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Commutativity

Commutativity

:For other meanings of commutation, see commutation (disambiguation).

Mathematical meaning

In mathematics, especially abstract algebra, a binary operation

\times

on a set S is commutative if :

x\times y = y\times x

for all x and y in S. Otherwise, the operation is noncommutative. Additionally, if :

x\times y = y\times x

for a particular pair of elements x and y, then x and y are said to commute. Every element commutes with itself and, in a group, every element commutes with the identity, with its own inverse, and with its powers. The most well-known examples of commutative binary operations are addition and multiplication of real numbers; for example:
- 4 + 5 = 5 + 4 (since both expressions evaluate to 9)
- 2 × 3 = 3 × 2 (since both expressions evaluate to 6) Further examples of commutative binary operations include addition and multiplication of complex numbers, addition of vectors, and intersection and union of sets. Among the noncommutative binary operations are subtraction (a − b), division (a/b), exponentiation (a^b), function composition (f o g), tetration (a↑↑b), matrix multiplication, and quaternion multiplication. An abelian group is a group whose group operation is commutative. A commutative ring is a ring whose multiplication is commutative. (Addition in a ring is always commutative.) In a field both addition and multiplication are commutative.

Neurophysiological meaning

In neurophysiology, commutative has much the same meaning as in algebra. Physiologist Douglas A. Tweed and coworkers consider whether certain neural circuits in the brain exhibit noncommutativity and state: :In non-commutative algebra, order makes a difference to multiplication, so that

a\times b\neq b\times a

. This feature is necessary for computing rotary motion, because order makes a difference to the combined effect of two rotations. It has therefore been proposed that there are non-commutative operators in the brain circuits that deal with rotations, including motor circuits that steer the eyes, head and limbs, and sensory circuits that handle spatial information. This idea is controversial: studies of eye and head control have revealed behaviours that are consistent with non-commutativity in the brain, but none that clearly rules out all commutative models. (Douglas A. Tweed and others, Nature 399, 261 - 263; 20 May 1999). Tweed goes on to demonstrate non-commutative computation in the vestibulo-ocular reflex by showing that subjects rotated in darkness can hold their gaze points stable in space---correctly computing different final eye-position commands when put through the same two rotations in different orders, in a way that is unattainable by any commutative system.

Commutation (disambiguation)

Commutation may mean:
- In Mathematics, commutation refers to a Commutative operation, where a x b = b x a
- In Law, commutation refers to a reduction in sentence for a criminal act.

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

Group (mathematics)

In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. For example, the set of integers is a group under the operation of addition. The branch of mathematics which studies groups is called group theory. The historical origin of group theory goes back to the works of Évariste Galois (1830), concerning the problem of when an algebraic equation is soluble by radicals. Previous to this work, groups were mainly studied concretely, in the form of permutations; some aspects of abelian group theory were known in the theory of quadratic forms. A great many of the objects investigated in mathematics turn out to be groups. These include familiar number systems, such as the integers, the rational numbers, the real numbers, and the complex numbers under addition, as well as the non-zero rationals, reals, and complex numbers, under multiplication. Another important example is given by non-singular matrices under multiplication, and more generally, invertible functions under composition. Group theory allows for the properties of these systems and many others to be investigated in a more general setting, and its results are widely applicable. Group theory is also a rich source of theorems in its own right. Groups underlie many other algebraic structures such as fields and vector spaces. They are also important tools for studying symmetry in all its forms; the principle that the symmetries of any object form a group is foundational for much mathematics. For these reasons, group theory is an important area in modern mathematics, and also one with many applications to mathematical physics (for example, in particle physics).

History

See Group theory.

