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Addition

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

Image:Addition01.svg

Licensing

Arithmetic

Arithmetic or arithmetics (from the Greek word αριθμός = number) in common usage is a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numerals, though in usage by professional mathematicians, it often is treated as a synonym for number theory. It is the oldest and simplest branch of mathematics, used widely by almost everyone from simple daily counting to more advanced science and business.

Arithmetic operations

The traditional arithmetic operations are addition, subtraction, multiplication and division, although more advanced operations (such as manipulations of percentages, square root, exponentiation, and logarithmic functions) are also sometimes included in this subject. Arithmetic is performed according to an order of operations. The arithmetic of natural numbers, integers, rational numbers (in the form of vulgar fractions), and real numbers (using the decimal place-value system known as algorism) is typically studied by schoolchildren, who learn manual algorithms for arithmetic. However, in adult life, many people prefer to use tools such as calculators, computers, or the abacus to perform the more complex arithmetical computations.

Number theory

The term arithmetic is also used to refer to number theory. This includes the properties of integers related to primality, divisibility, and the solution of equations by integers, as well as modern research which is an outgrowth of this study. It is in this context that one runs across the fundamental theorem of arithmetic and arithmetic functions. A Course in Arithmetic by Serre reflects this usage, as do such phrases as first order arithmetic or arithmetical algebraic geometry.

Summation

Summation is the addition of a set of numbers; the result is their sum. The "numbers" to be summed may be natural numbers, complex numbers, matrices, or still more complicated objects. An infinite sum is a subtle procedure known as a series.

Notation

The sum of 1, 2, and 4 is 1 + 2 + 4 = 7. Since addition is associative, it does not matter whether we interpret "1 + 2 + 4" as (1 + 2) + 4 or as 1 + (2 + 4); the result is the same, so parentheses are usually omitted in a sum. Addition is also commutative, so the order in which the numbers written does not affect its sum. If a sum has too many terms to write them all out individually, the sum may be written with an ellipsis to mark out the missing terms. Thus, the sum of all the natural numbers from 1 to 100 is 1 + 2 + … + 99 + 100 = 5050. Sums can be represented by the summation symbol, a capital Sigma. This is defined as: :

\sum_^ x_ = x_ + x_ + x_ + \cdots + x_ + x_.

The subscript gives the symbol for a dummy variable, i. Here, i represents the index of summation; m is the lower bound of summation, and n is the upper bound of summation. So, for example: :

\sum_^ x^ = 2^ + 3^ + 4^ + 5^ + 6^ = 90.

One often sees generalizations of this notation in which an arbitrary logical condition is supplied, and the sum is intended to be taken over all values satisfying the condition. For example: :

\sum_ f(x)

is the sum of f(x) over all (integer) x in the specified range, :

\sum_ f(x)

is the sum of f(x) over all integers x in the set S, and :

\sum_\;\mu(d)

is the sum of μ(d) over all integers d dividing n. There are also ways to generalize the use of many sigma signs. For example, :

\sum_

is the same as :

\sum_\ell\sum_.

Computerized notation

Summations can also be represented in a programming language.

Special cases

It's possible to add fewer than 2 numbers:
- If you add the single term x, then the sum is x.
- If you add zero terms, then the sum is zero, because zero is the identity for addition. This is known as the empty sum. These degenerate cases are usually only used when the summation notation gives a degenerate result in a special case. For example, if m = n in the definition above, then there is only one term in the sum; if m = n + 1, then there is none.

Approximation by definite integrals

Many such approximations can be obtained by the following connection between sums and integrals, which holds for any increasing function f: :

\int_^ f(s)\, ds \le \sum_^ f(i) \le \int_^ f(s)\, ds.

For more general approximations, see the Euler-Maclaurin formula. For functions that are integrable on the interval

[a,b]

, the Riemann sum can be used as an approximation of the definite integral. For example, the following formula is the left Riemann sum with equal partitioning of the interval :

\frac\sum_^ f\left(a+i\fracn\right)\approx \int_a^b f(x)\,dx

The accuracy of such approximation increases with the number of subintervals,

n

Identities

The following are useful identities: :

\sum_^ i = \frac

(see arithmetic series);

\sum_^n m = mn

\sum_^nk = \frac

\sum_^ (2i - 1) = n^2;

\sum_^ i^ = \frac;

\sum_^ i^ = \left(\frac\right)^;

\sum_^ x^ = \frac

(see geometric series); :

\sum_^ x^ = \frac

(special case of the above where

=0

) :

\sum_^ x^ = \frac;

(special case of the above,

\lim_

and

|x|<1

); :

\sum_^\infty ix^=\frac

(only for

|x|<1

); :

\sum_^  = 2^

(see binomial coefficient);

\sum_^  = .

\left(\sum_i a_i\right)\left(\sum_j b_j\right) = \sum_i\sum_j a_ib_j

^2 = 2\sum_i\sum_ a_ia_j + \sum_i a_i^2

In general, the sum of the first n mth powers is :

\sum_^n i^m = \frac + \sum_^m\frac(n+1)^,

where

B_k

is the kth Bernoulli number. The following are useful approximations (using theta notation): :

\sum_^ i^ = \Theta(n^)

for every real constant c greater than -1;

\sum_^ \frac = \Theta(\log);

\sum_^ c^ = \Theta(c^)

for every real constant c greater than 1;

\sum_^ \log(i)^ = \Theta(n \cdot \log(n)^)

for every nonnegative real constant c;

\sum_^ \log(i)^ \cdot i^ = \Theta(n^ \cdot \log(n)^)

for all nonnegative real constants c and d;

\sum_^ \log(i)^ \cdot i^ \cdot b^ = \Theta (n^ \cdot \log(n)^ \cdot b^)

for all nonnegative real constants b > 1, c, d.

