Addition

addition

The addition is a elementary Opération, describing the juxtaposition of objects or the addition of extensive sizes of comparable nature, like the lengths, the surfaces, or volumes. In particular in Physique, the addition of two sizes can be carried out numerically only if these sizes are expressed with same the Measuring unit.
the result of an addition is called a nap .

In Mathematical, the addition is developed on the whole of usual numbers but is also defined for other mathematical objects like the Vecteur S and the functions.
By analogy, one calls addition the Law of composition interns vector spaces and some abelian groups.

With other mathematical structures are also provided with called binary operations additions, but which always do not satisfy the properties of the usual addition.

Design

Meeting of quantities

The addition is conceived initially like the enumeration of a meeting of collections of objects, in three conditions:

• On the one hand, these objects should not be mixed by losing their individuality, as liquids or balls of modeling clay would do it.
• In addition, the elements “in double” appearing in several collections at the same time must be regarded as distinct and counted individually.
• Lastly, these objects must be of comparable nature, i.e. to answer a common denomination. Thus, to add with apples and pears, it is necessary to regard them overall as fruits, in order to express the result in many fruits .

The result of the addition is the total quantity of objects, which can be counted either by a counting, or by a mathematical calculation on the Nombre S describing the quantities starting.

In the same way, so that the addition can describe the meeting of fractional objects, like portions of circle or geometrical figures traced on a squaring, is needed that all the objects is evaluated starting from a common subdivision, a building bloc. Mathematically, this condition is interpreted as the search for a Dénominateur common to several fractions.

Certain physical sizes, but also geometrical or economic, can be also added by the meeting with the objects on which they are measured. But these sizes must then be evaluated relative with a Measuring unit common and a Nombre to the satisfactory precision.

This vision makes it possible to justify the fundamental properties intuitively of the addition: the order in which the quantities are given does not have an influence on the result.

Assessment of variations

The addition also seems an assessment of variations or displacements successive along a directed axis. Each term is then provided with a sign indicating its direction: positive for a profit, an increase or a displacement in the direction of the axis; negative for a loss, a reduction or a displacement in the contrary direction with that of the axis. The result of the operation is then called one algebraic sum .

The variations can still relate to there whole or fractional quantities, or any measured size.

For example, the addition of $-5$ and $+2$ translate a loss of five units and the profit of two units. The result of the addition, $-3$, corresponds to the total variation of the number of units: three units were lost.

It is this design which then makes it possible to define the addition of the vectors by juxtaposition of displacements or translations.

Formal construction

The mathematical formalization of the natural integers however privileges an ordinal definition of the addition, by Incrémentation reiterated. Thus, on the basis of the only operation “to add a ”, the addition of numbers 3 and 2 is conceived in the form “3 to which one adds a by twice” (3+1+1). In this context, the properties of commutation and associativeness are not then any more obvious and must be shown.

See also: Addition of the natural entireties

Numerical operation

Notation

The addition of two $a$ terms and $b$ notes usually $a+b$ and is read “$a$ more $b$”, sometimes “$a$ and $b$” or “$a$ added to $b$”.
the sign “+” replaces since the end of XVe century the symbol p for “more”.

This notation infix can be replaced in certain contexts by a functional notation $+\; (has,\; b)$ or by a postfixée notation $(a)\; (b)+$.
In the arborescent decomposition of an algebraical expression, the addition is represented by a trivalent node with two entries and an exit.

In a system of additive notation such as the unary System or the Egyptian Numeration, the sign “+” does not need to be indicated since the writing of the numbers already consists in breaking up the numbers into a sum of fixed numerical values.

In a system of positional Notation the such modern notation, the addition of several numbers is sometimes represented by the superposition of the writings of numbers, all the figures of the same position being aligned vertically. This provision facilitates the manual calculation sum of several numbers.

Properties

The addition of numbers has certain valid properties in all the whole of usual numbers:

• it is commutative, i.e. the order in which the terms of the addition are given does not have an influence on the result:

$a+b\; =\; b+a$;

• it is associative, i.e. there does not need to specify by brackets the order in which is carried out a succession of additions:

$(a+b)+c\; =\; a+\; (b+c)$, from where the notation without brackets $a+b+c$;

• it is that can be simplified, i.e. in an equality of additions, one can remove two identical terms on both sides equal sign:

if $x+a\; =\; y+a$ then $x=y$;

• the null element or Zero , noted 0, is neutral for the addition:

$a+0\; =\; has$.