Basic definitions

A group (G,
- ) is a nonempty set G together with a binary operation
- : G × G → G, satisfying the group axioms below. "a
- b" represents the result of applying the operation
- to the ordered pair (a, b) of elements of G. The group axioms are the following:
- Associativity: For all a, b and c in G, (a
- b)
- c = a
- (b
- c).
- Identity element: There is an element e in G such that for all a in G, e
- a = a
- e = a.
- Inverse element: For all a in G, there is an element b in G such that a
- b = b
- a = e, where e is the identity element from the previous axiom. You will often also see the axiom
- Closure: For all a and b in G, a
- b belongs to G. The way that the definition above is phrased, this axiom is not necessary, since binary operations are already required to satisfy closure. When determining if
- is a group operation, however, it is nonetheless necessary to verify that
- satisfies closure; this is part of verifying that it is in fact a binary operation. The above axioms are not strictly minimal from a logical viewpoint; they contain a small amount of redundancy. However, the difference is slight and in practice one usually just checks the above axioms. It should be noted that there is no requirement that the group operation be commutative, that is there may exist elements such that a
- b ≠ b
- a. A group G is said to be abelian (after the mathematician Niels Abel) (or commutative) if for every a, b in G, a
- b = b
- a. Groups lacking this property are called non-abelian. The order of a group G, denoted by |G| or o(G), is the number of elements of the set G. A group is called finite if it has finitely many elements, that is if the set G is a finite set. Note that we often refer to the group (G,
- ) as simply "G", leaving the operation
- unmentioned. But to be perfectly precise, different operations on the same set define different groups.

Notation for groups

Usually the operation, whatever it really is, is thought of as an analogue of multiplication, and the group operations are therefore written multiplicatively. That is:
- We write "a · b" or even "ab" for a
- b and call it the product of a and b;
- We write "1" for the identity element and call it the unit element;
- We write "a⁻¹" for the inverse of a and call it the reciprocal of a. However, sometimes the group operation is thought of as analogous to addition and written additively:
- We write "a + b" for a
- b and call it the sum of a and b;
- We write "0" for the identity element and call it the zero element;
- We write "−a" for the inverse of a and call it the opposite of a. Usually, only abelian groups are written additively, although abelian groups may also be written multiplicatively. When being noncommittal, one can use the notation (with "
- ") and terminology that was introduced in the definition, using the notation a⁻¹ for the inverse of a. If S is a subset of G and x an element of G, then, in multiplicative notation, xS is the set of all products ; similarly the notation Sx = ; and for two subsets S and T of G, we write ST for . In additive notation, we write x + S, S + x, and S + T for the respective sets.

Some elementary examples and nonexamples

An abelian group: the integers under addition

A group that we are introduced to in elementary school is the integers under addition. For this example, let Z be the set of integers, , and let the symbol "+" indicate the operation of addition. Then (Z,+) is a group (written additively). Proof:
- If a and b are integers then a + b is an integer. (Closure; + really is a binary operation)
- If a, b, and c are integers, then (a + b) + c = a + (b + c). (Associativity)
- 0 is an integer and for any integer a, 0 + a = a + 0 = a. (Identity element)
- If a is an integer, then there is an integer b := −a, such that a + b = b + a = 0. (Inverse element) This group is also abelian: a + b = b + a. The integers with both addition and multiplication together form the more complicated algebraic structure of a ring. In fact, the elements of any ring form an abelian group under addition, called the additive group of the ring.

Not a group: the integers under multiplication

On the other hand, if we consider the operation of multiplication, denoted by "·", then (Z,·) is not a group:
- If a and b are integers then a · b is an integer. (Closure)
- If a, b, and c are integers, then (a · b) · c = a · (b · c). (Associativity)
- 1 is an integer and for any integer a, 1 · a = a · 1 = a. (Identity element)
- However, it is not true that whenever a is an integer, there is an integer b such that ab = ba = 1. For example, a = 2 is a integer, but the only solution to the equation ab = 1 in this case is b = 1/2. We cannot choose b = 1/2 because 1/2 is not an integer. (Inverse element fails) Since not every element of (Z,·) has an inverse, (Z,·) is not a group. The most we can say is that it is a commutative monoid.

An abelian group: the nonzero rational numbers under multiplication

Consider the set of rational numbers Q, that is the set of numbers a/b such that a and b are integers and b is nonzero, and the operation multiplication, denoted by "·". Since the rational number 0 does not have a multiplicative inverse, (Q,·), like (Z,·), is not a group. However, if we instead use the set Q \ instead of Q, that is include every rational number except zero, then (Q \ ,·) does form an abelian group (written multiplicatively). The inverse of a/b is b/a, and the other group axioms are simple to check. We don't lose closure by removing zero, because the product of two nonzero rationals is never zero. Just as the integers form a ring, the rational numbers form the algebraic structure of a field. In fact, the nonzero elements of any given field form a group under multiplication, called the multiplicative group of the field.

