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Additive Inverse

Additive inverse

The additive inverse, or opposite, of a number n is the number which, when added to n, yields zero. The additive inverse of n is denoted −n. For example:
- The additive inverse of 7 is −7, because 7 + (−7) = 0;
- The additive inverse of −0.3 is 0.3, because −0.3 + 0.3 = 0. Thus by the last example, −(−0.3) = 0.3. The additive inverse of a number is its inverse element under the binary operation of addition. It can be calculated using multiplication by −1; that is, −n = −1 × n. Types of numbers with additive inverses include:
- Integers;
- Rational numbers;
- Real numbers;
- Complex numbers. Types of numbers without additive inverses (of the same type) include:
- Natural numbers;
- Cardinal numbers;
- Ordinal numbers. But note that we can construct the integers out of the natural numbers by formally including additive inverses. Thus we can say that natural numbers do have additive inverses, but because these additive inverses are not themselves natural numbers, the set of natural numbers is not closed under taking additive inverses.

General definition

The notation '+' is reserved for commutative binary operations, i.e. such that x + y = y + x, for all x,y. If such an operation admits a neutral element o (such that x + o (= o + x) = x for all x), then this element is unique (o' = o' + o = o). If then, for a given x, there exists x' such that x + x' (= x' + x) = o, then x' is called an additive inverse of x. If '+' is associative ((x+y)+z = x+(y+z) for all x,y,z), then an additive inverse is unique ( x" = x" + o = x" + (x + x') = (x" + x) + x' = o + x' = x' ) and denoted by (– x), and one can write x – y instead of x + (– y).

Other examples

All the following examples are in fact abelian groups:
- addition of real valued functions: here, the additive inverse of a function f is the function –f defined by (– f)(x) = – f(x), for all x, such that f + (–f) = o, the null function (constantly equal to zero, for all arguments).
- more generally, what precedes applies to all functions with values in an abelian group ('zero' meaning then the neutral element of this group): :
- complex valued functions, :
- vector space valued functions (not necessarily linear),
- sequences, matrices and nets are also special kinds of functions.
- In a vector space additive inversion corresponds to scalar multiplication by −1. For Euclidean space, it is inversion in the origin.

Universal construction

to do: symmetrization of an abelian semigroup

Inverse

Inverse typically means the opposite of something. See:
- Antonym - A word with the opposite meaning.
- Inverse multiplexer - Splits a signal into several signals, opposite of a multiplexer.
- Inverse (music) - Oppositional direction of voice movement.
- Inverse perspective - Also Byzantine perspective: the further the objects, the larger they are drawn.
- Inverse-square law - The magnitude of a force is proportional to the inverse square of the distance. See also inverse (mathematics) Inversion has different meanings in different fields of knowledge:
- Something that is inverted or the process by which an inverse is obtained.
- In music, see Inversion (music).
- In amusement rides, see Roller coaster inversions.
- In geophysical sciences, see inverse problem.
- In meteorology, see temperature inversion.
- In genetics, see chromosomal inversion
- In electrical systems, inversion is the process of converting direct current to alternating current, see inverter (electrical)
- In computer science, see priority inversion.
- In chemistry, see Nitrogen inversion.
- In computer graphics and digital image processing, see reverse video.
- In anatomy, see Anatomical terms of location Something that is inverted is something that is flipped over, around or otherwise appearing in an opposite manner than is normal, customary, or common. Examples:
- An inverted river delta is a river delta that has an mirror-imaged geometry compared to normal river deltas.
- Antimatter is sometimes called inverted matter.
- Inverting an object is often referred to flipping it upside down.
- Negative numbers are sometimes referred to as inverted numbers.
- Mirror images are called inverted.
- Inverting the colors of a photograph results in a negative. To invert means:
- to use an inverter
- to make something inverted
- the process of inversion In Freudian psychology, an invert is a homosexual. Invert is also the common name for a mixture of oil (petroleum) and diesel fuel. A by-product of oil well drilling, it is corrosive on clothing and skin, and highly flammable.

Addition

Addition is the most basic operation of arithmetic. In its simplest form, addition combines two numbers, the addends, into a single number, the sum. Adding more than two numbers can be viewed as repeated addition; this procedure is known as summation and includes ways to add infinitely many numbers in an infinite series. Repeated addition of the number one is the most basic form of counting. Addition can also be defined for mathematical objects other than numbers — for example, matrices or polynomials. Regardless of the nature and number of objects being added, the individual constituents of a sum typically are called summands or terms. (This is to be distinguished from factors, which are multiplied.)

Notation

multiplied Addition is written using the plus sign "+" between the terms. For example, :1 + 1 = 2 :2 + 2 = 4 :5 + 4 + 2 = 11 (see "associativity" below) :3 + 3 + 3 + 3 = 12 (see "multiplication" below) There are also situations where addition is "understood" even though no symbol appears:
- A column of numbers, with the last number in the column underlined, usually (but not always) indicates that the numbers in the column are to be added, with the sum written below the underlined number.
- A whole number followed immediately by a fraction indicates the sum of the two, called a mixed number. For example, ::3¹⁄₂ = 3 + ¹⁄₂ = 3.5. :This notation can cause confusion, since in most other contexts, juxtaposition denotes multiplication instead.

Interpretations

Addition is used to model countless physical processes. Even for the simple case of adding natural numbers, there are many possible interpretations and even more visual representations.

Combining sets

Possibly the most fundamental interpretation of addition lies in combining sets:
- When two or more collections are combined into a single collection, the number of objects in the single collection is the sum of the number of objects in the original collections. This interpretation is well-suited to quick proofs of the properties of natural number addition, and it is easy to visualize, with little danger of ambiguity. However, it is not obvious how one should extend this version of addition to include fractional numbers or negative numbers. See [http://arxiv.org/abs/math.QA/0004133 this article] for an example of the sophistication involved in adding with sets of "fractional cardinality". One possible fix is to consider collections of objects that can be easily divided, such as pies or, still better, segmented rods. Rather than just combining collections of segments, rods can be joined end-to-end. :This section is under construction.

Extending a measure

- When an original measure is extended by a given amount, the final measure is the sum of the original measure and the measure of the extension. Under this interpretation, the parts of a sum a + b play asymmetric roles; instead of calling both a and b addends, it is more appropriate to call a the augend, since a plays a passive role. In geometry, a might be a point and b a vector; their sum is then another point, the translation of a by b. In analytic geometry, a and b might both be represented by ordered pairs of numbers, but they remain conceptually different. Here, the addition operation is not so much a binary operation as a family of unary operations; the function (+b) is acting on a. The unary and binary views are formally equivalent, in that for any sets A and B there is a natural identification of sets of functions :

A^\cong \left(A^A\right)^B.

(This law of exponentiation may be more familiar for numbers.) The unary view is useful, for example, when discussing subtraction. Addition and subtraction are not inverses as binary operations, but they are inverses as families of unary operations. :This section is under construction.

Combining translations

- When two motions are performed in succession, the measure of the resulting motion is the sum of the measures of the original motions. :This section is under construction.

