the algebraic numbers allows calculations without rounding errors. However, if transcendent numbers like are to be used, algebraic numbers do not help. Often a number does not even have to be calculated exactly: It is then sufficient to estimate the rounding error, everyday routine in numerical error analysis . This can e.g. B. can be automated by interval arithmetic . In principle, all calculable numbers can also be displayed. ${\ displaystyle \ pi}$

Purely mathematical representations

Some representations are too cumbersome to use outside of math, but allow mathematical elegance and clear reasoning. Other representations, such as the infinite decimal fractions already mentioned, are abstract extensions of the established written representation into infinity .

Known representations of real numbers relate even to abstract objects, so for a Dedekind cut the representation of a partition or subset of the rational numbers is necessary. In an algebraic structure of a number can, in turn, by the links (i.e. , and ) are represented by other numbers within the structure indirectly (an algebraic structure can axiomatically be defined, for. Example, which can real numbers axiomatic as completely Archimedean-arranged body defined become). ${\ displaystyle +}$ ${\ displaystyle \ cdot}$ ${\ displaystyle \ leq}$

Examples

Representation of a natural number as ...
- ... successor of the successor of etc. the zero (corresponds to the Peano axioms ), corresponds to three , ${\ displaystyle 0 + 1 + \ dotsb +1}$ ${\ displaystyle 0 + 1 + 1 + 1}$

Representation of a real number as ...

... Cauchy sequence of rational numbers,
... interval nesting of rational numbers,
... Dedekind cut using a partition of rational numbers (unambiguous),
… Infinite decimal fraction (decimal numbers with an infinite number of places after the decimal point).

General representation ...
- ... regarding links in an algebraic structure ( is the unique element in every ring ). ${\ displaystyle 0}$ ${\ displaystyle x + x = x}$

literature

Georges Ifrah: Universal History of Numbers . Campus Verlag, Frankfurt / New York 1989, ISBN 3-593-34192-1 (French: Histoire Universelle des Chiffres . Translated by Alexander von Platen).
Jürgen Schmidt: Basic knowledge of mathematics . 2nd Edition. Springer, Berlin Heidelberg 2015, ISBN 978-3-662-43545-8 .
Peter Pepper: Basics of Computer Science . Oldenbourg, Munich / Vienna 1992, ISBN 3-486-21153-6 .
Dirk W. Hoffmann: Fundamentals of technical computer science . 4th edition. Hanser, 2014, ISBN 978-3-446-44251-1 , numbers and codes, p. 59-88 , doi : 10.3139 / 9783446442481 .
Hermann Maier: Didactics of the representation of numbers - A workbook for lesson planning . Schöningh, Paderborn 1992, ISBN 3-506-37487-7 .

Individual evidence

↑ Josef Stoer, Roland W. Freund, Ronald HW Hoppe, R. Bulirsch: Numerical Mathematics . 10th edition. tape 1 . Springer, Berlin / Heidelberg / New York 2007, ISBN 978-3-540-45389-5 .
↑ ^a ^b See Ifrah (1989), p. 47 ff.
^ Karl Menninger : Number and number. A cultural history of the number . 2nd Edition. tape 1 . Vandenhoeck & Ruprecht, Göttingen 1958, ISBN 3-525-40701-7 ( Digitale-sammlungen.de ). P. 64.
^ Menninger: number and number. 1958, p. 65.
↑ ^a ^b K. Döhmann: About inconsistencies and anomalies in the linguistic representation of numbers . In: The pyramid . tape 3 , no. 11, 12 . Innsbruck December 1953, p. 233-235 .
^ Friedhelm Padberg / Rainer Dankwerts / Martin Stein: number ranges . Spectrum Academic Publishing House, Heidelberg / Berlin / Oxford 1995, ISBN 3-86025-394-8 . P. 159 ff.
↑ There is no evidence for the use of the word “number circle” in mathematics, in Timo Leuders: Erlebnis Arithmetik . Spectrum, Heidelberg 2012, ISBN 978-3-8274-2414-3 , p. 145 . the term "circular number line" is used instead.
↑ Ehrhard Behrends: Analysis Volume 1 . 6th edition. Springer, Wiesbaden 2015, ISBN 978-3-658-07122-6 , doi : 10.1007 / 978-3-658-07123-3 . Pp. 52-58.

[1] Josef Stoer, Roland W. Freund, Ronald HW Hoppe, R. Bulirsch: Numerical Mathematics . 10th edition. tape 1 . Springer, Berlin / Heidelberg / New York 2007, ISBN 978-3-540-45389-5 .

[Ifrah-Zahlzeiche-Symbole-2] See Ifrah (1989), p. 47 ff.

[3] Karl Menninger : Number and number. A cultural history of the number . 2nd Edition. tape 1 . Vandenhoeck & Ruprecht, Göttingen 1958, ISBN 3-525-40701-7 ( Digitale-sammlungen.de ). P. 64.

[4] Menninger: number and number. 1958, p. 65.

[Sprachliche_Inkonsequenz-5] K. Döhmann: About inconsistencies and anomalies in the linguistic representation of numbers . In: The pyramid . tape 3 , no. 11, 12 . Innsbruck December 1953, p. 233-235 .

[6] Friedhelm Padberg / Rainer Dankwerts / Martin Stein: number ranges . Spectrum Academic Publishing House, Heidelberg / Berlin / Oxford 1995, ISBN 3-86025-394-8 . P. 159 ff.

[7] There is no evidence for the use of the word “number circle” in mathematics, in Timo Leuders: Erlebnis Arithmetik . Spectrum, Heidelberg 2012, ISBN 978-3-8274-2414-3 , p. 145 . the term "circular number line" is used instead.

[8] Ehrhard Behrends: Analysis Volume 1 . 6th edition. Springer, Wiesbaden 2015, ISBN 978-3-658-07122-6 , doi : 10.1007 / 978-3-658-07123-3 . Pp. 52-58.