A finite nonabelian group: permutations of a set

For a more concrete example, consider three colored blocks (red, green, and blue), initially placed in the order RGB. Let a be the action "swap the first block and the second block", and let b be the action "swap the second block and the third block". field In multiplicative form, we traditionally write xy for the combined action "first do y, then do x"; so that ab is the action RGB → RBG → BRG, i.e., "take the last block and move it to the front". If we write e for "leave the blocks as they are" (the identity action), then we can write the six permutations of the set of three blocks as the following actions:
- e : RGB → RGB
- a : RGB → GRB
- b : RGB → RBG
- ab : RGB → BRG
- ba : RGB → GBR
- aba : RGB → BGR Note that the action aa has the effect RGB → GRB → RGB, leaving the blocks as they were; so we can write aa = e. Similarly,
- bb = e,
- (aba)(aba) = e, and
- (ab)(ba) = (ba)(ab) = e; so each of the above actions has an inverse. By inspection, we can also determine associativity and closure; note for example that
- (ab)a = a(ba) = aba, and
- (ba)b = b(ab) = bab. This group is called the symmetric group on 3 letters, or S₃. It has order 6 (or 3 factorial), and is non-abelian (since, for example, ab ≠ ba). Since S₃ is built up from the basic actions a and b, we say that the set generates it. Every group can be expressed in terms of permutation groups like S₃; this result is Cayley's theorem and is studied as part of the subject of group actions.

Further examples

For some further examples of groups from a variety of applications, see Examples of groups and List of small groups.

Simple theorems

- A group has exactly one identity element.
- Every element has exactly one inverse.
- You can perform division in groups; that is, given elements a and b of the group G, there is exactly one solution x in G to the equation x
- a = b and exactly one solution y in G to the equation a
- y = b.
- The expression "a₁
- a₂
- ···
- a_n" is unambiguous, because the result will be the same no matter where we place parentheses.
- (Socks and shoes) The inverse of a product is the product of the inverses in the opposite order: (a
- b)⁻¹ = b⁻¹
- a⁻¹. These and other basic facts that hold for all individual groups form the field of elementary group theory.

Constructing new groups from given ones

#If a subset H of a group (G,
- ) together with the operation
- restricted on H is itself a group, then it is called a subgroup of (G,
- ). #The direct product of two groups (G,
- ) and (H,•) is the set G×H together with the operation (g₁,h₁)(g₂,h₂) = (g₁
- g₂,h₁•h₂). The product can also be defined with an infinite number of terms. #The direct external sum of a family of groups is the subgroup of the product constituted by elements that have a finite number of non zero terms. If the family is finite the direct sum and the product are of course the same. #Given a group G and a normal subgroup N, the quotient group is the set of cosets of G/N together with the operation (gN)(hN)=ghN.

Identity

- In creative works:
- Identity is a novel written by Milan Kundera.
- Identity is a movie starring John Cusack.
- Identity (music). In music, identity is a transformation of pitches.
- In business:
- Corporate identity is the physical manifestation of a business brand.
- Identity theft is the deliberate appropriation of someone else's identity (without that person's permission) for criminal purposes.
- In social science and psychology:
- Identity (social science). In the social sciences, identity has specific meanings, stemming from cognitive theory, sociology, politics, and psychology. See social identity and identity politics.
- Ethnic identity is a person's self-affiliation (or categorization by others) as a member of an ethnic group.
- Gender identity is the gender with which a person identifies (or is identified by others).
- Digital identity is the representation of identity in terms of digital information.
- Online identity is the digital identity established by computer network users.
- In philosophy:
- Identity is the sameness of two things (also see law of identity).
- In the philosophy of mind, the identity theory of mind holds that the mind is identical to the brain.
- Philosophy is also concerned with personal identity.
- In mathematics:
- An identity is an equality that holds regardless of the values of its variables.
- An identity object is an entity that does not change other objects: see identity function, identity element, identity matrix, and identity morphism.
- In computer science:
- Identity (object-oriented programming). In object-oriented programming, identity is a property of objects that allows those objects to be distinguished from each other.

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.

Exponentiation

In mathematics, exponentiation is a process generalized from repeated (or iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition. (The next operation after exponentiation is sometimes called tetration; repeating this process leads to the Ackermann function.)