Basic properties

Commutivity

subtraction Addition is commutative, meaning that one can reverse the terms in a sum left-to-right, and the result will be the same. Symbolically, if a and b are any two numbers, then :a + b = b + a. The fact that addition is commutative is known as the "commutative law of addition". This phrase suggests that there are other commutative laws: for example, there is a commutative law of multiplication. However, many binary operations are not commutative, such as subtraction and division, so it is misleading to speak of an unqualified "commutative law".

Associativity

binary operation A somewhat subtler property of addition is associativity, which comes up when one tries to define repeated addition. Should the expression :"a + b + c" be defined to mean (a + b) + c or a + (b + c)? That addition is associative tells us that the choice of definition is irrelevant. For any three numbers a, b, and c, it is true that :(a + b) + c = a + (b + c). Not all operations are associative, so in expressions with operations other than addition, it is important to specify the order of operations.

Zero and one

order of operations If one adds zero to any number, the quantity won't change; zero is the identity element for addition. In symbols, for any a, :a + 0 = 0 + a = a. The sum of any number and its additive inverse (in contexts where such a thing exists) is zero. In the context of integers, addition of one plays a special role: for any integer a, the integer (a + 1) is the least integer greater than a, also known as the successor of a.

Units

In order to numerically add certain types of numbers, such as vulgar fractions and physical quantities with units, they must first be expressed with a common denominator. For example, if a measure of 5 feet is extended by 2 inches, the sum is 62 inches, since 60 inches is another name for 5 feet. On the other hand, it is usually meaningless to try to add 3 meters and 4 square meters, since those units are incomparable; this sort of consideration is fundamental in dimensional analysis.

Generalizations

:There are many things that can be added: numbers, vectors, matrices, spaces, shapes, sets, functions, equations, strings, chains... —[http://www.cut-the-knot.org/do_you_know/addition.shtml Alexander Bogomolny] Addition is first defined on the natural numbers. In set theory, addition is then extended to larger sets that include the natural numbers: the integers, the rational numbers, and the real numbers. (In mathematics education, positive fractions are added before negative numbers are even considered; this is also the historical route.) In turn, real addition extends to addition operations on even larger sets, such as the set of complex numbers or a many-dimensional vector space in linear algebra.

In algebra

There are many more sets that support an operation called addition. There are already infinitely many natural numbers, and the set of real numbers is even larger. It is also useful to study addition on smaller sets, even finite ones. In modular arithmetic, the set of integers modulo 12 has twelve elements; it inherits an addition operation from the integers that is central to musical set theory. The set of integers modulo 2 has just two elements; the addition operation it inherits is known in Boolean logic as "exclusive or". The ideas of extending and compacting sets can be combined. In geometry, the sum of two angles is often taken to be their sum as two real numbers modulo 2π. This amounts to an addition operation on the circle, which in turn generalizes to addition operations on many-dimensional tori. A general form of addition occurs in abstract algebra, where addition may be almost any well-defined binary operation on a set. For an operation to be called "addition" in abstract algebra, it is required to be associative and commutative.

Addition of sets

One extraordinary generalization of the addition of natural numbers is the addition of ordinal numbers. Unlike most addition operations, ordinal addition is not commutative. However, passing to the "smaller" class of cardinal numbers, we recover a commutative operation. Cardinal addition is closely related to the disjoint union of two sets. In category theory, the disjoint union is a kind of coproduct, so coproducts are perhaps the most abstract of all the generalizations of addition. Some coproducts are named to evoke their connection with addition; see Direct sum and Wedge sum.

Related operations

- Incrementation, also known as the successor operation, is the addition of 1 to a number. In formal treatments of addition, such as the Peano axioms, the successor is an elementary operation, and addition is defined from successors through recursion.
- Summation describes the addition of arbitrarily many numbers, usually more than just two. It includes the idea of the sum of a single number, which is itself, and the empty sum, which is 0. An infinite summation is known as a series.
- Counting is an intuitive procedure that can be formalized as the summation of 1 over some finite domain. In everyday counting, the domain is typically a small set of physical objects; in mathematics it may be large and abstract, as it is for the prime counting function.
- Integration is a kind of "summation" over a continuum, or more precisely and generally, over a differentiable manifold. Integration over a zero-dimensional manifold reduces to summation.
- Subtraction can be thought of as a kind of addition—that is, the addition of an additive inverse. Subtraction is itself a sort of inverse to addition, in that adding x and subtracting x are inverse functions.
- Multiplication can be thought of as repeated addition. If a single term x appears in a sum n times, then the sum is the product of n and x. If n is not a natural number, the product may still make sense; for example, multiplication by −1 yields the additive inverse of a number. In many contexts, multiplication can be transformed into addition, and vice versa, through exponentials and logarithms. In general, multiplication operations always distribute over addition.
- Linear combinations combine multiplication and summation; they are sums in which each term has a multiplier, usually a real or complex number. Linear combinations are especially useful in contexts where straightforward addition would violate some normalization rule, such as mixing of strategies in game theory or superposition of states in quantum mechanics.
- Convolution is used to add two independent random variables defined by distribution functions. Its usual definition combines integration, subtraction, and multiplication. In general, convolution is useful as a kind of domain-side addition; by contrast, vector addition is a kind of range-side addition.

Notes

# Begle (p.57) and Johnson (p.119) prefer "addends" and "sum". Calling both inputs "addends" emphasizes the symmetry of addition; see the section on #Extending a measure for a context in which "augend" is more appropriate. # Devine et al p.263 # Adding it up (p.73) compares adding measuring rods to adding sets of cats: "For example, inches can be subdivided into parts, which are hard to tell from the wholes, except that they are shorter; whereas it is painful to cats to divide them into parts, and it seriously changes their nature." # Stewart makes the distinction by writing angle brackets for vectors and parentheses for points, although this notation is not widely used. See the chapter Vectors. # Weaver (p.62) argues for the importance of contrasting the two views, going so far as to term the version of commutivity satisfied by unary addition "pseudocommutivity". # Enderton (p.142, Theorem 6I) discusses this relationship in the context of cardinal arithmetic identities. # Enderton chapters 4 and 5, for example, follow this development. # California standards; see grades [http://www.cde.ca.gov/be/st/ss/mthgrade2.asp 2], [http://www.cde.ca.gov/be/st/ss/mthgrade3.asp 3], and [http://www.cde.ca.gov/be/st/ss/mthgrade4.asp 4]. # Baez (p.37) explains the historical development, in "stark contrast" with the set theory presentation: "Apparently, half an apple is easier to understand than a negative apple!"