External links

- Category:Arithmetic Category:Mathematical notation

Series (mathematics)

In mathematics, a series is the sum of a sequence of terms. That is, a series is a list of numbers with addition operations between them, e.g, :1 + 2 + 3 + 4 + 5 + ... which may or may not be meaningful. In most cases of interest the terms of the sequence are produced according to a certain rule, e.g., by a formula, by an algorithm, by a sequence of measurements, or even by a random number generator. Series may be finite, or infinite; in the first case they may be handled with elementary algebra, but infinite series require tools from mathematical analysis if they are to be applied in anything more than a tentative way. Examples of simple series include the arithmetic series which is a sum of an arithmetic progression, written as: :

\sum_^k (an+b);

and finite geometric series, a sum of a geometric progression, which can be written as: :

\sum_^k a^.

Infinite series

The sum of an infinite series is a limit of partial sums of infinitely many terms. Such a limit can have a finite value; if it has, the series is said to converge; if it does not, it is said to diverge. The fact that infinite series can converge resolves several of Zeno's paradoxes. The simplest convergent infinite series is perhaps :

1+\frac+\frac+\frac+\frac+\cdots

It is possible to "visualize" its convergence on the real number line: we can imagine a line of length 2, with successive segments marked off of lengths 1, 1/2, 1/4, etc. There is always room to mark the next segment, because the amount of line remaining is always the same as the last segment marked: when we have marked off 1/2, we still have a piece of length 1/2 unmarked, so we can certainly mark the next 1/4. This argument does not prove that the sum is equal to 2 (although it is), but it does prove that it is at most 2 — in other words, the series has an upper bound. This series is a geometric series and mathematicians usually write it as: :

\sum_^\infty 2^=2.

An infinite series is formally written as :

\sum_^\infty a_n

where the elements a_n are real (or complex) numbers. We say that this series converges towards S, or that its value is S, if the limit :

\lim_\sum_^N a_n

exists and is equal to S. If there is no such number, then the series is said to diverge. The sequence of partial sums is defined as the sequence :

\sum_^N a_n

indexed by N. Then, the definition of series convergence simply says that the sequence of partial sums has limit S, as N → ∞.

Formal definition

Mathematicians usually define a series as the above sequence of partial sums. The notation :

\sum_^\infty a_n

represents then a priori this sequence, which is always well defined, but which may or may not converge. Only in the latter case, i.e., if this sequence has a limit, the notation is also used to denote the limit of this sequence. To make a distinction between these two completely different objects (sequence vs. numerical value), one may sometimes omit the limits (atop and below the sum's symbol) in the former case, although it is usually clear from the context which one is meant. Also, different notions of convergence of such a sequence do exist (absolute convergence, summability., etc). In case the elements of the sequence (and thus of the series) are not simple numbers, but, for example, functions, still more types of convergence can be considered (pointwise convergence, uniform convergence, etc.; see below).

History of the theory of infinite series

Convergence criteria

The investigation of the validity of infinite series is considered to begin with Gauss. Euler had already considered the hypergeometric series :

1 + \fracx + \fracx^2 + \cdots.

on which Gauss published a memoir in 1812. It established simpler criteria of convergence, and the questions of remainders and the range of convergence. Cauchy (1821) insisted on strict tests of convergence; he showed that if two series are convergent their product is not necessarily so, and with him begins the discovery of effective criteria. The terms convergence and divergence had been introduced long before by Gregory (1668). Euler and Gauss had given various criteria, and Maclaurin had anticipated some of Cauchy's discoveries. Cauchy advanced the theory of power series by his expansion of a complex function in such a form. Abel (1826) in his memoir on the series :

1 + \fracx + \fracx^2 + \cdots

corrected certain of Cauchy's conclusions, and gave a completely scientific summation of the series for complex values of

m

and

x

. He showed the necessity of considering the subject of continuity in questions of convergence. Cauchy's methods led to special rather than general criteria, and the same may be said of Raabe (1832), who made the first elaborate investigation of the subject, of De Morgan (from 1842), whose logarithmic test DuBois-Reymond (1873) and Pringsheim (1889) have shown to fail within a certain region; of Bertrand (1842), Bonnet (1843), Malmsten (1846, 1847, the latter without integration); Stokes (1847), Paucker (1852), Tchebichef (1852), and Arndt (1853). General criteria began with Kummer (1835), and have been studied by Eisenstein (1847), Weierstrass in his various contributions to the theory of functions, Dini (1867), DuBois-Reymond (1873), and many others. Pringsheim's (from 1889) memoirs present the most complete general theory.

Uniform convergence

The theory of uniform convergence was treated by Cauchy (1821), his limitations being pointed out by Abel, but the first to attack it successfully were Stokes and Seidel (1847-48). Cauchy took up the problem again (1853), acknowledging Abel's criticism, and reaching the same conclusions which Stokes had already found. Thomé used the doctrine (1866), but there was great delay in recognizing the importance of distinguishing between uniform and non-uniform convergence, in spite of the demands of the theory of functions.

Semi-convergence

Semi-convergent series were studied by Poisson (1823), who also gave a general form for the remainder of the Maclaurin formula. The most important solution of the problem is due, however, to Jacobi (1834), who attacked the question of the remainder from a different standpoint and reached a different formula. This expression was also worked out, and another one given, by Malmsten (1847). Schlömilch (Zeitschrift, Vol.I, p. 192, 1856) also improved Jacobi's remainder, and showed the relation between the remainder and Bernoulli's function

F(x) = 1^n + 2^n + \cdots + (x -
1)^n

. Genocchi (1852) has further contributed to the theory. Among the early writers was Wronski, whose "loi suprême" (1815) was hardly recognized until Cayley (1873) brought it into prominence.