Positive integer exponents

The simplest case involves a positive integer exponent: For example, 3⁵ = 3 × 3 × 3 × 3 × 3 = 243. Here, 3 is the base, 5 is the exponent (written as a superscript), and 243 is 3 raised to the 5th power or 3 raised to the power 5. (The word "raised" is usually omitted, and most often "power" as well, so 3⁵ is typically pronounced "three to the fifth" or "three to the five".) Notice that the base 3 appears 5 times in the repeated multiplication, because the exponent is 5.

Notation

In contexts where superscripts are not available, such as computer languages and e-mail, 3⁵ is commonly written "3^5" (with a caret see Bc_%28Unix%29), and sometimes as "3
- 5" (with two asterisks, see Fortran). Another way of writing it, requiring Unicode encoding, is "3↑5" (with an up-arrow; HTML ↑). The exponent 1 is not normally written, since any number to the power 1 is itself. The exponents 2 and 3 occur so commonly that there are short words for them: the powers are called the square and cube of the base, respectively. 3² is pronounced "three squared," and 3³ is "three cubed." The exponents 1/2 and 1/3 occur so commonly that there are short words for them too: the powers are called the Square root and Cube root of the base, respectively.

Computation

aⁿ can be computed by means of n-1 multiplications, but, if n is a large number, the work required can be much reduced by the following trick. :aⁿ= a^b(a²)^c where :n=b+2c ; b=n mod 2 ; c=(n-b)/2. For example :2¹³=(2)(4)⁶=(2)(16)³=(2)(16)(256)¹=(32)(256)=8192 The number of multiplications performed was 5, rather than 12. The article Exponentiation by squaring provides more detail.

Zero exponents

The meaning of 3⁵ may also be viewed as 1 × 3 × 3 × 3 × 3 × 3: the starting value 1 (the identity element of multiplication) is multiplied by the base, as many times as indicated by the exponent. With this definition in mind, it is easy to see how to generalize exponentiation to zero exponents: any number to the 0 power is 1. :

\ x^0=1

0⁰ is sometimes taken as undefined, but is sensibly defined as 1. See Empty_product#0_raised_to_the_0th_power.

Negative integer exponents

A negative exponent indicates repeated division by the base. Thus 3^-5 = 1 ÷ 3 ÷ 3 ÷ 3 ÷ 3 ÷ 3 = 1/243. Raising any nonzero number to the -1 power produces its reciprocal. :

\ x^=\frac

\ x^=(x^n)^=\frac

Raising 0 to a negative power would imply division by 0, and so is undefined.

Identities and properties

Important identities satisfied by exponentiation include:
- x^m+n = x^mxⁿ
- x^m-n = x^m/xⁿ
- (x^m)ⁿ = x^mn Whereas addition or multiplication are commutative (for example, 2+3 = 5 = 3+2 and 2×3 = 6 = 3×2), this is not true of exponentiation: 2³ = 8 while 3² = 9. Similarly, whereas addition or multiplication are associative (for example, (2+3)+4 = 9 = 2+(3+4) and (2×3)×4 = 24 = 2×(3×4)), this is not true of exponentiation either: 2³ to the 4th power is 8⁴ or 4,096, while 2 to the 3⁴ power is 2⁸¹ or 2,417,851,639,229,258,349,412,352.

Powers of ten

Powers of 10 are easy to compute because we use a base ten number system: for example 10⁶ = 1 million, which is 1 followed by 6 zeros. Exponentiation with base 10 is often used in the physical sciences to describe large or small numbers in scientific notation; for example, 299792458 (the speed of light in a vacuum, in meters per second) can be written as 2.99792458 × 10⁸ and then approximated as 2.998 × 10⁸ if this is useful. SI prefixes are also used to describe small or large quantities, and these are also based on powers of 10; for example, the prefix kilo means 10³ = 1000, so a kilometre is 1000 metres.

Powers of two

Powers of 2 are important in computer science; for example, there are 2ⁿ possible values for a variable that takes n bits to store in memory. They occur so commonly that SI prefixes are commonly reinterpreted to refer to them: 1 kilobyte = 2¹⁰ = 1024 bytes. As the standard meanings of the prefixes also occur, confusion may result, and in 1998 the International Electrotechnical Commission approved a set of binary prefixes. For instance, the prefix for multiples of 1024 is kibi-, so 1024 bytes is 1 kibibyte. Other prefixes are mebi-, gibi-, and tebi-.

