In

(1.000_{2}×2^{0} +
1.000_{2}×2^{0}) +
1.000_{2}×2^{4} =
1.000_{2}×2^{} +
1.000_{2}×2^{4} =
1.00_{2}×2^{4}

1.000_{2}×2^{0} +
(1.000_{2}×2^{0} +
1.000_{2}×2^{4}) =
1.000_{2}×2^{} +
1.000_{2}×2^{4} =
1.00_{2}×2^{4}
Even though most computers compute with a 24 or 53 bits of mantissa, this is an important source of rounding error, and approaches such as the

)

Using Order of Operations and Exploring Properties

section 9Bronstein: :de:Taschenbuch der Mathematik, pages 115-120, chapter: 2.4.1.1, ::$x-y-z=(x-y)-z$ ::$x/y/z=(x/y)/z$ * Function application: ::$(f\; \backslash ,\; x\; \backslash ,\; y)\; =\; ((f\; \backslash ,\; x)\; \backslash ,\; y)$ :This notation can be motivated by the

Codeplea. 23 August 2016. Retrieved 20 September 2016. ::$(x^\backslash wedge\; y)^\backslash wedge\; z\backslash ne\; x^\backslash wedge(y^\backslash wedge\; z)$ * Knuth's up-arrow operators: ::$a\; \backslash uparrow\; \backslash uparrow\; (b\; \backslash uparrow\; \backslash uparrow\; c)\; \backslash ne\; (a\; \backslash uparrow\; \backslash uparrow\; b)\; \backslash uparrow\; \backslash uparrow\; c$ ::$a\; \backslash uparrow\; \backslash uparrow\; \backslash uparrow\; (b\; \backslash uparrow\; \backslash uparrow\; \backslash uparrow\; c)\; \backslash ne\; (a\; \backslash uparrow\; \backslash uparrow\; \backslash uparrow\; b)\; \backslash uparrow\; \backslash uparrow\; \backslash uparrow\; c$ * Taking the

mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...

, the associative property is a property of some binary operation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s, which means that rearranging the parentheses in an expression will not change the result. In propositional logic
Propositional calculus is a branch of logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, ...

, associativity is a valid rule of replacement
In logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, a ...

for expressions
Expression may refer to:
Linguistics
* Expression (linguistics), a word, phrase, or sentence
* Fixed expression, a form of words with a specific meaning
* Idiom, a type of fixed expression
* Metaphor#Common types, Metaphorical expression, a parti ...

in logical proofs.
Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations
Operation or Operations may refer to:
Science and technology
* Surgical operation
Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, meaning "hand work". is a medical or dental specialty that ...

are performed does not matter as long as the sequence of the operand
In mathematics an operand is the object of a mathematical operation, i.e., it is the object or quantity that is operated on.
Example
The following arithmetic expression shows an example of operators and operands:
:3 + 6 = 9
In the above example, ...

s is not changed. That is, (after rewriting the expression with parentheses and in infix notation if necessary) rearranging the parentheses
A bracket is either of two tall fore- or back-facing punctuation
Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding ...

in such an expression will not change its value. Consider the following equations:
:$(2\; +\; 3)\; +\; 4\; =\; 2\; +\; (3\; +\; 4)\; =\; 9\; \backslash ,$
:$2\; \backslash times\; (3\; \backslash times\; 4)\; =\; (2\; \backslash times\; 3)\; \backslash times\; 4\; =\; 24\; .$
Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any real number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s, it can be said that "addition and multiplication of real numbers are associative operations".
Associativity is not the same as commutativity
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, which addresses whether the order of two operand
In mathematics an operand is the object of a mathematical operation, i.e., it is the object or quantity that is operated on.
Example
The following arithmetic expression shows an example of operators and operands:
:3 + 6 = 9
In the above example, ...

s affects the result. For example, the order does not matter in the multiplication of real numbers, that is, , so we say that the multiplication of real numbers is a commutative operation. However, operations such as function composition
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

and matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the nu ...

are associative, but (generally) not commutative.
Associative operations are abundant in mathematics; in fact, many algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s (such as semigroups
In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative binary operation.
The binary operation of a semigroup is most often denoted multiplication, multiplicatively: ''x''·''y'', o ...

and categories
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the human ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such ...

) explicitly require their binary operations to be associative.
However, many important and interesting operations are non-associative; some examples include subtraction
Subtraction is an arithmetic operation
Arithmetic (from the Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located ...

, exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " raised to the power of ". When is a positive integer
An integer (from the Latin wikt: ...