Each $x$ number has symmetrical for the addition, called “opposed” and noted $-x$, i.e. such as $x+\; (-\; X)\; =\; -\; X\; +\; X\; =\; 0$.
The whole of numbers $\backslash \; mathbb\; \{Z\}$, $\backslash \; mathbb\; \{D\}$, $\backslash \; mathbb\; \{Q\}$ and $\backslash \; R$ has all the opposites of their numbers, but the unit $\backslash \; N$ does not have the opposites of the strictly positive integers.

The addition with symmetrical makes it possible to define the Soustraction by $x\; -\; there\; =\; X\; +\; (there)$.

Process of calculation

The evaluation of the result of an addition depends on the Numbering system employed, i.e. in the manner of representing the numbers.

In an additive system, it is enough to juxtapose the writings then to simplify the expression by gathering the of the same symbols value to replace of it a part by symbols of higher value when it is possible. In a general way, the numbering systems not quantified could develop a technique of addition by the practice of the abacus.

In a system of quantified positional numeration, the calculation of a sum of entireties passes by the use of a Table of addition.

This one makes it possible to find the sum of the figures on each position. The writing of the result is made the lowest position with the highest position (from right to left in modern notation). For each position, one registers the figure of the units of the sum of the figures and one defers a reserve on the following position if this sum is larger than the base. Each figure of the result is then incremented possible reserve.

This method spreads for the decimal numbers by aligning the commas vertically.

The addition of fractions of entireties passes by a setting to the same denominator, then an addition of the numerators and finally by a possible simplification of the fraction obtained.

$\backslash \; frac\; \{7^\; \{(\backslash \; times\; 3)\}\}\; \{10^\; \{(\backslash \; times\; 3)\}\}\; +\; \backslash \; frac\; \{5^\; \{(\backslash \; times\; 5)\}\}\; \{6^\; \{(\backslash \; times\; 5)\}\}\; =\; \backslash \; frac\; \{21\}\; \{30\}\; +\; \backslash \; frac\; \{25\}\; \{30\}\; =\; \backslash \; frac\; \{21+25\}\; \{30\}\; =\; \backslash \; frac\; \{46^\; \{(\backslash \; div\; 2)\}\}\; \{30^\; \{(\backslash \; div\; 2)\}\}\; =\; \backslash \; frac\; \{23\}\; \{15\}$

As for the addition of the Egyptian fractions of numerators unit and distinct denominators all, it calls upon an iterative process of simplification of the fractions appearing in double.

The sums of entireties, decimal and rational can be always calculated in reduced form, i.e. without using the sign “+”. On the other hand, a sum of realities always does not admit a reduced form: one cannot simplify the writing of 1 +.

See also: Technical of the addition, Technical of addition

Iteration

By choosing a constant term $r$, the addition makes it possible to define a function $x\; \backslash \; to\; x+r$ which one can reiterate to build of the arithmetic continuations of reason $r$.
such continuations $(u\_n)$ check for entire positive $n$ the relation $u\_\; \{n+1\}\; =\; u\_n+r$. They are written then in the form:
$(u\_0,\; u\_0+r,\; u\_0+r+r,\; u\_0+r+r+r,\; \backslash \; dowries)$.

These repetitions of addition make it possible to define the Multiplication par
$p\; \backslash \; times\; R\; =\; \backslash \; underbrace\; \{R\; +\; \backslash \; dowries\; +\; R\}\; \_\; \{\backslash \; textstyle\; p\; \backslash \; \backslash \; mbox\; \{time\}\}$.

The addition of a finished succession of numbers defined by a general formula (for example, addition of the odd entireties from 1 to 99) uses specific processes which leave the operational field of the addition. The study of the continuations and associated series provides to more effective methods for calculation of such sums.

See also: Somme (arithmetic)

Popular culture

The addition gives place to certain plays like the Mourre, which consists in guessing the sum of small numbers given simultaneously by the two adversaries.

In poetry, it is evoked by the Page of writing of Jacques Prévert.