Powers of e

A big power of a number close to one can be written in the form :

\ \left(1+\frac\right)^n

Here

\ n

is an integer, so the power is well defined. Substituting :

\ n=mx

gives :

\ \left(1+\frac\right)^n= \left(1+\frac\right)^= \left( \left(1+\frac \right) ^m\right) ^x

. The limiting value when

\ m

goes to infinity has got the name e :

\ e=\lim_ \left(1+\frac \right) ^m

. Any power of e is defined by :

\ e^x=\lim_ \left(1+\frac \right) ^n

. (For further reading, see Exponential function).

Powers of one

The above definition of

e^x

also apply when the exponent

x

is a complex number. If the exponent is an imaginary number,

i x

, then

e^

is a complex number on the unit circle, a direction. (See Euler's formula). The real number

x

is an angle measured in radian. The angle

2\pi

radian is a turn: :

\ e^=1

2\pi i

is a solution to the equation

e^x=1

. It is not the only one. The solutions are simply

2\pi i n

where

n

is an integer (see Pi). The expression :

\ e^

is an x 'th power of one. If x is an integer the result is 1. If x is a rational number the result is a root of unity. If x is a real number the result is a direction.

Powers of zero

If the exponent

\ x

is positive, the power of zero is zero:

\ 0^x=0

. If the exponent is negative, the power of zero

\ 0^

is undefined. If the exponent is zero, the power is always one.

\ 0^0=1

. (See Empty_product#0_raised_to_the_0th_power)

Fractional exponents

Exponentiation with a fractional exponent

\ \frac

, where

n\ge 2

, is a solution to the equation :

\ x^n = a.

For

a \ne 0

this equation has

n

solutions. If

\ a

is a positive real number, then one of the solutions is also a positive real number. The positive solution to

\ x^n = a

is called a radical,

\sqrt[n]

for

a\ne 1.

The positive solution to

\ x^n = 1

is (of course)

\ 1

. All the solutions are given by: :

e^\sqrt[n]

for

k=0,\ldots,n-1

. If

a

is a complex number which is not a positive real number, then

\sqrt[n]

is a multivalued function of

a

. Exponentiation with a fractional exponent

m/n

a^ = \left( \sqrt[n]\right)^m.

For example: 8^2/3 = 4. It was once believed that exponentiation with fractional exponents leads to finding roots of polynomials. That this is not true in general is the assertion of the Abel-Ruffini theorem. For example, the solutions of the equation :

\ x^5=x+1

cannot be expressed in terms of fractional exponents. For solving any equation of the n^th degree, see Root-finding algorithm#How to solve an algebraic equation.

Arbitrary real and complex exponents

Exponentiation to an arbitrary real exponent can then be defined by continuity. With real numbers, the exponential function exp is the same as raising the transcendental number e to the indicated power: exp x = e^x. Exponentiation of real numbers, and even complex numbers, can be understood with the aid of the exponential function and its inverse, the natural logarithm; in general, we can define : x^y = exp(y ln x). However, in the complex number field, it should be noted that logarithms are always multi-valued functions, usually with an imaginary periodicity. Therefore, expressions such as x^y are always similarly multi-valued. A "branch cut" may be created in the complex plane, if a single-valued logarithm or power is desired. Most often, this branch cut is made along the negative real axis. The use of e^x in this context is usually assumed to use this "principal branch" of the logarithm, so the results correspond with that of the exponential function which satisfies analyticity constraints. For more on exponents in real and complex numbers, and other situations relevant to mathematical analysis, see Exponential function. That article also lists certain exponential laws (more general than the algebraic laws listed below) that apply in these situations.

Exponents on function names

When the name or symbol of a function is given an integer superscript, as if being raised to a power, this commonly refers to repeated function composition rather than repeated multiplication. Thus f³(x) may mean f(f(f(x))); in particular, f^-1(x) usually denotes f's inverse function. A special syntax applies to the trigonometric functions: a positive exponent applied to the function's abbreviation means that the result is raised to that power, while an exponent of -1 indicates the inverse function. That is, sin²x is just a shorthand way to write (sin x)² without using parentheses, whereas sin^-1x refers to the inverse function of the sine, also called arcsin x. There is no need for a shorthand of this kind for reciprocal trigonometric functions since they each have their own name and abbreviation already: (sin x)^-1 is normally just written as csc x.