Fourier series

Fourier series were being investigated as the result of physical considerations at the same time that Gauss, Abel, and Cauchy were working out the theory of infinite series. Series for the expansion of sines and cosines, of multiple arcs in powers of the sine and cosine of the arc had been treated by Jakob Bernoulli (1702) and his brother Johann Bernoulli (1701) and still earlier by Viète. Euler and Lagrange simplified the subject, as did Poinsot, Schröter, Glaisher, and Kummer. Fourier (1807) set for himself a different problem, to expand a given function of

x

in terms of the sines or cosines of multiples of

x

, a problem which he embodied in his Théorie analytique de la Chaleur (1822). Euler had already given the formulas for determining the coefficients in the series; Fourier was the first to assert and attempt to prove the general theorem. Poisson (1820-23) also attacked the problem from a different standpoint. Fourier did not, however, settle the question of convergence of his series, a matter left for Cauchy (1826) to attempt and for Dirichlet (1829) to handle in a thoroughly scientific manner (see convergence of Fourier series). Dirichlet's treatment (Crelle, 1829), of trigonometric series was the subject of criticism and improvement by Riemann (1854), Heine, Lipschitz, Schläfli, and DuBois-Reymond. Among other prominent contributors to the theory of trigonometric and Fourier series were Dini, Hermite, Halphen, Krause, Byerly and Appell.

Some types of infinite series

- A geometric series is one where each successive term is produced by multiplying the previous term by a constant number. Example: ::

1 +  +  +  +  + \cdots=\sum_^\infty.

:In general, the geometric series ::

\sum_^\infty z^n

:converges if and only if |z| < 1.
- The harmonic series is the series ::

1 +  +  +  +  + \cdots =\sum_^\infty .

- An alternating series is a series where terms alternate signs. Example: ::

1 -  +  -  +  - \cdots =\sum_^\infty (-1)^ .

- The series ::

\sum_^\infty\frac

:converges if r > 1 and diverges for r ≤ 1, which can be shown with the integral criterion 5) from below, in convergence tests. As a function of r, the sum of this series is Riemann's zeta function.
- A telescoping series ::

\sum_^\infty (b_n-b_)

:converges if the sequence b_n converges to a limit L as n goes to infinity. The value of the series is then b₁ − L.

Absolute convergence

:Main article: absolute convergence. A series :

\sum_^\infty a_n

is said to converge absolutely if the series of absolute values :

\sum_^\infty \left|a_n\right|

converges. In this case, the original series, and all reorderings of it, converge, and converge towards the same sum. The Riemann series theorem says that if a series converges, but not absolutely, then one can always find a reordering of the terms so that the reordered series diverges. Moreover, if the a_n are real and S is any real number, one can find a reordering so that the reordered series converges with limit S.

Convergence tests

- Comparison test 1: If ∑b_n is an absolutely convergent series such that |a_n | ≤ C |b_n | for some number C and for sufficiently large n , then ∑a_n converges absolutely as well. If ∑|b_n | diverges, and |a_n | ≥ |b_n | for all sufficiently large n , then ∑a_n also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the a_n alternate in sign).
- Comparison test 2: If ∑b_n is an absolutely convergent series such that |a_n+1 /a_n | ≤ C |b_n+1 /b_n | for some number C and for sufficiently large n , then ∑a_n converges absolutely as well. If ∑|b_n | diverges, and |a_n+1 /a_n | ≥ |b_n+1 /b_n | for all sufficiently large n , then ∑a_n also fails to converge absolutely (though it could still be conditionally convergent, e.g. if the a_n alternate in sign).
- Ratio test: If |a_n+1/a_n| < 1 for all sufficiently large n, then ∑ a_n converges absolutely. When the ratio is 1, convergence can sometimes be determined as well.
- Root test: If there exists a constant C < 1 such that |a_n|^1/n ≤ C for all sufficiently large n, then ∑ a_n converges absolutely.
- Integral test: if f(x) is a positive monotone decreasing function defined on the interval [1, ∞) with f(n) = a_n for all n, then ∑ a_n converges if and only if the integral ∫₁^∞ f(x) dx is finite.
- Alternating series test: A series of the form ∑ (−1)ⁿ a_n (with a_n ≥ 0) is called alternating. Such a series converges if the sequence a_n is monotone decreasing and converges to 0. The converse is in general not true.
- For some specific types of series there are more specialized convergence tests, for instance for Fourier series there is the Dini test.

Power series

Several important functions can be represented as Taylor series; these are infinite series involving powers of the independent variable and are also called power series. For example, the series :

\sum_^\infty\frac

converges to

e^x

for all x. See also radius of convergence. Historically, mathematicians such as Leonhard Euler operated liberally with infinite series, even if they were not convergent. When calculus was put on a sound and correct foundation in the nineteenth century, rigorous proofs of the convergence of series were always required. However, the formal operation with non-convergent series has been retained in rings of formal power series which are studied in abstract algebra. Formal power series are also used in combinatorics to describe and study sequences that are otherwise difficult to handle; this is the method of generating functions.

Generalizations

Asymptotic series, otherwise asymptotic expansions, are infinite series that do not converge. But they are useful as sequences of approximations, each of which provides a value close to the desired answer for a finite number of terms. The difference is that an asymptotic series cannot be made to produce an answer as exact as desired, the way that convergent series can. In fact, after a certain number of terms, a typical asymptotic series reaches its best approximation; if more terms are included, most such series will produce worse answers. The notion of series can be defined in every abelian topological group; the most commonly encountered case is that of series in a Banach space. There is no serious definition for an infinite sum over an uncountable set. For example if X is a set and f a function on X taking non-negative real values, such that :

\sum_ f(y) for any countable subset

Y of X, with A an absolute constant, it follows that f(x) = 0 for all x outside some countable subset of X. In other words, infinite sums of uncountably many non-negative reals make sense only in the case that this is a conventional convergent infinite series, extended by the value 0 to an uncountable set.