References

- Preprint available [http://arxiv.org/abs/math.QA/0004133 here] on arXiv.
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- [http://www.cde.ca.gov/be/st/ss/mthmain.asp California State Board of Education mathematics content standards] Adopted December 1997, accessed December 2005.
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- Available [http://www.nap.edu/books/0309069955/html/index.html here] from the publisher.
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External links

;General
- [http://www.cut-the-knot.org/do_you_know/addition.shtml Addition on cut-the-knot.org] An exploration of various kinds of addition. ;Methods and practice
- [http://www.mathsisfun.com/worksheets/addition.php Addition Worksheets or Online Practice]
- [http://www.apples4theteacher.com/flash-cards.html Addition Flash Cards]
- [http://webhome.idirect.com/~totton/abacus/pages.htm#Addition1 Addition on a Japanese abacus] selected from [http://webhome.idirect.com/~totton/abacus/ Abacus: Mystery of the Bead] Category:Arithmetic ja:総和 ko:덧셈 simple:Addition th:การบวก

0 (number)

:This page is about the number and numeral 0. For other uses of 0 or "zero", see 0 (disambiguation) 0 (zero), alternatively called naught or nought, is both a number and a numeral. It was the last numeral to be created in most numerical systems, as it is not a counting number (which is to say, one begins counting at the number 1) and was in many eras and places represented only by a gap or mark very different from the other numerals.

0 as a number

0 is the integer that precedes the positive 1, and all positive integers, and follows -1, and all negative integers. In most (if not all) numerical systems, 0 was identified before the idea of 'negative integers' was accepted. Zero is a number which means nothing, null, void or an absence of value. For example, if the number of one's brothers is zero, then that person has no brothers. If the difference between the number of pieces in two piles is zero, it means the two piles have an equal number of pieces. Almost all historians omit the year zero from the proleptic Gregorian and Julian calendars, but astronomers include it in these same calendars. However, the phrase Year Zero may be used to describe any event considered so significant that it virtually starts a new time reckoning.

0 as a numeral

Year Zero The modern numeral 0 is normally written as a circle or (rounded) rectangle. On the seven-segment displays of calculators, watches, etc., 0 is usually written with six line segments (at right), though on some historical calculator models it was written with four line segments. This variant glyph has not caught on. It is important to distinguish the number zero (as in the "zero brothers" example above) from the numeral or digit zero, used in numeral systems where the position of a digit signifies its value. Successive positions of digits have higher values, so the digit zero is used to skip a position and give appropriate value to the preceding and following digits. The Babylonian numeral system used two narrow slanting wedges, similar to \\, for the equivalent of a positional zero numeral starting in about 400BC. A zero digit is not always necessary in a positional number system: decimal without a zero provides a possible counterexample. In fonts with text figures, 0 is usually the same height as a lowercase X, for example, Image:TextFigs036.png.

History

Etymology

The word zero comes ultimately from the Arabic sifr (صفر) meaning empty or vacant, a literal translation of the Sanskrit meaning void or empty. Through transliteration this became zephyr or zephyrus in Latin. The word zephyrus already meant "west wind" in Latin; the proper noun Zephyrus was the Roman god of the west wind (after the Greek god Zephyros). With its new use for the concept of zero, zephyr came to mean a light breeze—"an almost nothing" (Ifrah 2000; see References). The word zephyr survives with this meaning in English today. The Italian mathematician Fibonacci (c.1170-1250), who grew up in Arab North Africa and is credited with introducing the Arabic decimal system to Europe, used the term zephyrum. This became zefiro in Italian, which was contracted to zero in the Venetian dialect, giving the modern English word. As the decimal zero and its new mathematics spread through a Europe that was still in the Middle Ages, words derived from sifr and zephyrus came to refer to calculation, as well as to privileged knowledge and secret codes. According to Ifrah (2000), "in thirteenth-century Paris, a 'worthless fellow' was called a... cifre en algorisme, i.e., an 'arithmetical nothing.' " (algorithm is also a borrowing from the Arabic, in this case from the name of the 9th-century mathematician al-Khwarizmi.) The Arabic root gave rise to the modern French chiffre, which means digit, figure, or number; chiffrer, to calculate or compute; and chiffré, encrypted; as well as to the English word cipher. Today, the word in Arabic is still sifr, and cognates of sifr are common throughout the languages of Europe. A few additional examples follow.
- Polish: cyfra, digit; szyfrować, to encrypt
- German: Ziffer, digit, figure, numeral, cypher
- French: zéro, zero
- Spanish: cifra, figure, numeral, cypher, code; cero, zero
- Swedish: siffra, numeral, sum, digit Note that zero in Greek is translated as Μηδέν (Meithen).

Babylonians and Greeks

By the mid second millennium BC, Babylonians had a sophisticated sexagesimal positional numeral system. The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. By 300 BC a punctuation symbol (two slanted wedges) was co-opted as a placeholder in the same Babylonian system. However, "... a tablet found at Kish ... thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place" ([http://www-history.mcs.st-and.ac.uk/history/HistTopics/Zero.html] and natural number). Records show that the Ancient Greeks seemed unsure about the status of zero as a number: they 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 and the vacuum. The paradoxes of Zeno of Elea depend in large part on the uncertain interpretation of zero. (The ancient Greeks even questioned that 1 was a number.)

First use of the number

An early use of zero by the Indian mathematician Pingala (possibly 5th-3rd century BC), implied at first glance by [http://home.ica.net/~roymanju/Binary.htm Binary Numbers in Ancient India], is only the modern binary representation using 0 and 1 of Pingala's binary system, which used short and long syllables (the latter equal in length to two short syllables) as described in [http://www.sju.edu/~rhall/Rhythms/Poets/arcadia.pdf Math for Poets and Drummers] (pdf), making it similar to Morse code. In Pingala's system, four short syllables meant one, not zero. Nevertheless, he does use the Sanskrit word Shunya to refer to the concept of void, which was fairly similar to the concept of zero [http://sunsite.utk.edu/math_archives/.http/hypermail/historia/apr99/0197.html]. The late Olmec people of south-central Mexico began to use a true zero (a shell glyph) in the New World possibly by the 4th century BC but certainly by 40 BC, which became an integral part of Maya numerals, but did not influence Old World numeral systems. By 130, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero (a small circle with a long overbar) within a sexagesimal numeral system otherwise using alphabetic Greek numerals. Because it was used alone, not as just a placeholder, this Hellenistic zero was the first documented use of a true zero in the Old World. In later Byzantine manuscripts of his Syntaxis Mathematica (Almagest), the Hellenistic zero had morphed into the Greek letter omicron (otherwise meaning 70). Another true zero was used in tables alongside Roman numerals by 525 (first known use by Dionysius Exiguus), but as a word, nulla meaning nothing, not as a symbol. When division produced zero as a remainder, nihil, also meaning nothing, was used. These medieval zeros were used by all future medieval computists (calculators of Easter). An isolated use of their initial, N, was used in a table of Roman numerals by Bede or a colleague about 725, a true zero symbol.

Zero as a decimal digit

The earliest known decimal digit zero is thought to have been introduced by Indian mathematicians sometime around the 3rd century. It was written in the shape of a dot, and consequently called "dot". An early documented use of the zero by Brahmagupta dates to 628. He treated zero as a number and discussed operations involving it. By this time (7th century) the concept had clearly reached Cambodia, and documentation shows the idea later spreading to China and the Islamic world. The Hindu-Arabic number system reached Europe in the late 11th century, via Andalusia, together with knowledge of astronomy and instruments like the astrolabe. The Italian mathematician Fibonacci was instrumental in bringing the system into European mathematics around 1200, though he spoke of the "sign" zero, not as a number. It was not until the 1600s that decimal notation began to come into widespread use in the Occident.