Geometrical constructions

The numbers intervening in an addition represent sometimes geometrical magnitudes: length of a segment, measures of an angle (directed or not), surface of a square surface. In each one of these cases, the calculation of the sum can be illustrated by a geometrical construction with the rule and the compass. There also exists in each case a construction of the Soustraction which allows starting from the size summons and of one of the starting sizes to find the other starting size.

Lengths

To trace the sum of two Length S, it is enough to prolong with the rule one these two lengths beyond one of its ends, then to trace a circle centered in this end of ray the other length. The intersection of the circle with the prolongation defines the new end prolonged length.

Geometrical angles

Being given two angular sectors traced in the plan, it is possible to build an angular sector whose measurement of the angle is the sum of measurements of the angles given. It is enough for that to initially trace a Isosceles triangle whose principal top and its adjacent sides constitute one of the angular sectors, then to build a isometric triangle of principal top at the point of the other angular sector with an adjacent side jointly and the other side outside the angular sector. The two external sides delimit the angle then summons.

In the event of addition of angles with important measurements, the angle nap can have a measurement of more than 360°.

This procedure, applied to the angles of a triangle, makes it possible to check that the sum of its angles measures 180° well.

Directed angles, angles of vectors

The addition of directed angles is made in a way similar to that of the geometrical angles, with the difference that the first side of the second angle must be superimposed at the second side of the first angle.

Construction can then be described in terms of transformations of the plan. If the first directed angle is determined by a couple of vectors represented starting from the same origin $O$ and of respective ends $A$ and $B$, it is enough to build the image $B\text{'}$ $B$ by the rotation of center $O$ and angle the second directed angle. The of the same vectors origin $O$ and of $A$ ends and $B\text{'}$ then define the angle directed nap.

By applying this operation to the angles of vectors of the form $(\backslash \; vec\; \{\backslash \; imath\};\; \backslash \; overrightarrow\; \{OM\})$, where $M$ is a point of the trigonometrical Cercle, the angular addition defines an operation on the points of the circle which corresponds to the multiplication of the complex numbers.

Surfaces of square surfaces

Being given two Square S traced in the plan, it is possible to build a square whose surface is the sum of the surfaces of the initial squares. Indeed, if the two initial squares can be traced in order to have a joint top and two perpendicular sides, the triangle formed by these two east coasts then a right-angled triangle. The Théorème of Pythagore then makes it possible to show that the square formed on the third side of the triangle has as a surface the sum of the surfaces of the initial squares.

The operation thus defined over the lengths on the sides of the squares is the addition pythagorician which is expressed (on the couples of positive realities) by:

$(has,\; b)\; \backslash \; mapsto\; \backslash \; sqrt\; \{a^2+b^2\}$.

This problem of construction generalizes that of the duplication of the square, where the initial squares have same dimension.

Vectorial addition

Vectors of a space closely connected

Being given four points $A$, $B$, $C$, $D$ of a Espace refines such as the plan or Euclidean space, the addition of both Vecteur S $\backslash \; overrightarrow\; \{AB\}$ and $\backslash \; overrightarrow\; \{CD\}$ is built by defining a point $E$ such as $\backslash \; overrightarrow\; \{BE\}\; =\; \backslash \; overrightarrow\; \{CD\}$ (by tracing the parallelogram $BCDE$).
The vector summons $\backslash \; overrightarrow\; \{AB\}\; +\; \backslash \; overrightarrow\; \{CD\}$ is identified then with the vector $\backslash \; overrightarrow\; \{AE\}$.

The addition of vectors satisfies all the properties of the numerical addition. Its neutral is the null vector and the opposite of a vector is a of the same vector direction and even standard but of opposite direction.

When the vectors are defined on the same line provided with a reference mark, the addition of the vectors are identified with that of the algebraic measurements.

Coordinates and components

The Coordonnée S of the vectors in a Cartesian Repère make it possible to represent the vectorial addition into addition of numbers. Indeed, if two vectors of the plan have as respective coordinates $(X;\; there)$ and $(x\text{'};\; y\text{'})$, the vector nap will have as coordinates $(x+x\text{'};\; y+y\text{'})$.

In usual space, the addition is represented by the operation on the triplets of coordinates $(X;\; there;\; Z)\; +\; (x\text{'};\; y\text{'};\; z\text{'})\; =\; (x+x\text{'};\; y+y\text{'};\; z+z\text{'})$.