ONE

: This article is about the dog food brand ONE, by Ralston Purina Company. ONE is also the name of the American version of the Make Poverty History campaign. ----- ONE is a trademark of Nestlé Purina PetCare for a line of pet food products (acronym for the "optimum nutritional effectiveness" they are claimed to possess).

Counting

Counting is the mathematical action of repeatedly adding (or subtracting) one, usually to find out how many objects there are or to set aside a desired number of objects (starting with one for the first object and proceeding with a one-to-one correspondence). However, counting is also used (primarily by children) to demonstrate knowledge of the number names and the number system. Counting sometimes involves numbers other than one -- for example, when counting money, or counting out change, or when "counting by twos" (2, 4, 6, 8, 10, 12, ...) or when "counting by fives" ( 5, 10, 15, 20, 25 ...). There is archeological evidence that humans have been counting for at least 50,000 years [1 - p.9]; the development of counting led to the development of mathematical notation and numeral systems. Counting was primarily used by ancient cultures to keep track of economic data such as debts and capital.

References

# An Introduction to the History of Mathematics (6th Edition) by Howard Eves (1990) Category:Elementary mathematics Category:Numeration

Polynomial

In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. Here, simple means they are constructed using only multiplication and addition. Smooth means they are infinitely differentiable, i.e., they have derivatives of all finite orders. Because of their simple structure, polynomials are very easy to evaluate, and are used extensively in numerical analysis for polynomial interpolation or to numerically integrate more complex functions. In linear algebra, the characteristic polynomial of a square matrix encodes several important properties of the matrix. In graph theory the chromatic polynomial of a graph encodes the different ways to vertex color the graph using x colors. With the advent of computers, polynomials have been replaced by splines in many areas in numerical analysis. Splines are piecewise defined polynomials and provide more flexibility than ordinary polynomials when defining simple and smooth functions. They are used in spline interpolation and computer graphics.

History

Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. Some polynomials, such as f(x) = x² + 1, do not have any roots among the real numbers. If, however, the set of allowed candidates is expanded to the complex numbers, every (non-constant) polynomial has a root: this is the statement of the fundamental theorem of algebra. There is a difference between approximating roots and finding concrete closed formulas for them. Formulas for the roots of polynomials of degree up to 4 have been known since the 16th century (see quadratic equation, Gerolamo Cardano, Niccolo Fontana Tartaglia). But formulas for degree 5 eluded researchers for a long time. In 1824, Niels Henrik Abel proved the striking result that there can be no general formula (involving only the arithmetical operations and radicals) for the roots of a polynomial of degree 5 or greater in terms of its coefficients (see Abel-Ruffini theorem). This result marked the start of Galois theory which engages in a detailed study of relations among roots of polynomials. The difference engine of Charles Babbage was designed to create large tables of values of logarithms and trigonometric functions automatically, by evaluating approximating polynomials at many points using Newton's difference method.

Definition

For given constants (i.e., numbers) a₀, …, a_n in some field (possibly but not limited to R or C) with a_n non-zero, for n > 0, then a polynomial (function) of degree n is a function of the form :

f(x) = a_0 + a_1 x + \cdots + a_ x^ + a_n x^n.

More concisely, the polynomial can be written in sigma notation as :

f(x) = \sum_^ a_ x^.

The constants a₀, …, a_n are called the coefficients of the polynomial. a₀ is called the constant coefficient and a_n is called the leading coefficient. When the leading coefficient is 1, the polynomial is called monic or normed. Each summand a_i xⁱ of the polynomial is called a term. A polynomial with one, two or three terms is called monomial, binomial or trinomial respectively. Polynomial functions of
- degree 0 are called constant functions (excluding the zero polynomial, which has indeterminate degree),
- degree 1 are called linear functions,
- degree 2 are called quadratic functions,
- degree 3 are called cubic functions,
- degree 4 are called quartic functions and
- degree 5 are called quintic functions.

Graphs

- The graph of a constant function :

f(x) = a_0

is a horizontal line with y-intercept

a_0

.
- The graph of a degree 1 polynomial function (or linear function) :

f(x) = a_0 + a_1 x

, where

a_1 \neq 0

is an oblique line with y-intercept

a_0

and slope

a_1

.
- The graph of a degree 2 or higher polynomial function :

f(x) = a_0 + a_1 x + \cdots + a_ x^ + a_n x^n,

where

a_n \neq 0

and

n \geq 2

is a continuous non-linear curve. The best way to analyze the graph of a degree 2 or higher polynomial function is by its end behavior, the number of x-intercepts and the number of turning points. End behavior There are four end behaviors which are direct results of whether

a_n

, the leading coefficient, is positive or negative and whether

n

, the degree of the polynomial, is even or odd.
- If

a_n

is positive and

n

is even, the right end of the polynomial is in quadrant I while the left end is in quadrant II.
- If

a_n

is negative and

n

is even, the right end is in quadrant IV while the left end is in quadrant III.
- If

a_n

is positive and

n

is odd, the right end is in quadrant I while the left end is in quadrant III.
- If

a_n

is negative and

n

is odd, the right end is in quadrant IV while the left end is in quadrant II. Number of x-intercepts From the Fundamental theorem of algebra, a polynomial of degree

n

has exactly

n

complex roots, which may or may not be real. Therefore, the number of x-intercepts can't exceed

n

. It also follows from the Fundamental Theorem of Algebra that the complex roots of a polynomial must exist in conjugate pairs. This implies that an even-degree polynomial may have no x-intercepts (because all its roots may be complex); an odd-degree polynomial, on the other hand, must have at least one x-intercept, since any pairing of roots into conjugate pairs will necessarily leave at least one unpaired for odd

n

. These "unpaired" roots must therefore be real. For example, a degree 4 polynomial function can have 0, 2 or 4 x-intercepts whereas a degree 5 polynomial function can have 1, 3 or 5 x-intercepts. Number of turning points The number of turning points of an even-degree polynomial is any odd number less than the degree, while the number of turning points of an odd-degree polynomial is any even number less than the degree. For example, a degree 4 polynomial function can have 1 or 3 turning points whereas a degree 5 polynomial function can have 0, 2, or 4 turning points. The following are some examples of polynomials of low degree.