In mathematics

Zero (0) is both a number and a numeral. The natural number following zero is one and no natural number precedes zero. Zero may or may not be counted as a natural number, depending on the definition of natural numbers. Zero is neither prime nor composite. In set theory, the number zero is the cardinality of the empty set: if one does not have any apples, then one has zero apples. In fact, in certain axiomatic developments of mathematics from set theory, zero is defined to be the empty set. The following are some basic rules for dealing with the number zero, first described in Brahmasphutasiddhanta. These rules apply for any complex number x, unless otherwise stated.
- Addition: x + 0 = x and 0 + x = x. (That is, 0 is an identity element with respect to addition.)
- Subtraction: x − 0 = x and 0 − x = − x.
- Multiplication: x · 0 = 0 · x = 0.
- Division: 0 / x = 0, for nonzero x. But x / 0 is undefined, because 0 has no multiplicative inverse, a consequence of the previous rule. For positive x, as y in x / y approaches zero from positive values, its quotient increases toward positive infinity, but as y approaches zero from negative values, the quotient increases toward negative infinity. The different quotients confirms that division by zero is undefined.
- Exponentiation: x⁰ = 1, except that the case x = 0 may be left undefined in some contexts. For all positive real x, 0^x = 0. The expression "0/0" is an "indeterminate form". That does not simply mean that it is undefined; rather, it means that if f(x) and g(x) both approach 0 as x approaches some number, then f(x)/g(x) could approach any finite number or ∞ or −∞; it depends on which functions f and g are. See L'Hopital's rule. The sum of 0 numbers is 0, and the product of 0 numbers is 1.

Extended use of zero in mathematics

- Zero is the identity element in an additive group or the additive identity of a ring.
- A zero of a function is a point in the domain of the function whose image under the function is zero. See zero (complex analysis).
- In geometry, the dimension of a point is 0.
- In analytic geometry, 0 is the origin.
- In nonstandard analysis the number zero is taken as an infinitesimal element of a non-principal ultrafilter.
- The concept of "almost" impossible in probability. More generally, the concept of almost nowhere in measure theory.
- A zero function is a constant function with 0 as its only possible output value; i.e.,

f(x) = 0

. A particular zero function is a zero morphism. A zero function is the identity in the additive group of functions.
- The zero of a function is a preimage of zero, also called the root of a function.
- Zero is one of three possible return values of the Möbius function. Passed an integer x² or x²y, the Möbius function returns zero.
- It is the number of n×n magic squares for n = 2.
- It is the number of n-queens problem solutions for n = 2, 3.
- Zero is neither a prime nor a composite number.

In physics

The value zero plays a special role for a large number of physical quantities. For some quantities, the zero level is naturally distinguished from all other levels, where as it for others is more or less arbitrarily chosen. For example, on the kelvin temperature scale, zero is the coldest possible temperature (so that negative temperatures are non-existent), where as on the celsius scale, 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, e.g. at a value for the threshold of hearing.

In computer science

Numbering from 1 or 0?

Human beings usually number things starting from one, not zero. Yet in computer science zero has become the popular indication for a starting point. For example, in almost all old programming languages, an array starts from 1 by default, which is natural for humans. As programming languages have developed, it has become more common that an array starts from zero by default (zero-based). One reason for this convention is that modular arithmetic normally describes a set of N numbers as containing 0,1,2,...N-1 in order to contain the additive identity. Because of this, many arithmetic concepts (such as hash tables) are less elegant to express in code unless the array starts at zero. Another reason to use zero-based array indices is that it can improve efficiency under certain circumstances. To illustrate, suppose a is the memory address of the first element of an array, and i is the index of the desired element. In this fairly typical scenario, it is quite common to want the address of the desired element. If the index numbers count from 1, the desired address is computed by this expression: :

a + s \times (i-1)

where s is the size of each element. In contrast, if the index numbers count from 0, the expression becomes this: :

a + s \times i

This simpler expression can be more efficient to compute in certain situations. Note, however, that a language wishing to index arrays from 1 could simply adopt the convention that every "array address" is represented by

a'=a-s

; that is, rather than using the address of the first array element, such a language would use the address of an imaginary element located immediately before the first actual element. The indexing expression for a 1-based index would be the following: :

a' + s \times i

Hence, the efficiency benefit of zero-based indexing is not inherent, but is an artifact of the decision to represent an array by the address of its first element. This situation can lead to some confusion in terminology. In a zero-based indexing scheme, the first element is "element number zero"; likewise, the twelfth element is "element number eleven". For this reason, the first element is often referred to as the zeroth element to eliminate any possible doubt (though, strictly speaking, this is unnecessary and arguably incorrect, since the meanings of the ordinal numbers are not ambiguous).

Null value

In databases a field can have a null value. This is equivalent to the field not having a 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. Asking for all records with value 0 or value not equal 0 will not yield all records, since the records with value null are excluded. This is owing to the notion that records in a relational database are a set of key/value tuples. A null value, notionally, indicates not that the record has some particular value – "null" – for a given column, but rather that the record has no value at all for that particular column.

Null pointer

A null pointer is a pointer in a computer program that does not point to any object or function. In C, 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), and has no particular association with zero.

Negative zero

In some signed number representations (but not the two's complement representation predominant today) and most floating point number representations, zero has two distinct representations, one grouping it with the positive numbers and one with the negatives; this latter representation is known as negative zero. Representations with negative zero can be troublesome, because the two zeroes will compare equal but may be treated differently by some operations.

Distinguishing zero from O

negative zero The oval-shaped zero (appearing like a rugby ball stood on end) and circular letter O together came into use on modern character displays. The zero with a dot in the centre seems to have originated as an option on IBM 3270 controllers (this has the problem that it looks like the Greek letter Theta). The slashed zero, looking identical to the letter O other than the slash, is used in old-style ASCII graphic sets descended from the default typewheel on the venerable ASR-33 Teletype. This format causes problems for certain Scandinavian languages which use Ø as a letter. The convention which has the letter O with a slash and the zero without was used at IBM and a few other early mainframe makers; this is even more problematic for Scandinavians because it means two of their letters collide. Some Burroughs/Unisys equipment displays a zero with a reversed slash. And yet another convention common on early line printers left zero unornamented but added a tail or hook to the letter-O so that it resembled an inverted Q or cursive capital letter-O. The typeface used on some European number plates for cars distinguish the two symbols by making the O rather egg-shaped and the zero more circular, but most of all by opening the zero on the upper right side, so here the circle is not closed any more (as in German plates). In paper writing one may not distinguish the 0 and O at all, or may add a slash across it in order to show the difference, although this sometimes causes ambiguity in regard to the symbol for the null set.