The principle of the term addition in the long term is taken again for other mathematical structures such as the unit of the $n$- Uplet S of numbers and the continuations: $(x\_1,\; x\_2,\; x\_3,\; \backslash \; dowries)\; +\; (y\_1,\; y\_2,\; y\_3,\; \backslash \; dowries)\; =\; (x\_1+y\_1,\; x\_2+y\_2,\; x\_3+y\_3,\; \backslash \; dowries)$.

The matrices of the same size and the applications to numerical value are also added with this manner.

Extensions

Other mathematical structures extend certain whole of numbers and are provided with a binary operation which prolongs the usual addition, but which always does not have all its properties.

Functions

If the applications definite on a common given unit and with numerical value can be added simply component by component like vectors, it is not the same for the functions which have a Domaine of clean definition.

Being given two functions $f$ and $g$ definite on the respective fields $D\_f$ and $D\_g$ (for example of the real intervals), the function $f+g$ has for field the intersection $D\_\; \{f+g\}\; =\; D\_f\; \backslash \; course\; D\_g$ and for expression the usual addition $(f+g)\; (X)\; =f\; (X)\; +\; G\; (X)$.

This addition is associative and commutative. Its neutral is the function defined everywhere and constantly null, but the addition of a “opposite function” does not make it possible to extend the field of definition. For example, the sum of the functions $x\; \backslash \; mapsto\; \backslash \; sqrt\; \{X\}$ and $x\; \backslash \; mapsto\; -\; \backslash \; sqrt\; \{X\}$ is the null function only definite on positive realities .

In certain contexts, as in the addition of the functions méromorphes, the obliteration of the singularities however makes it possible to evacuate the problem of the field of definition of the sum.

Independent random variables

In elementary probabilities, being given two random variable independent being able to take only one finished number of values, the addition is calculated by building a table with a line by value of the first variable and a column by value of the second variable.

Each box of the table is filled with on the one hand the nap of the values of the line and the corresponding column, on the other hand the produces corresponding probabilities . Then, it is enough for each value appearing in the table to make the sum of the probabilities boxes which contain it.

In continuous probabilities, the Densité of probability of a sum of two independent random variables is given by the Produit convolution of the densities of probabilities initiales.
$f\_\; \{X+Y\}\; =\; f\_X\; *\; f\_Y\; \backslash \; colonist\; X\; \backslash \; mapsto\; \backslash \; int\; f\_X\; (T)\; f\_Y\; (x-t)\; \backslash \; mathrm\; dt$.

This presentation extends to the random variables whose function of density is a distribution.

This operation is associative and commutative. The neutral is the always null random variable, but only the numbers, represented by the constant random variables admit opposites. There does not exist opposite with the nonconstant random variables they are then of wide strictly positive, but the extent of a sum is then the sum of the extents.

Real limits

The limiting of continuations or functions with actual value can be taken in the continued line $\backslash \; overline\; \{\backslash \; R\}\; =\; \backslash \; R\; \backslash \; cup\; \backslash \; left\; \backslash \; \{-\; \backslash \; infty,\; +\; \backslash \; infty\; \backslash \; right\; \backslash \}$. The addition of the numbers can then extend partially under the infinite terms by:

for all $x$ real, $x+\; (+\; \backslash \; infty)\; =\; (+\; \backslash \; infty)\; +x\; =\; (+\; \backslash \; infty)\; +\; (+\; \backslash \; infty)\; =\; +\; \backslash \; infty$ and $x+\; (-\; \backslash \; infty)\; =\; (-\; \backslash \; infty)\; +x\; =\; (-\; \backslash \; infty)\; +\; (-\; \backslash \; infty)\; =\; -\; \backslash \; infty$.

This operation keeps properties of commutation and of associativeness but for the couples $(-\; \backslash \; infty\; is\; not\; defined;\; +\; \backslash \; infty)$ and $(+\; \backslash \; infty;\; -\; \backslash \; infty)$.

According to the cases, the sum of two continuations or functions admitting of the opposite infinite limits can have a finished, infinite limit or not of limit of the whole.

This extension of the addition is used in particular in Théorie of measurement to satisfy the additivity of measurement on spaces of infinite measurement.