Examples

Fundamental theorem of algebra	Fundamental theorem of algebra
Fundamental theorem of algebra	Fundamental theorem of algebra

The function :

f(x)= -7x^3 + \begin\frac\end x^2 - 5x + 3

is an example of a cubic function with leading coefficient −7 and constant coefficient 3.

Notes

The polynomials up to degree n form a vector space of dimension n + 1, which is sometimes called

\Pi_n

K_n[x]

(where K indicates the field of coefficients, e.g. K=R or C). In this article polynomials are written using the (canonical) monomial basis (i.e. 1, x, x², …, xⁿ), but it should be mentioned that other bases exist, for example the Chebyshev polynomials, which may be preferable depending on the problem domain.

Roots

A root or zero of a polynomial f is a number ζ so that f(ζ) = 0. The fundamental theorem of algebra states that a polynomial of degree n over the complex numbers has exactly n complex roots (not necessarily distinct ones). Therefore a polynomial can be factorized as :

f(x) = a_n(x-\zeta_1)\cdots(x-\zeta_)

where each

\zeta_i

is a root of the polynomial f. The Abel-Ruffini theorem in algebra states that generally there is no closed formula to calculate the roots of a polynomial of degree 5 or higher. Closed formula means a formula constructed using only the coefficients of the polynomial and the operations of addition, multiplication and exponentiation (and their inverse operations).

Numerical analysis

Polynomials and calculus

One important aspect of calculus is the project of analyzing complicated functions by means of approximating them with polynomials. The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial, and the Stone-Weierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial. Polynomials are also frequently used to interpolate functions. Quotients of polynomials are called rational functions. Piecewise rationals are the only functions that can be evaluated directly on a computer, since typically only the operations of addition, multiplication, division and comparison are implemented in hardware. All the other functions that computers need to evaluate, such as trigonometric functions, logarithms and exponential functions, must then be approximated in software by suitable piecewise rational functions.

Evaluation of polynomials

The fast and numerically stable evaluation of a polynomial for a given x is a very important topic in numerical analysis. Several different algorithms have been developed for this problem. Which algorithm is used for a given polynomial depends on the form of the polynomial and the chosen x. To evaluate a polynomial in monomial form one can use the Horner scheme. For a polynomial in Chebyshev form the Clenshaw algorithm can be used. If several equidistant x_n have to be calculated one would use Newton's difference method.

Finding roots

As there is no general closed formula to calculate the roots of a polynomial of degree 5 and higher, root-finding algorithms are used in numerical analysis to approximate the roots. Approximations for the real roots of a given polynomial can be found using Newton's method, or more efficiently using Laguerre's method which employs complex arithmetic and can locate all complex roots.

Several variables

In multivariate calculus, polynomials in several variables play an important role. These are the simplest multivariate functions and can be defined using addition and multiplication alone. An example of a polynomial in the variables x, y, and z is :

f(x, y, z) = 2 x^2 y z^3 - 3 y^2 + 5 y z - 2. \,

The total degree of such a multivariate polynomial can be gotten by adding the exponents of the variables in every term, and taking the maximum. The above polynomial f(x, y, z) has total degree 6.

Abstract algebra

: Main article: polynomial ring.

In abstract algebra, one must take care to distinguish between polynomials and polynomial functions. A polynomial f is defined to be a formal expression of the form :

f = a_n X^n + a_ X^ + \cdots + a_1 X + a_0

where the coefficients a₀, ..., a_n are elements of some ring R and X is considered to be a formal symbol. Two polynomials are considered to be equal if and only if the sequences of their coefficients are equal. Polynomials with coefficients in R can be added by simply adding corresponding coefficients and multiplied using the distributive law and the rules :

X \; a = a \; X

for all elements a of the ring R :

X^k \; X^l = X^

for all natural numbers k and l. One can then check that the set of all polynomials with coefficients in the ring R forms itself a ring, the ring of polynomials over R, which is denoted by R[X]. If R is commutative, then R[X] is an algebra over R. One can think of the ring R[X] as arising from R by adding one new element X to R and only requiring that X commute with all elements of R. In order for R[X] to form a ring, all sums of powers of X have to be included as well. Formation of the polynomial ring, together with forming factor rings by factoring out ideals, are important tools for constructing new rings out of known ones. For instance, the clean construction of finite fields involves the use of those operations, starting out with the field of integers modulo some prime number as the coefficient ring R (see modular arithmetic). To every polynomial f in R[X], one can associate a polynomial function with domain and range equal to R. One obtains the value of this function for a given argument r by everywhere replacing the symbol X in f's expression by r. The reason that algebraists have to distinguish between polynomials and polynomial functions is that over some rings R (for instance, over finite fields), two different polynomials may give rise to the same polynomial function. This is not the case over the real or complex numbers and therefore many analysts often don't separate the two concepts.