In other fields

- In some countries, 0 on a telephone calls for operator assistance. On the BlackBerry the 0 key also functions as a spacebar.
- In Braille, the numeral 0 has the same dot configuration as the letter J.
- DVDs that can be played in any region are sometimes referred to as being "region 0".

References

- [http://www.amazon.com/exec/obidos/ASIN/0471393401/qid=1124292648/sr=2-2/ref=pd_bbs_b_2_2/102-7275474-2228915 The Universal History of Numbers: From Prehistory to the Invention of the Computer.] Georges Ifrah. Wiley (2000)
- [http://www.mediatinker.com/blog/archives/008821.html A Brief History of Zero] - Kristen McQuillin, July 1997 (revised January 2004)
- [http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Zero.html A history of Zero]
- [http://home.ubalt.edu/ntsbarsh/zero/ZERO.HTM Zero Saga]
- [http://www.neo-tech.com/zero/part6.html The Discovery of the Zero]
- Charles Seife (2000). [http://www.amazon.com/exec/obidos/tg/detail/-/0140296476/qid=1111606043/sr=8-1/ref=sr_8_xs_ap_i1_xgl14/104-6166861-2891133?v=glance&s=books&n=507846 "Zero: The Biography of a Dangerous Idea".] Publisher: Penguin USA (Paper). ISBN 0140296476 Category:Elementary arithmetic 0 Category:Integers ko:0 ja:0 simple:Zero th:0 (จำนวน)

Binary operation

In mathematics, a binary operation is a calculation involving two input quantities. Binary operations can be accomplished using either a binary function or binary operator. Binary operations are sometimes called dyadic operations in order to avoid confusion with the binary numeral system. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division. More precisely, a binary operation on a set S is a binary function from S and S to S, in other words a function f from the Cartesian product S × S to S. Sometimes, especially in computer science, the term is used for any binary function. That f takes values in the same set S that provides its arguments is the property of closure. Binary operations are the keystone of algebraic structures studied in abstract algebra: they form part of groups, monoids, semigroups, rings, and more. Most generally, a magma is a set together with any binary operation defined on it. Many binary operations of interest in both algebra and formal logic are commutative or associative. Many also have identity elements and inverse elements. Typical examples of binary operations are the addition (+) and multiplication (
- ) of numbers and matrices as well as composition of functions on a single set. Examples of operations that are not commutative are subtraction (-), division (/), exponentiation(^), and super-exponentiation(@). Binary operations are often written using infix notation such as a
- b, a + b, or a · b rather than by functional notation of the form f(a,b). Sometimes they are even written just by juxtaposition: ab. They can also be expressed using prefix or postfix notations. A prefix notation, Polish notation, dispenses with parentheses; it is probably more often encountered now in its postfix form, reverse Polish notation.

External binary operations

An external binary operation is a binary function from K and S to S. This differs from a binary operation in the strict sense in that K need not be S; its elements come from outside. An example of an external binary operation is scalar multiplication in linear algebra. Here K is a field and S is a vector space over that field. An external binary operation may alternatively be viewed as an action; K is acting on S. Category:Algebra Category:Abstract algebra ja:二項演算

Integer

The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. They are also known as the whole numbers, although that term is also used to refer only to the positive integers (with or without zero). Like the natural numbers, the integers form a countably infinite set. The set of all integers is usually denoted in mathematics by a boldface Z (or blackboard bold,

\mathbb

), which stands for Zahlen (German for "numbers"). The term rational integer is used, in algebraic number theory, to distinguish these 'ordinary' integers, in the rational numbers, from other concepts such as the Gaussian integers.

Algebraic properties

Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers, and, importantly, zero, Z (unlike the natural numbers) is also closed under subtraction. Z is not closed under the operation of division, since the quotient of two integers (e.g., 1 divided by 2), need not be an integer. The following table lists some of the basic properties of addition and multiplication for any integers a, b and c. In the language of abstract algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, since every nonzero integer can be written as a finite sum 1 + 1 + ... 1 or (−1) + (−1) + ... + (−1). In fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, note that not every integer has a multiplicative inverse; e.g. there is no integer x such that 2x = 1, because the left hand side is even, while the right hand side is odd. This means that Z under multiplication is not a group. All the properties from the above table taken together say that Z together with addition and multiplication is a commutative ring with unity. In fact, Z provides the motivation for defining such a structure. The lack of multiplicative inverses, which is equivalent to the fact that Z is not closed under division, means that Z is not a field. The smallest field containing the integers is the field of rational numbers. This process can be mimicked to form the quotient field of any integral domain, where an integral domain is a commutative ring with unity such that whenever ab = 0, either a = 0 or b = 0. Although ordinary division is not defined on Z, it does possess an important property called the division algorithm: that is, given two integers a and b with b ≠ 0, there exist unique integers q and r such that a = q × b + r and 0 ≤ r < |b|, where |b| denotes the absolute value of b. The integer q is called the quotient and r is called the remainder, resulting from division of a by b. This is the basis for the Euclidean algorithm for computing greatest common divisors. Again, in the language of abstract algebra, the above says that Z is a Euclidean domain. This implies that Z is a principal ideal domain and any positive integer can be written as the products of primes in an essentially unique way. This is the fundamental theorem of arithmetic.

Order-theoretic properties

Z is a totally ordered set without upper or lower bound. The ordering of Z is given by : ... < −2 < −1 < 0 < 1 < 2 < ... An integer is positive if it is greater than zero and negative if it is less than zero. Zero is defined as neither negative nor positive. The ordering of integers is compatible with the algebraic operations in the following way: # if a < b and c < d, then a + c < b + d # if a < b and 0 < c, then ac < bc. (From this fact, one can show that if c < 0, then ac > bc.)

Integers in computing

An integer (sometimes known as an "int", from the name of a datatype in the C programming language) is often a primitive datatype in computer languages. However, integer datatypes can only represent a subset of all integers, since practical computers are of finite capacity. Variable-length representations of integers, such as bignums, can store any integer that fits in the computers memory. Other integer datatypes are implemented with a fixed size, usually a number of bits which is a power of 2 (4, 8, 16, etc.) or a memorable number of decimal digits (e.g., 9 or 10). In contrast, theoretical models of digital computers, such as Turing machines, typically do have infinite (but only countable) capacity.