, and the vector cross product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. In contrast to the theoretical properties of real numbers, the addition of floating point
In computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and soft ...

numbers in computer science is not associative, and the choice of how to associate an expression can have a significant effect on rounding error.
Definition

Formally, abinary operation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

∗ on a set ''S'' is called associative if it satisfies the associative law:
:(''x'' ∗ ''y'') ∗ ''z'' = ''x'' ∗ (''y'' ∗ ''z'') for all ''x'', ''y'', ''z'' in ''S''.
Here, ∗ is used to replace the symbol of the operation, which may be any symbol, and even the absence of symbol (juxtaposition
Juxtaposition is an act or instance of placing two elements close together or side by side. This is often done in order to compare/contrast the two, to show similarities or differences, etc.
Speech
Juxtaposition in literary terms is the showing ...

) as for multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary Operation (mathematics), mathematical operations of arithmetic, with the ...

.
:(''xy'')''z'' = ''x''(''yz'') = ''xyz'' for all ''x'', ''y'', ''z'' in ''S''.
The associative law can also be expressed in functional notation thus: .
Generalized associative law

250px, In the absence of the associative property, five factors ''a, b, c, d, e'' result in a of order four, possibly different products.">Tamari lattice of order four, possibly different products. If a binary operation is associative, repeated application of the operation produces the same result regardless of how valid pairs of parentheses are inserted in the expression. This is called the generalized associative law. For instance, a product of four elements may be written, without changing the order of the factors, in five possible ways: : $((ab)c)d$ : $(ab)(cd)$ : $(a(bc))d$ : $a((bc)d)$ : $a(b(cd))$ If the product operation is associative, the generalized associative law says that all these formulas will yield the same result. So unless the formula with omitted parentheses already has a different meaning (see below), the parentheses can be considered unnecessary and "the" product can be written unambiguously as :$abcd.$ As the number of elements increases, the number of possible ways to insert parentheses grows quickly, but they remain unnecessary for disambiguation. An example where this does not work is thelogical biconditional
In logic and mathematics, the logical biconditional, sometimes known as the material biconditional, is the logical connective (\leftrightarrow) used to conjoin two statements and to form the statement " if and only if ", where is known as the ...

$\backslash leftrightarrow$. It is associative, thus A$\backslash leftrightarrow$(B$\backslash leftrightarrow$C) is equivalent to (A$\backslash leftrightarrow$B)$\backslash leftrightarrow$C, but A$\backslash leftrightarrow$B$\backslash leftrightarrow$C most commonly means (A$\backslash leftrightarrow$B and B$\backslash leftrightarrow$C), which is not equivalent.
Examples

Some examples of associative operations include the following. * Theconcatenation
In formal language theory and computer programming
Computer programming is the process of designing and building an executable computer program to accomplish a specific computing result or to perform a specific task. Programming involves ...

of the three strings `"hello"`

, `" "`

, `"world"`

can be computed by concatenating the first two strings (giving `"hello "`

) and appending the third string (`"world"`

), or by joining the second and third string (giving `" world"`

) and concatenating the first string (`"hello"`

) with the result. The two methods produce the same result; string concatenation is associative (but not commutative).
* In arithmetic
Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cra ...

, addition
Addition (usually signified by the plus symbol
The plus and minus signs, and , are mathematical symbol
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object
A mathematical object is an ...

and multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary Operation (mathematics), mathematical operations of arithmetic, with the ...

of real number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s are associative; i.e.,
:: $\backslash left.\; \backslash begin\; (x+y)+z=x+(y+z)=x+y+z\backslash quad\; \backslash \backslash \; (x\backslash ,y)z=x(y\backslash ,z)=x\backslash ,y\backslash ,z\backslash qquad\backslash qquad\backslash qquad\backslash quad\backslash \; \backslash \; \backslash ,\; \backslash end\; \backslash right\backslash \}\; \backslash mboxx,y,z\backslash in\backslash mathbb.$
:Because of associativity, the grouping parentheses can be omitted without ambiguity.
* The trivial operation (that is, the result is the first argument, no matter what the second argument is) is associative but not commutative. Likewise, the trivial operation (that is, the result is the second argument, no matter what the first argument is) is associative but not commutative.
* Addition and multiplication of complex number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s and quaternion
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s are associative. Addition of octonion
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s is also associative, but multiplication of octonions is non-associative.
* The greatest common divisor
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

and least common multiple
In arithmetic
Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'a ...

functions act associatively.
:: $\backslash left.\; \backslash begin\; \backslash operatorname(\backslash operatorname(x,y),z)=\; \backslash operatorname(x,\backslash operatorname(y,z))=\; \backslash operatorname(x,y,z)\backslash \; \backslash quad\; \backslash \backslash \; \backslash operatorname(\backslash operatorname(x,y),z)=\; \backslash operatorname(x,\backslash operatorname(y,z))=\; \backslash operatorname(x,y,z)\backslash quad\; \backslash end\; \backslash right\backslash \}\backslash mboxx,y,z\backslash in\backslash mathbb.$
* Taking the intersection
The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points.
In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...