Exponentiation in abstract algebra

Exponentiation can also be understood purely in terms of abstract algebra, if we limit the exponents to integers. Specifically, suppose that X is a set with a power-associative binary operation, which we will write multiplicatively. In this very general situation, we can define xⁿ for any element x of X and any nonzero natural number n, by simply multiplying x by itself n times; by definition, power associativity means that it doesn't matter in which order we perform the multiplications. Now additionally suppose that the operation has an identity element 1. Then we can define x⁰ to be equal to 1 for any x. Now xⁿ is defined for any natural number n, including 0. Finally, suppose that the operation has inverses, and that the multiplication is associative (so that the magma is a group). Then we can define x^-n to be the inverse of xⁿ when n is a natural number. Now xⁿ is defined for any integer n. Exponentiation in this purely algebraic sense satisfies the following laws (whenever both sides are defined):
-

\ x^=x^mx^n

\ x^=x^m/x^n

\ x^=1/x^n

\ x^0=1

\ x^1=x

\ x^=1/x

\ (x^m)^n=x^

Here, we use a division slash ("/") to indicate multiplying by an inverse, in order to reserve the symbol x^-1 for raising x to the power -1, rather than the inverse of x. However, as one of the laws above states, x^-1 is always equal to the inverse of x, so the notation doesn't matter in the end. If in addition the multiplication operation is commutative (so that the magma is an abelian group), then we have some additional laws:
- (xy)ⁿ = xⁿyⁿ
- (x/y)ⁿ = xⁿ/yⁿ Notice that in this algebraic context, 0⁰ is always equal to 1. When 0⁰ is attained as a limit, however, it may be more useful to leave 0⁰ undefined. However, when exponentiation is purely algebraic, that is when the exponents are taken only to be integers, then it is generally most useful to let 0⁰ be 1, just like every other case of x⁰. For example, if you expand (0 + x)ⁿ using the binomial theorem, you'll want to use 0⁰ = 1. If we take this whole theory of exponentiation in an algebraic context but write the binary operation additively, then "exponentiation is repeated multiplication" can be reinterpreted as "multiplication is repeated addition". Thus, each of the laws of exponentiation above has an analogue among laws of multiplication. When one has several operations around, any of which might be repeated using exponentiation, it is common to indicate which operation is being repeated by placing its symbol in the superscript. Thus, x^{- n} is x
- ···
- x, while x^#n is x # ··· # x, whatever the operations
- and # might be. Exponential notation is also used, especially in group theory, to indicate conjugation. That is, g^h = h^-1gh, where g and h are elements of some group. Although conjugation obeys some of the same laws as exponentiation, it is not an example of repeated multiplication in any sense. A quandle is an algebraic structure in which these laws of conjugation play a central role.

Exponentiation over sets

The above algebraic treatment of exponentiation builds a finitary operation out of a binary operation. In more general contexts, one may be able to define an infinitary operation directly on an indexed set. For example, in the arithmetic of cardinal numbers, it makes sense to say :

\prod_ k_

for any index set I and cardinal numbers k_i. By taking k_i = k for every i, this can be interpreted as a repeated product, and the result is k^I. In fact, this result depends only on the cardinality of I, so we can define exponentiation of cardinal numbers so that k^l is k^I for any set I whose cardinality is l. This can be done even for operations on sets or sets with extra structure. For example, in linear algebra, it makes sense to index direct sums of vector spaces over arbitrary index sets. That is, we can speak of :

\bigoplus_ V_,

where each V_i is a vector space. Then if V_i = V for each i, the resulting direct sum can be written in exponential notation as V^(+)I, or simply V^I with the understanding that the direct sum is the default. We can again replace the set I with a cardinal number k to get V^k, although without choosing a specific standard set with cardinality k, this is defined only up to isomorphism. Taking V to be the field R of real numbers (thought of as a vector space over itself) and k to be some natural number n, we get the vector space that is most commonly studied in linear algebra, the Euclidean space Rⁿ. If the base of the exponentiation operation is itself a set, then by default we assume the operation to be the Cartesian product. In that case, S^I becomes simply the set of all functions from I to S. This fits in with the exponentiation of cardinal numbers once gain, in the sense that |S^I| = |S|^|I|, where |X| is the cardinality of X. When I=2=, we have |2^X| = 2^|X|, where 2^X, usually denoted by PX, is the power set of X. (This is where the term "power set" comes from.) Note that exponentiation of cardinal numbers doesn't match up with exponentiation of ordinal numbers, which is defined by a limit process. In the ordinal numbers, a^b is the smallest ordinal number greater than a^c for c < b when b is a limit ordinal, and of course a^b+1 := a^ba. In category theory, we learn to raise any object in a wide variety of categories to the power of a set, or even to raise an object to the power of an object, using the exponential.