Quotations

God invented the integers, all else is the work of man. Kronecker

External links

- [http://www.positiveintegers.org The Positive Integers - divisor tables and numeral representation tools] Category:Elementary mathematics Category:Group theory Category:Integers Category:Elementary number theory Category:Set theory ko:정수 ja:整数 th:จำนวนเต็ม

Real number

In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to "imaginary number". Real numbers may be rational or irrational; algebraic or transcendental; and positive, negative, or zero. Real numbers measure continuous quantities. They may in theory be expressed by decimal fractions that have an infinite sequence of digits to the right of the decimal point; these are often (mis-)represented in the same form as 324.823211247… The three dots indicate that there would still be more digits to come, no matter how many more might be added at the end. Measurements in the physical sciences are almost always conceived as approximations to real numbers. Writing them as decimal fractions (which are rational numbers that could be written as ratios, with an explicit denominator) is not only more compact, but to some extent conveys the sense of an underlying real number. The real numbers are the central object of study in real analysis. A real number is said to be computable if there exists an algorithm that yields its digits. Because there are only countably many algorithms, but an uncountable number of reals, most real numbers are not computable. Some constructivists accept the existence of only those reals that are computable. The set of definable numbers is broader, but still only countable. Computers can only approximate most real numbers with rational numbers; these approximations are known as floating point numbers or fixed-point numbers; see real data type. Computer algebra systems are able to treat some real numbers exactly by storing an algebraic description (such as "sqrt(2)") rather than their decimal approximation. Mathematicians use the symbol R (or alternatively,

\Bbb

, the letter "R" in blackboard bold) to represent the set of all real numbers. The notation Rⁿ refers to an n-dimensional space of real numbers; for example, a value from R³ consists of three real numbers and specifies a location in 3-dimensional space. In mathematics, real is used as an adjective, meaning that the underlying field is the field of real numbers. For example real matrix, real polynomial and real Lie algebra.

History

Vulgar fractions had been used by the Egyptians around 1000 BC; around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers. Negative numbers were invented by Indian mathematicians around 600 AD, and then possibly reinvented in China shortly after. They were not used in Europe until the 1600s, but even in the late 1700s, Leonhard Euler discarded negative solutions to equations as unrealistic. The development of calculus in the 1700s used the entire set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871.

Definition

Construction from the rational numbers

The real numbers can be constructed as a completion of the rational numbers. For details and other construction of real numbers, see construction of real numbers.

Axiomatic approach

Let R denote the set of all real numbers. Then:
- The set R is a field, meaning that addition and multiplication are defined and have the usual properties.
- The field R is ordered, meaning that there is a total order ≥ such that, for all real numbers x, y and z:
- if x ≥ y then x + z ≥ y + z;
- if x ≥ 0 and y ≥ 0 then xy ≥ 0.
- The order is Dedekind-complete, i.e., every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R. The last property is what differentiates the reals from the rationals. For example, the set of rationals with square less than 2 has a rational upper bound (e.g., 1.5) but no rational least upper bound, because the square root of 2 is not rational. The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind complete ordered fields R₁ and R₂, there exists a unique field isomorphism from R₁ to R₂, allowing us to think of them as essentially the same mathematical object.

Properties

Completeness

The main reason for introducing the reals is that the reals contain all limits. More technically, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section). This means the following: A sequence (x_n) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |x_n − x_m| is less than ε provided that n and m are both greater than N. In other words, a sequence is a Cauchy sequence if its elements x_n eventually come and remain arbitrarily close to each other. A sequence (x_n) converges to the limit x if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |x_n − x| is less than ε provided that n is greater than N. In other words, a sequence has limit x if its elements eventually come and remain arbitrarily close to x. It is easy to see that every convergent sequence is a Cauchy sequence. An important fact about the real numbers is that the converse is also true: :Every Cauchy sequence of real numbers is convergent. That is, the reals are complete. Note that the rationals are not complete. For example, the sequence (1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ...) is Cauchy but it does not converge to a rational number. (In the real numbers, in contrast, it converges to the square root of 2.) The existence of limits of Cauchy sequences is what makes calculus work and is of great practical use. The standard numerical test to determine if a sequence has a limit is to test if it is a Cauchy sequence, as the limit is typically not known in advance. For example, the standard series of the exponential function :

\mathrm^x = \sum_^ \frac

converges to a real number because for every x the sums :

\sum_^ \frac

can be made arbitrarily small by choosing N sufficiently large. This proves that the sequence is Cauchy, so we know that the sequence converges even if we do not know ahead of time what the limit is.

"The complete ordered field"

The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways. First, an order can be lattice-complete. It is easy to see that no ordered field can be lattice-complete, because it can have no largest element (given any element z, z + 1 is larger), so this is not the sense that is meant. Additionally, an order can be Dedekind-complete, as defined in the section Axioms. The uniqueness result at the end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way. These two notions of completeness ignore the field structure. However, an ordered group (and a field is a group under the operations of addition and subtraction) defines a uniform structure, and uniform structures have a notion of completeness (topology); the description in the section Completeness above is a special case. (We refer to the notion of completeness in uniform spaces rather than the related and better known notion for metric spaces, since the definition of metric space relies on already having a characterisation of the real numbers.) It is not true that R is the only uniformly complete ordered field, but it is the only uniformly complete Archimedean field, and indeed one often hears the phrase "complete Archimedean field" instead of "complete ordered field". Since it can be proved that any uniformly complete Archimedean field must also be Dedekind complete (and vice versa, of course), this justifies using "the" in the phrase "the complete Archimedean field". This sense of completeness is most closely related to the construction of the reals from Cauchy sequences (the construction carried out in full in this article), since it starts with an Archimedean field (the rationals) and forms the uniform completion of it in a standard way. But the original use of the phrase "complete Archimedean field" was by David Hilbert, who meant still something else by it. He meant that the real numbers form the largest Archimedean field in the sense that every other Archimedean field is a subfield of R. Thus R is "complete" in the sense that nothing further can be added to it without making it no longer an Archimedean field. This sense of completeness is most closely related to the construction of the reals from surreal numbers, since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield.

Advanced properties

The reals are uncountable; that is, there are strictly more real numbers than natural numbers, even though both sets are infinite. This is proved with Cantor's diagonal argument. In fact, the cardinality of the reals is 2^ω, i.e., the cardinality of the set of subsets of the natural numbers. Since only a countable set of real numbers can be algebraic, almost all real numbers are transcendental. The non-existence of a subset of the reals with cardinality strictly between that of the integers and the reals is known as the continuum hypothesis. The continuum hypothesis can neither be proved nor be disproved; it is independent from the axioms of set theory. The real numbers form a metric space: the distance between x and y is defined to be the absolute value |x − y|. By virtue of being a totally ordered set, they also carry an order topology; the topology arising from the metric and the one arising from the order are identical. The reals are a contractible (hence connected and simply connected), separable metric space of dimension 1, and are everywhere dense. The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, continuous, and separable order topologies are necessarily homeomorphic to the reals. Every nonnegative real number has a square root in R, and no negative number does. This shows that the order on R is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one root: these two properties make R the premier example of a real closed field. Proving this is the first half of one proof of the fundamental theorem of algebra. The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalised such that the unit interval [0,1] has measure 1. The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement. It is not possible to characterize the reals with first-order logic alone: the Löwenheim-Skolem theorem implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first order logic as the real numbers themselves. The set of hyperreal numbers is much bigger than R but also satisfies the same first order sentences as R. Ordered fields that satisfy the same first-order sentences as R are called nonstandard models of R. This is what makes nonstandard analysis work; by proving a first-order statement in some nonstandard model (which may be easier than proving it in R), we know that the same statement must also be true of R.