Divisibility

In commutative algebra, one major focus of study is divisibility among polynomials. If R is an integral domain and f and g are polynomials in R[X], it is said that f divides g if there exists a polynomial q in R[X] such that f q = g. One can then show that "every zero gives rise to a linear factor", or more formally: if f is a polynomial in R[X] and r is an element of R such that f(r) = 0, then the polynomial (X − r) divides f. The converse is also true. The quotient can be computed using the Horner scheme. If F is a field and f and g are polynomials in F[X] with g ≠ 0, then there exist unique polynomials q and r in F[X] with :

f = q \, g + r

and such that the degree of r is smaller than the degree of g. The polynomials q and r are uniquely determined by f and g. This is called "division with remainder" or "polynomial long division" and shows that the ring F[X] is a Euclidean domain. Analogously, polynomial "primes" (more correctly, irreducible polynomials) can be defined which cannot be factorized into the product of two polynomials of lesser degree. It is not easy to determine if a given polynomial is irreducible. One can start by simply checking if the polynomial has linear factors. Then, one can check divisibility by some other irreducible polynomials. Eisenstein's criterion can also be used in some cases to determine irreducibility.

More variables

One also speaks of polynomials in several variables, obtained by taking the ring of polynomials of a ring of polynomials: R[X,Y] = (R[X])[Y] = (R[Y])[X]. These are of fundamental importance in algebraic geometry which studies the simultaneous zero sets of several such multivariate polynomials. Polynomials are frequently used to encode information about some other object. The characteristic polynomial of a matrix or linear operator contains information about the operator's eigenvalues. The minimal polynomial of an algebraic element records the simplest algebraic relation satisfied by that element. Other related objects studied in abstract algebra are formal power series, which are like polynomials but may have infinite degree, and the rational functions, which are ratios of polynomials.

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

Plus and minus signs

The plus (+) and minus (−) signs are used universally to represent the notions of positive and negative as well as the operations of addition and subtraction. Their use has been extended to many other meanings, more or less analogous.
Plus and Minus are Latin terms.

History

Though the signs now seem as familiar as the alphabet or the Hindu-Arabic numerals, they are not of great antiquity. The Egyptian hieroglyphic sign for addition, for example, resembled a pair of legs walking in the direction in which the text was written (Egyptian was written in varying directions), with the reverse sign indicating subtraction: In Europe in the early 15th century the letters P and M were generally used. The earliest print appearance of the modern signs seems to come from a book on "Behende und hüpsche Rechenung auff allen Kauffmanschafft" or Mercantile Arithmetic by Johannes Widmann in 1489, used to indicate surpluses and deficits. The + is a simplification of the Latin "et" (comparable to the ampersand &). The − may be derived from a tilde written over m when used to indicate subtraction; or it may come from a shorthand version of the letter m itself. According to the [http://members.aol.com/jeff570/operation.html Earliest Uses of Various Mathematical Symbols] website, a book published by Henricus Grammateus in 1518 is the earliest found to use + and − for addition and subtraction. Robert Recorde, the designer of the equals sign introduced plus and minus to the UK in 1557 in The Whetstone of Witte. Recorde wrote, There be other 2 signes in often use of which the first is made thus + and betokeneth more: the other is thus made − and betokeneth lesse.

Alternate uses

The plus sign can mean many different operations, depending on the mathematical system under consideration. Many algebraic structures have some operation which is called, or equivalent to, addition. Moreover, the symbolism has been extended to very different operations, such as concatenation of strings of characters. Plus can mean:
- exclusive or (usually written ⊕): 1+1=0
- logical conjunction (usually written ∧): 1+1=1
- logical disjunction (usually written ∨): 1+1=1
- concatenation of string literals is sometimes written: "1"+"1"="11"

Three uses of the minus sign

The short dash we call a minus sign has three uses in mathematics, always to indicate an opposite. #The original use is to indicate the operation of subtraction, as in 5 − 3 = 2. Subtraction is the opposite of addition. #The second use is as a prefix to a number indicating that the number is negative, as in −4. A negative number always indicates the opposite of a positive number. If 4 means four steps right, then −4 means four steps left. If $10 means a deposit of ten dollars, then −$10 means a withdrawal of ten dollars. #Finally, in algebra, the sign means "the opposite of", for example, −x means the opposite of x. Much confusion arises for those not familiar with this notation, since −x may be a positive number, if x itself is a negative number. Properly, 5 − 3 should be read "five minus three," −4 should be read "negative four," and −x should be read "the opposite of x," but not even professional mathematicians do this, usually reading all three signs as "minus." Because of the special use of the minus sign in mathematics, written mathematics should avoid as much as possible hyphens and dashes that could be misread as minus signs.

Plus and minus signs in other cultures

A Jewish tradition dating at least from the 19th century is to write plus using a symbol like an inverted T. This practice was adopted into Israeli schools in the 1970s and is now commonplace in most elementary schools (including secular schools) and some secondary schools. It is also used occasionally in books by religious authors, but most books for adults use the international symbol "+". The usual explanation for the practice is that it avoids the writing of a symbol "+" that looks like a Christian cross. Unicode has this symbol at position U+FB29 "Hebrew letter alternative plus sign" ().

In computing

In Unicode, the plus sign (+) has the code point U+002B. The minus sign (−) is U+2212. In HTML it can be entered using the character entity − or one of the numeric forms − or −. The Unicode minus sign is designed to be the same length and height as the plus and equals signs. In most fonts these are usually monospaced along with the numbers for ease when being used in tabular formats. The hyphen-minus sign (-) is U+002D. This is the ASCII version of the minus sign, and doubles as a hyphen. It is usually shorter in length than the plus sign and sometimes at a different height. It should be used for the minus sign only when the character set is limited to ASCII, or with fixed-width fonts.