Syntax in some common languages / applications

- x ^ y: Basic, Matlab and many others
- x
- y: Fortran, Perl, Python, Ruby
- Power(x, y): Pascal
- pow(x, y): C, C++
- Math.pow(x, y): Java, JavaScript Note that C, C++, Java and JavaScript represent bitwise XOR with ^ .

Table

Table of kⁿ, with k on the left and n at the top.

External links

- [http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0to0/ sci.math FAQ: What is 0⁰?]
- Category:Exponentials ja:冪乗

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.
-
- [http://www.cde.ca.gov/be/st/ss/mthmain.asp California State Board of Education mathematics content standards] Adopted December 1997, accessed December 2005.
-
-
-
- Available [http://www.nap.edu/books/0309069955/html/index.html here] from the publisher.
-
-
-

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

Multiplication

:This article is about multiplication in mathematics. For multiplication in music, see multiplication (music). In its simplest form, multiplication is the sum of a list of identical numbers. For example, the product 7 × 4 is 7 + 7 + 7 + 7. The numbers being multiplied are called the multiplicand and multiplier or the factors.

Notation

Multiplication can be denoted in several equivalent ways. All of the following mean, "5 times 2": :5×2 :5·2 :(5)2, 5(2), (5)(2), 5[2], [5]2, [5][2] :5
- 2 The asterisk (
- ) is often used on computers because it is a symbol on every keyboard, but it is rarely used when writing math by hand. This usage originated in the FORTRAN programming language. Frequently, multiplication is implied by Juxtaposition rather than shown in a notation. This is standard in algebra, taking forms like :5x and xy This is potentially confusing if variables are permitted to have names longer than one letter. The notation is not used with numbers alone: 52 never means 5 × 2. If the terms are not written out individually, then the product may be written with an ellipsis to mark out the missing terms, as with other series operations (like sums). Thus, the product of all the natural numbers from 1 to 100 can be written

1 \cdot 2 \cdot \ldots \cdot 99 \cdot 100

. This can also be written with the ellipsis vertically placed in the middle of the line, as

1 \cdot 2 \cdot \cdots \cdot 99 \cdot 100

. Alternatively, the product can be written with the product symbol, which derives from the capital letter Π (Pi) in the Greek alphabet. This is defined as: :

\prod_^ x_ := x_ \cdot x_ \cdot x_ \cdot \cdots \cdot x_ \cdot x_.

The subscript gives the symbol for a dummy variable (

i

in our case) and its lower value (

m

); the superscript gives its upper value. So for example: :

\prod_^ \left(1 + \right) = \left(1 + \right) \cdot \left(1 + \right) \cdot \left(1 + \right) \cdot \left(1 + \right) \cdot \left(1 + \right) = .

One may also consider products of infinitely many terms; these are called infinite products. Notationally, we would replace n above by the infinity symbol (∞). The product of such a series is defined as the limit of the product of the first

n

terms, as

n

grows without bound. That is: :

\prod_^ x_ := \lim_ \prod_^ x_.

One can similarly replace

m

with negative infinity, and :

\prod_^\infty x_i := \left(\lim_\prod_^m x_i\right) \cdot \left(\lim_\prod_^n x_i\right),

for some integer

m

, provided both limits exist.

Definition

As for what multiplication means, the product of two whole numbers n and m is: :

mn := \sum_^n m

This is just a shorthand for saying, "Add m to itself n times." Expanding the above to make its meaning more clear: :m × n = m + m + m + ... + m such that there are n m's added together. So for instance:

5 × 2 = 5 + 5 = 10
2 × 5 = 2 + 2 + 2 + 2 + 2 = 10
4 × 3 = 4 + 4 + 4 = 12
m × 6 = m + m + m + m + m + m