Generalizations and extensions

The real numbers can be generalized and extended in several different directions. Perhaps the most natural extension are the complex numbers which contain solutions to all polynomial equations. However, the complex numbers are not an ordered field. Ordered fields extending the reals are the hyperreal numbers and the surreal numbers; both of them contain infinitesimal and infinitely large numbers and thus are not Archimedean. Occasionally, the two formal elements +∞ and −∞ are added to the reals to form the extended real number line, a compact space which is not a field but retains many of the properties of the real numbers. Self-adjoint operators on a Hilbert space (for example, self-adjoint square complex matrices) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvalues are real and they form a real associative algebra. Positive-definite operators correspond to the positive reals and normal operators correspond to the complex numbers. Category:Elementary mathematics Category:Real numbers Category:Set theory ko:실수 ja:実数 th:จำนวนจริง

Complex number

In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for the square root of minus one (−1), which cannot be represented by any real number. For example, :3 + 2i is a complex number, where 3 is called the real part and 2 the imaginary part. Since a complex number a + bi is uniquely specified by an ordered pair (a, b) of real numbers, the complex numbers are in one-to-one correspondence with points on a plane, called the complex plane. The set of all complex numbers is usually denoted by C, or in blackboard bold by

\mathbb

. It includes the real numbers because every real number can be regarded as complex: a = a + 0i. Complex numbers are added, subtracted, and multiplied by formally applying the associative, commutative and distributive laws of algebra, together with the equation i ² = −1: :(a + bi) + (c + di) = (a+c) + (b+d)i :(a + bi) − (c + di) = (a−c) + (b−d)i :(a + bi)(c + di) = ac + bci + adi + bd i ² = (ac−bd) + (bc+ad)i Division of complex numbers can also be defined (see below). Thus the set of complex numbers forms a field which, in contrast to the real numbers, is algebraically closed. In mathematics, the adjective "complex" means that the field of complex numbers is the underlying number field considered, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra.

Definition

The complex number field

Formally, the complex numbers can be defined as ordered pairs of real numbers (a, b) together with the operations:
-

( a , b ) + ( c , d ) = ( a + c , b + d ) \,

( a , b ) \cdot ( c , d ) = ( ac - bd , bc + ad ). \,

So defined, the complex numbers form a field, the complex number field, denoted by C. We identify the real number a with the complex number (a, 0), and in this way the field of real numbers R becomes a subfield of C. The imaginary unit i is the complex number (0, 1). In C, we have:
- additive identity ("zero"): (0, 0)
- multiplicative identity ("one"): (1, 0)
- additive inverse of (a,b): (−a, −b)
- multiplicative inverse (reciprocal) of non-zero (a, b):

\left(,\right).

C can also be defined as the topological closure of the algebraic numbers or as the algebraic closure of R, both of which are described below.

The complex plane

image:complex.png

A complex number can be viewed as a point or a position vector on a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram (named after Jean-Robert Argand). The Cartesian coordinates of the complex number are the real part x and the imaginary part y, while the circular coordinates are r = |z|, called the absolute value or modulus, and φ = arg(z), called the complex argument of z (mod-arg form). Together with Euler's formula we have :

z = x + iy = r (\cos \phi + i\sin \phi ) = r e^. \,

Additionally the notation r cis φ is sometimes used. Note that the complex argument is unique modulo 2π, that is, if any two values of the complex argument exactly differ by an integer multiple of 2π, they are considered equivalent. By simple trigonometric identities, we see that :

r_1 e^ \cdot r_2 e^ 
= r_1 r_2 e^ \,

and that :

\frac

= \frac e^. \,

Now the addition of two complex numbers is just the vector addition of two vectors, and the multiplication with a fixed complex number can be seen as a simultaneous rotation and stretching. Multiplication with i corresponds to a counter clockwise rotation by 90 degrees (

\pi/2

radians). The geometric content of the equation i² = −1 is that a sequence of two 90 degree rotations results in a 180 degree (

\pi

radians) rotation. Even the fact (−1) · (−1) = +1 from arithmetic can be understood geometrically as the combination of two 180 degree turns.

Absolute value, conjugation and distance

The absolute value (or modulus or magnitude) of a complex number z = r e^iφ is defined as |z| = r. Algebraically, if z = a + ib, then

| z | = \sqrt.

One can check readily that the absolute value has three important properties: :

| z | = 0 \,

iff

z = 0 \,

| z + w | \leq | z | + | w | \,

| z w | = | z | \; | w | \,

for all complex numbers z and w. It then follows, for example, that

| 1 | = 1

and

|z/w|=|z|/|w|

. By defining the distance function d(z, w) = |z − w| we turn the complex numbers into a metric space and we can therefore talk about limits and continuity. The addition, subtraction, multiplication and division of complex numbers are then continuous operations. Unless anything else is said, this is always the metric being used on the complex numbers. The complex conjugate of the complex number z = a + ib is defined to be a - ib, written as

\bar

z^ - \,

. As seen in the figure,

\bar

is the "reflection" of z about the real axis. The following can be checked: :

\overline = \bar + \bar

\overline = \bar\bar

\overline = \bar/\bar

\bar=z

\bar=z

iff z is real :

|z|=|\bar|

|z|^2 = z\bar

z^ = \bar|z|^

if z is non-zero. The latter formula is the method of choice to compute the inverse of a complex number if it is given in rectangular coordinates. That conjugation commutes with all the algebraic operations (and many functions; e.g.

\sin\bar z=\overline

) is rooted in the ambiguity in choice of i (−1 has two square roots); note, however, that conjugation is not differentiable (see holomorphic).

Complex number division

Given a complex number (a + ib) which is to be divided by another complex number (c + id) whose magnitude is non-zero, there are two ways to do this; in either case it is the same as multiplying the first by the multiplicative inverse of the second. The first way has already been implied: to convert both complex numbers into exponential form, from which their quotient is easy to derive. The second way is to express the division as a fraction, then to multiply both numerator and denominator by the complex conjugate of the denominator. This causes the denominator to simplify into a real number: :

=  =

:::

= \left(\right) + i\left(  \right).

Matrix representation of complex numbers

While usually not useful, alternative representations of complex fields can give some insight into their nature. One particularly elegant representation interprets every complex number as 2×2 matrix with real entries which stretches and rotates the points of the plane. Every such matrix has the form :

\begin
  a &   -b  \\
  b & \;\; a  
\end

with real numbers a and b. The sum and product of two such matrices is again of this form. Every non-zero such matrix is invertible, and its inverse is again of this form. Therefore, the matrices of this form are a field. In fact, this is exactly the field of complex numbers. Every such matrix can be written as :

\begin
  a &     -b  \\
  b & \;\; a  
\end
=
a \begin
  1 & \;\; 0  \\
  0 & \;\; 1 
\end
+
b \begin
  0 &     -1  \\
  1 & \;\; 0 
\end

which suggests that we should identify the real number 1 with the matrix :

\begin
  1 & \;\; 0  \\
  0 & \;\; 1 
\end

and the imaginary unit i with :

\begin
  0 &     -1  \\
  1 & \;\; 0  
\end

a counter-clockwise rotation by 90 degrees. Note that the square of this latter matrix is indeed equal to −1. The absolute value of a complex number expressed as a matrix is equal to the square root of the determinant of that matrix. If the matrix is viewed as a transformation of a plane, then the transformation rotates points through an angle equal to the argument of the complex number and scales by a factor equal to the complex number's absolute value. The conjugate of the complex number z corresponds to the transformation which rotates through the same angle as z but in the opposite direction, and scales in the same manner as z; this can be described by the transpose of the matrix corresponding to z. If the matrix elements are themselves complex numbers, then the resulting algebra is that of the quaternions. In this way, the matrix representation can be seen as a way of expressing the Cayley-Dickson construction of algebras.