External links

- [http://www.roma.unisa.edu.au/07305/symbols.htm#Plus The History of Mathematical Symbols - Plus and Minus]
- [http://members.aol.com/jeff570/mathsym.html Earliest uses of various mathematical symbols] Category:Elementary arithmetic Category:Mathematical notation ja:符号 (数学)

Associativity

:This article is about associativity in mathematics. For associativity in central processor unit memory cache architecture see CPU cache. In mathematics, associativity is a property that a binary operation can have. It means that the order of evaluation is immaterial if the operation appears more than once in an expression. Put another way, no parentheses are required for an associative operation. Consider for instance the equation :(5+2)+1 = 5+(2+1) Adding 5 and 2 gives 7, and adding 1 gives an end result of 8 for the left hand side. To evaluate the right hand side, we start with adding 2 and 1 giving 3, and then add 5 and 3 to get 8, again. So the equation holds true. In fact, it holds true for all real numbers, not just for 5, 2 and 1. We say that "addition of real numbers is an associative operation". Associative operations are abundant in mathematics, and in fact most algebraic structures explicitly require their binary operations to be associative. However, many important and interesting operations are non-associative; one common example would be the vector cross product.

Definition

Formally, a binary operation

-

on a set S is called associative if it satisfies the associative law: :

(x - y) - z=x - (y - z)\qquad\mboxx,y,z\in S.

The evaluation order does not affect the value of such expressions, and it can be shown that the same holds for expressions containing any number of

-

operations. Thus, when

-

is associative, the evaluation order can therefore be left unspecified without causing ambiguity, by omitting the parentheses and writing simply: :

x - y - z.

Examples

Some examples of associative operations include the following.
- In arithmetic, addition and multiplication of real numbers are associative; i.e., ::

\left.
    \begin
     (x+y)+z=x+(y+z)=x+y+z\quad
    \\
     (x\,y)z=x(y\,z)=x\,y\,z\qquad\qquad\qquad\quad\ \ \,
    \end
   \right\{for all {R{matrix{gcd{gcd{gcd{gcd{gcd{lcm{lcm{lcm{lcm{lcm{matrix

Fraction (mathematics)

: In mathematics, a fraction is a way of expressing a quantity based on an amount that is divided into a number of equal-sized parts. For example, each part of a cake split into four equal parts is called a quarter (and represented numerically as ¹⁄₄); two quarters is half the cake, and eight quarters would make two cakes. Mathematically, a fraction is a quotient of numbers, like ³⁄₄, or more generally, an element of a quotient field. In our cake example above, where a quarter is represented numerically as ¹⁄₄ the bottom number, (called the denominator) is the total number of equal parts making up the cake as a whole, and the top number (called the numerator) is the number of these parts we have. For example, the fraction ³⁄₄ represents three quarters. The numerator and denominator are the terms of the fraction. The word "numerator" is related to the word "enumerate." To enumerate means to "tell how many"; thus the numerator tells us how many fractional parts we have in the indicated fraction. To denominate means to "give a name" or "tell what kind"; thus the denominator tells us what kind of parts we have (halves, thirds, fourths, etc.). The word is also used in related expressions, like continued fraction, see Special cases below.

Arithmetic

There are four basic arithmetic operations, which in order of simplicity for fractions, includes (1) Multiplication (2) Addition (3) Subtraction (4) Division.

Addition

Adding Fractions

Adding fractions can be a little tricky, since you cannot simply add the numerators and denominators. For example, if we had a cake divided into three pieces, each piece would be 1/3. Then, if we try to add one piece from the cake divided into four pieces, and one piece from the cake divided into three pieces, what would be have? Well, we would have, um, 1/4 + 1/3 = ??? You can see this is NOT equal to 1/7 or 2/7 !! To add fractions together, they must be changed to equivalent values having the same fractional unit -- the same denominator -- in this case 1/12. How do we do this? By multiplying each fraction by 1. By one? Yes. 1 = 3/3 and 1 = 4/4. Now watch: 1/4 = 1/4 x 1 = 1/4 x 3/3 = 3/12. And 1/3 = 1/3 x 1 = 1/3 x 4/4 = 4/12. So now 1/4 + 1/3 = 3/12 + 4/12 = 7/12 and we have the correct result. Notice that we only add the numerators together. The denominator does not change, since we are working with the same fractional unit. Another way to see this is: 1/4 + 1/3 = 3/12 + 4/12 = 1/12 x (3 + 4) = 1/12 x 7 = 7/12. Lets take another example. If you add a half dollar to a quarter, what will you get? You know it's 75 cents, right? When we say 75 cents we have automatically, in our mind, changed each coin into cents (pennies): One half dollar = 50 cents; one quarter = 25 cents; so 1/2 + 1/4 = 50/100 + 25/100 = 75/100 or 75 cents. Of course, we could use a smaller denominator since we know one half dollar equals two quarters. I.e., 1/2 + 1/4 = 2/4 + 1/4 = 3/4. In words, one half plus one quarter equals two quarters plus one quarter equals three quarters, or 75 cents. So the trick is to find a common fractional unit -- a common denominator -- that will let us simply add the numerators together. Let's take one more example. Find 2/3 + 1/2. We see that the denominators are 3 and 2. We need to find a value that each denominator can be multiplied by to give a common value. Well, it's easy to see that we can multiply 3 by 2, and 2 by 3, to give a common denominator of 6. But remember, you cannot change the value of each fraction, so we must multiply both numerator and denominator by the same number. We now have: 2/3 + 1/2 = 2/2 x 2/3 + 3/3 x 1/2 = 4/6 + 3/6 = 7/6 or 1 + 1/6.