Ordinal and ordered units

The class of the ordinal extends the whole of the natural whole by the transfinite numbers. The addition extends thus in an operation on the ordinal numbers which is associative but noncommutative. For example, first the ordinal infinite one, noted $\backslash \; omega$, check the relation $1+\; \backslash \; Omega\; =\; \backslash \; omega$ but .

Element 0 remains neutral for the addition but there is not the ordinal negative one, although one can define a difference between two ordinal.

See also: ordinal Number

This operation extends to the ordered units in general, the addition of two ordered units $(E,\; \backslash \; it)$ and $(F,\; \backslash \; it)$ having for result the disjoined union $E\; \backslash \; sqcup\; F$ in which the order of the elements is preserved inside each starting whole and all the elements of $E$ are lower than all the elements of $F$.

Surreal numbers

A surreal Nombre is a generalization of the concept of number in the shape of a couple of units being written $X\; =\; \backslash \; left\; \backslash \; \{X\_L\; |\; X\_R\; \backslash \; right\; \backslash \}$, in which each element of the whole of left is smaller than any element of the whole of right-hand side.

The addition is formulated then in a recursive way by

$X+Y\; =\; \backslash \; left\; \backslash \; \{X\_L+Y\; \backslash \; cup\; X+Y\_L\; |\; X\_R+Y\; \backslash cup\; X+Y\_R\; \backslash right\backslash \}$

with $A+Y\; =\; \backslash \; left\; \backslash \; \{a+Y/has\; \backslash \; in\; has\; \backslash \; right\; \backslash \}$ and $X+B\; =\; \backslash \; left\; \backslash \; \{X+b/B\; \backslash \; in\; B\; \backslash \; right\; \backslash \}$.

See also: surreal Number

Other additions

Addition with modulo

Since the Parity of a sum depending only on the parity of the operands, it can be defined an addition on the parities.

This operation spreads for entire strictly positive $m$ in a addition modulo $m$ on the figures of 0 with $m-1$, in which each number is replaced by the remainder of its Euclidean Division by $m$.
the addition on the parities is then represented by the addition modulo 2, where the even numbers are replaced by 0 and the odd numbers by 1.

See also: Congruence on the entireties

Boolean addition

The addition Booléen is not the writing of the logical connector “OR” with figures 0 for FALSE and 1 for TRUE . It is thus given by the table of following addition:

The operation is associative and commutative, element 0 is neutral but element 1 does not have opposite.

See also: Boolean algebra (logical)

Vector addition on a cubic curve

On certain curves, one can define an addition geometrically. It is possible in particular on cubic curved , i.e. plane curves defined by an equation of the 3ère degree. More precisely, by calling $x$ and $y$ the coordinates in the real plan, the points of the curve are the points $P$ whose coordinates $x,\; y$ check an equation $F\; (X,\; there)\; =0$, for a polynomial $F$ of the third degree with real coefficients given. One also supposes that the curve does not have a singular points, i.e. here of double point or points of reflection; the tangent is thus well defined in each point. To standardize constructions, one ad infinitum adds also a Point.

That is to say maintaining two points unspecified of the curve, $P$ and $Q$. The line which them joint recuts the curve in a third point $R$ (if $P=Q$, one takes as right-hand side uniting them the tangent in $P$). This property comes from what the intersection of a line and a curve of degree 3 has 1 or 3 real solutions (one can also prove it starting from the equation $F$ curve and of that of the right-hand side $PQ$). This process (which one calls “the method of the tangents and the secants”) defines a binary operation well on the curve. It does not have yet the awaited properties of an addition: for example, there is no neutral element. To cure it, one fixes a point at the choice on the curve, which one notes $P\_0$, and one considers the line passing by $P\_0$ and $R$: it still cuts the cubic one in a third point. It is this point which one calls “nap of $P$ and $Q$” (and one notes it $P\; +\; Q$).

The selected point $P\_0$ is the neutral element (the “zero”) for this operation. As for the “opposite” of a point $P$, it is the third point of intersection with the curve of the right-hand side passing by $P$ and $P\text{'}\_0$, where $P\text{'}\_0$ is the third point of intersection with the curve of the tangent to the curve in $P\_0$.

See also: elliptic Curve

See too

Related articles

• Arithmetic
• Operation (mathematics)
• Additive (page of homonymy)
• Plus signs and less
• Technical Table of addition
• of the addition

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