Geometric interpretation of the operations on complex numbers

Cayley-Dickson construction Choose a point in the plane which will be the origin,

0

. Given two points A and B in the plane, their sum is the point X in the plane such that the triangles with vertices 0, A, B and X, B, A are similar. similar Choose in addition a point in the plane different from zero, which will be the unity, 1. Given two points A and B in the plane, their product is the point X in the plane such that the triangles with vertices 0, 1, A, and 0, B, X are similar. similar Given a point A in the plane, its complex conjugate is a point X in the plane such that the triangles with vertices 0, 1, A and 0, 1, X are mirror image of each other.

Some properties

Real vector space

C is a two-dimensional real vector space. Unlike the reals, complex numbers cannot be ordered in any way that is compatible with its arithmetic operations: C cannot be turned into an ordered field. R-linear maps C → C have the general form :

f(z)=az+b\overline

with complex coefficients a and b. Only the first term is C-linear; also only the first term is holomorphic; the second term is real-differentiable, but does not satisfy the Cauchy-Riemann equations. The function :

f(z)=az\,

corresponds to rotations combined with scaling, while the function :

f(z)=b\overline

corresponds to reflections combined with scaling.

Solutions of polynomial equations

A root of the polynomial p is a complex number z such that p(z) = 0. A most striking result is that all polynomials of degree n with real or complex coefficients have exactly n complex roots (counting multiple roots according to their multiplicity). This is known as the fundamental theorem of algebra, and shows that the complex numbers are an algebraically closed field. Indeed, the complex number field is the algebraic closure of the real number field, and Cauchy constructed complex numbers in this way. It can be identified as the quotient ring of the polynomial ring R[X] by the ideal generated by the polynomial X² + 1: :

\mathbb = \mathbb[ X ] / ( X^2 + 1). \,

This is indeed a field because X² + 1 is irreducible, hence generating a maximal ideal, in R[X]. The image of X in this quotient ring becomes the imaginary unit i.

Algebraic characterization

The field C is (up to field isomorphism) characterized by the following three facts:
- its characteristic is 0
- its transcendence degree over the prime field is the cardinality of the continuum
- it is algebraically closed Consequently, C contains many proper subfields which are isomorphic to C. Another consequence of this characterization is that the Galois group of C over the rational numbers is enormous, with cardinality equal to that of the power set of the continuum.

Characterization as a topological field

As noted above, the algebraic characterization of C fails to capture some of its most important properties. These properties, which underpin the foundations of complex analysis, arise from the topology of C. The following properties characterize C as a topological field:
- C is a field.
- C contains a subset P of nonzero elements satisfying:
- P is closed under addition, multiplication and taking inverses.
- If x and y are distinct elements of P, then either x-y or y-x is in P
- If S is any nonempty subset of P, then S+P=x+P for some x in C.
- C has a nontrivial involutive automorphism x->x
- , fixing P and such that xx
- is in P for any nonzero x in C. Given these properties, one can then define a topology on C by taking the sets
-

B(x,p) = \

as a base, where x ranges over C, and p ranges over P. To see that these properties characterize C as a topological field, one notes that P ∪ ∪ -P is an ordered Dedekind-complete field and thus can be identified with the real numbers R by a unique field isomorphism. The last property is easily seen to imply that the Galois group over the real numbers is of order two, completing the characterization. Pontryagin has shown that the only connected locally compact topological fields are R and C. This gives another characterization of C as a topological field, since C can be distinguished from R by noting the nonzero complex numbers are connected whereas the nonzero real numbers are not.

Complex analysis

The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example). Unlike real functions which are commonly represented as two dimensional graphs, complex functions have four dimensional graphs and may usefully be illustrated by color coding a three dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.

Applications

Control theory

In control theory, systems are often transformed from the time domain to the frequency domain using the Laplace transform. The system's poles and zeros are then analyzed in the complex plane. The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane. In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes, i.e. have real part greater than or less than zero. If a system has poles that are
- in the right half plane, it will be unstable,
- all in the left half plane, it will be stable,
- on the imaginary axis, it will be marginally stable. If a system has zeros in the right half plane, it is a nonminimum phase system.

Signal analysis

Complex numbers are used in signal analysis and other fields as a convenient description for periodically varying signals. The absolute value |z| is interpreted as the amplitude and the argument arg(z) as the phase of a sine wave of given frequency. If Fourier analysis is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as the real part of complex valued functions of the form :

f ( t ) = z e^ \,

where ω represents the angular frequency and the complex number z encodes the phase and amplitude as explained above. In electrical engineering, the Fourier transform is used to analyze varying voltages and currents. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. (Electrical engineers and some physicists use the letter j for the imaginary unit since i is typically reserved for varying currents and may come into conflict with i.) This use is also extended into digital signal processing and digital image processing, which utilize digital versions of Fourier analysis (and Wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, and video signals.

Improper integrals

In applied fields, the use of complex analysis is often used to compute certain real-valued improper integrals, by means of complex-valued functions. Several methods exist to do this, see methods of contour integration.

Quantum mechanics

The complex number field is also of utmost importance in quantum mechanics since the underlying theory is built on (infinite dimensional) Hilbert spaces over C.

Relativity

In special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time variable to be imaginary.

Applied mathematics

In differential equations, it is common to first find all complex roots r of the characteristic equation of a linear differential equation and then attempt to solve the system in terms of base functions of the form f(t) = e^rt.

Fluid dynamics

In fluid dynamics, complex functions are used to describe potential flow in 2d.

Fractals

Certain fractals are plotted in the complex plane e.g. Mandelbrot set and Julia set.

History

The earliest fleeting reference to square roots of negative numbers occurred in the work of the Greek mathematician and inventor Heron of Alexandria in the 1st century AD, when he considered the volume of an impossible frustum of a pyramid. They became more prominent when in the 16th century closed formulas for the roots of third and fourth degree polynomials were discovered by Italian mathematicians (see Niccolo Fontana Tartaglia, Gerolamo Cardano). It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. For example, Tartaglia's cubic formula gives the following solution to the equation

x^3-x=0

: :

\frac\left(\sqrt^+\frac\right).

At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation

z^3=i

has solutions −i,

\frac+\fraci

and

\frac+\fraci

. Substituting these in turn for

\sqrt^

into the cubic formula and simplifying, one gets 0,