Using this definition, it is easy to prove some interesting properties of multiplication. As the first two examples above hint at, the order in which two numbers are multiplied does not matter. This is called the commutative property and it turns out to be true in general that for any two numbers x and y, :x · y = y · x. Multiplication also has what is called the associative property. The associative property means that for any three numbers x, y, and z, :(x · y)z = x(y · z). Note from algebra: the parentheses mean that the operations inside the parentheses must be done before anything outside the parentheses is done. Multiplication also has what is called a distributive property with respect to the addition, because :x(y + z) = xy + xz. Also of interest is that any number times 1 is equal to itself, thus, :1 · x = x. and this is called the identity property What about zero? Well, we have: :m · 0 = m + m + m +...+ m where there are zero m's added together. The sum of zero m's is zero, so :m · 0 = 0 no matter what m is (as long as it is finite). Multiplication with negative numbers also requires a little thought. First consider negative 1. For any positive integer m: :(−1)m = (−1) + (−1) +...+ (−1) = −m This is an interesting fact that shows that any negative number is just negative one multiplied by a positive number. So multiplication with any integers can be represented by multiplication of whole numbers and (−1)'s. All that remains is to explicitly define (−1)(−1): :(−1)(−1) = −(−1) = 1 In this way, the multiplication of any two integers is defined. The definitions can be extended to larger and larger sets of numbers: first to vulgar fractions called the rational numbers, then to infinitely long decimals called real numbers, and then to the complex numbers. Students are sometimes mystified when told that the result of multiplying no numbers is 1. A formal recursive definition of multiplication can be given by the rules: : x · 0 = 0 : x · y = x + x·(y − 1) where x is a real number, and y is a natural number. Once multiplication has been defined for natural numbers, it can be extended to include integers, and then to real and complex numbers.

Computation

For fast ways to compute products of large numbers, see multiplication algorithms. Some algorithms are suitable for multiplying numbers using pencil and paper. Most, such as lattice multiplication, require a multiplication table of memorized or consulted products of small numbers (typically any two numbers from 0 to 9); the peasant multiplication algorithm does not.

External links

- [http://www.cut-the-knot.org/do_you_know/multiplication.shtml Multiplication] at cut-the-knot
- [http://www.mathsisfun.com/multiplying-negatives.html Multiplying Negative Numbers]
- [http://www.cut-the-knot.org/blue/SysTable.shtml Arithmetic Operations In Various Number Systems] at cut-the-knot
- [http://webhome.idirect.com/~totton/abacus/pages.htm#Multiplication1 Multiplication on a Japanese abacus] selected from [http://webhome.idirect.com/~totton/abacus/ Abacus: Mystery of the Bead]
- [http://webhome.idirect.com/~totton/suanpan/mod_mult/ Modern Chinese Multiplication Techniques on an Abacus] Category:Elementary arithmetic ko:곱셈 ja:乗法 simple:Multiplication th:การคูณ

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

Expression (mathematics)

An expression combines numbers, operators, and/or free variables and bound variables: bound variables are defined in the expression (they are for internal use), free variables are taken from the context. For a given combination of values for the free variables, an expression may be evaluated to a value, and is said to have that value, although for some combinations of values of the free variables, the expression may be undefined. Thus an expression represents a function of the values for the free variables. The evaluation of an expression is dependent on the definition of the mathematical operators and system of values that forms the context of an expression. Two expressions are said to be equivalent if, for each combination of values for the free variables, they have the same value, i.e., they represent the same function. Example: The expression :

\sum_^ 2xy

has free variable x, bound variable y, constants 1, 2, and 3, two occurrences of an implicit multiplication operator, and a summation operator. The expression is equivalent with the simpler expression 12x. The value for x=3 is 36. Expressions and their evaluation were formalised by Alonzo Church and Stephen Kleene in the 1930s in their lambda calculus. The lambda calculus has been a major influence in the development of modern mathematics and computer programming languages. One of the more interesting results of the lambda calculus is that the equivalence of two expressions in the lambda calculus is in some cases undecidable. This is also true of any expression in any system that has power equivalent to the lambda calculus.

External links

- [http://www.mathematics21.org/theory-of-formulas-index.html Axiomatic Theory of Formulas] - theory of expressions on high abstraction level.
- [http://www.algebra.com/services/rendering/ Plot mathematical expressions] this system plots math equations, graphs, diagrams, and even animated cartoons of transformation of math expressions and arithmetic operations. Knowledge of TeX not required. Category:Abstract algebra Category:Algebra ja:数式

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