Multiplication

By whole numbers

If you consider the cake example above, if you have a quarter of the cake, and you multiple the amount by three, then you end up with three quarters. We can write this numerically as follows: :

3 \times  =

As another example, suppose that five people work for three hours out of a seven hour day (ie. for three seventh of the work day). In total, they will have worked for 15 hours (5 x 3 hours each), or 15 sevenths of a day. Since 7 seventh of a day is a whole day, 14 sevenths is two days, then in total, they will have worked for 2 days and a seventh of day. Numerically: :

5 \times  =  = 2

By fractions

If you consider the cake example above, if you have a quarter of the cake, and you multiple the amount by a third, then you end up with a twelfth of the cake. In other words, a third of a quarter (or a third times a quarter), is a twelfth. Why? Because we are splitting each quarter into three pieces, and four quarters times three makes 12 parts (or twelfths). We can write this numerically as follows: :

\times  =

As another example, suppose that five people do an equal amount work that totals three hours out of a seven hour day. Each person will have done a fifth of the work, so they will have worked for a fifth of three sevenths of a day. Numerically: :

\times  =

General rule

You may have noticed that when we multiply fractions, we simply multiply the two numerators (the top numbers), and multiply the two denominators) (the bottom numbers). For example: :

\times  =  =

By mixed whole number/fractions

If we are multiplying fractions that include a whole number component, then it is best to convert the whole number into a fraction. For example: :

3 \times 2 = 3 \times \left ( \right ) = 3 \times  =  = 8

In other words,

2

is the same as

\left ( \right )

, making 11 quarters in total (because 2 cakes, each split into quarters makes 8 quarters total). And 33 quarters is

8

since 8 cakes, each made of quarters, is 32 quarters in total.

Commutativity

It is also worth recalling that multiplication is commutative which just means that the order of the numbers we are multiplying does not matter. In other words, three lots of a quarters is equivalent to a quarter of three; numerically: :

3 \times  =  \times 3 =

Note that when talking, we say "three times a quarter", but "a quarter of three", the implication being that in the latter example, we are talking about a fractional part of a larger number.

Special cases

- A vulgar fraction (or common fraction) is a rational number written as one integer (the numerator) divided by a non-zero integer (the denominator). The line that separates the numerator and the denominator is called the vinculum. Rational numbers are the quotient field of integers.

Particular vulgar fractions

- irreducible fraction: a vulgar fraction "in lowest terms", where the numerator is an integer, the denominator is a positive integer, and the highest common factor of the numerator and the denominator is 1;
- proper fraction: a vulgar fraction with (absolute) value between 0 and 1;
- improper fraction: a vulgar fraction with a (absolute) value greater than 1;
- unit fraction: a vulgar fraction with a numerator of 1;
- Egyptian fraction: the sum of distinct unit fractions;
- decimal fraction: a vulgar fraction where the denominator is a power of 10;
- dyadic fraction: a vulgar fraction in which the denominator is a power of two.

Other fractions

- A mixed fraction: A mixed fraction is an integer plus a proper fraction.
- A compound fraction is a fraction where the numerator or denominator (or both) contain fractions.
- Rational functions are the quotient field of polynomials (over some integral domain). Let us end with the only example on this page where the "fraction" is not an element of a quotient field:
- A continued fraction is an expression such as

a_0 + \frac

, where the a_i are integers. The term partial fraction is used in algebra, when decomposing rational functions. However, a partial fraction is an expression of a particular decomposition, and so is more than just an element of a quotient field. The term irrational fraction is sometimes used to indicate a magnitude whose quotient with another fixed magnitude is irrational, e.g. "1 is an irrational fraction of 2π". "Fraction", in this sense, simply means "a part of the whole", not a strict ratio in the mathematical sense. Taking the latter meaning, the term is an oxymoron.

Pedagogical tools

In Primary Schools, fractions have been demonstrated through Cuisenaire rods. See also the external links below.

External links

- [http://www.mathfactcafe.com Curricula for Creating Fractions]
- [http://www.ericdigests.org/2000-2/fractions.htm Curricula for Teaching about Fractions]
- [http://www.ericdigests.org/2004-1/fractions.htm Teaching Fractions: New Methods, New Resources]
- [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1551&CurriculumID=4&Method=Worksheet Worksheets: Identifying Fractions]
- [http://www.kwiznet.com/p/takeQuiz.php?ChapterID=1364&CurriculumID=4&Method=Worksheet Worksheets: Improper Fractions to Mixed Numbers]
- [http://www.math-lessons.ca Curricula for Teaching about Equivalent Fractions]
- [http://www.quiz-tree.com/Fractions_Practice_main.html Free online quizzes about Fractions]
- Category:Mathematical disambiguation Category:Elementary arithmetic Category:Numbers ja:分数

Juxtaposition

Juxtaposition is an act or instance of placing two things close together or side by side. This is often done in order to compare/contrast the two, to show similarities or differences, etc. In logic, juxtaposition is a logical fallacy on the part of the observer, where two items placed next to each other imply a correlation, when none is actually claimed. For example, an illustration of a politician and Adolf Hitler on the same page would imply that the politician had a common ideology with Hitler. Similarly, saying "Hitler was in favor of gun control, and so are you" would have the same effect. In music, it is an abrupt change of elements. In film, the position of shots next to one another is intended to create meaning within the audience's mind. In literature, a juxtaposition occurs when two images that are otherwise not commonly brought together appear side by side or structurally close together, thereby forcing the reader to stop and reconsider the meaning of the text through the contrasting images, ideas, motifs, etc. For example, "He was slouched alertly" is a juxtaposition. Modern poetry plays extensively with juxtaposing images, inserting unrelated fragments together in order to create wonder and interest in readers. In mathematics, juxtaposition of symbols is commonly used for multiplication: ax denotes the product of a with x, or a times x. It is also used for scalar multiplication, matrix multiplication, and function composition. In numeral systems, juxtaposition of digits has a specific meaning. In geometry, juxtaposition of names of points represents lines or line segments. In physics, juxtaposition is also used for "multiplication" of a numerical value and a physical unit, and of two physical units, for example, three Newton-meters, would be written as 3Nm. Category:Literature Category:Mathematical notation Category:Logical fallacies

Multiplication

Notation

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) = .

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

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