or the union of sets:
:: $\backslash left.\; \backslash begin\; (A\backslash cap\; B)\backslash cap\; C=A\backslash cap(B\backslash cap\; C)=A\backslash cap\; B\backslash cap\; C\backslash quad\; \backslash \backslash \; (A\backslash cup\; B)\backslash cup\; C=A\backslash cup(B\backslash cup\; C)=A\backslash cup\; B\backslash cup\; C\backslash quad\; \backslash end\; \backslash right\backslash \}\backslash mboxA,B,C.$
* If ''M'' is some set and ''S'' denotes the set of all functions from ''M'' to ''M'', then the operation of function composition
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

on ''S'' is associative:
:: $(f\backslash circ\; g)\backslash circ\; h=f\backslash circ(g\backslash circ\; h)=f\backslash circ\; g\backslash circ\; h\backslash qquad\backslash mboxf,g,h\backslash in\; S.$
* Slightly more generally, given four sets ''M'', ''N'', ''P'' and ''Q'', with ''h'': ''M'' to ''N'', ''g'': ''N'' to ''P'', and ''f'': ''P'' to ''Q'', then
:: $(f\backslash circ\; g)\backslash circ\; h=f\backslash circ(g\backslash circ\; h)=f\backslash circ\; g\backslash circ\; h$
: as before. In short, composition of maps is always associative.
* Consider a set with three elements, A, B, and C. The following operation:
:
:is associative. Thus, for example, A(BC)=(AB)C = A. This operation is not commutative.
* Because matrices
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...

represent linear function
In mathematics, the term linear function refers to two distinct but related notions:
* In calculus and related areas, a linear function is a function (mathematics), function whose graph of a function, graph is a straight line, that is, a polynomial ...

s, and matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix (mathematics), matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the nu ...

represents function composition, one can immediately conclude that matrix multiplication is associative.
Propositional logic

Rule of replacement

In standard truth-functional propositional logic, ''association'', or ''associativity'' are two valid rules of replacement. The rules allow one to move parentheses inlogical expressions
Logic (from Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 milli ...

in logical proofs. The rules (using logical connectives notation) are:
:$(P\; \backslash lor\; (Q\; \backslash lor\; R))\; \backslash Leftrightarrow\; ((P\; \backslash lor\; Q)\; \backslash lor\; R)$
and
:$(P\; \backslash land\; (Q\; \backslash land\; R))\; \backslash Leftrightarrow\; ((P\; \backslash land\; Q)\; \backslash land\; R),$
where "$\backslash Leftrightarrow$" is a metalogic
Metalogic is the study of the metatheory
A metatheory or meta-theory is a theory
A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational th ...

al symbol
A symbol is a mark, sign, or word
In linguistics, a word of a spoken language can be defined as the smallest sequence of phonemes that can be uttered in isolation with semantic, objective or pragmatics, practical meaning (linguistics), meanin ...

representing "can be replaced in a proof
Proof may refer to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Formal sciences
* Formal proof, a construct in proof theory
* Mathematical proof, a co ...

with."
Truth functional connectives

''Associativity'' is a property of somelogical connective
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, la ...

s of truth-functional propositional logic
Propositional calculus is a branch of logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, ...

. The following logical equivalence
Logic (from Greek: grc, λογική, label=none, lit=possessed of reason
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=n ...

s demonstrate that associativity is a property of particular connectives. The following are truth-functional tautologies.
Associativity of disjunction:
:$((P\; \backslash lor\; Q)\; \backslash lor\; R)\; \backslash leftrightarrow\; (P\; \backslash lor\; (Q\; \backslash lor\; R))$
:$(P\; \backslash lor\; (Q\; \backslash lor\; R))\; \backslash leftrightarrow\; ((P\; \backslash lor\; Q)\; \backslash lor\; R)$
Associativity of conjunction:
:$((P\; \backslash land\; Q)\; \backslash land\; R)\; \backslash leftrightarrow\; (P\; \backslash land\; (Q\; \backslash land\; R))$
:$(P\; \backslash land\; (Q\; \backslash land\; R))\; \backslash leftrightarrow\; ((P\; \backslash land\; Q)\; \backslash land\; R)$
Associativity of equivalence:
:$((P\; \backslash leftrightarrow\; Q)\; \backslash leftrightarrow\; R)\; \backslash leftrightarrow\; (P\; \backslash leftrightarrow\; (Q\; \backslash leftrightarrow\; R))$
:$(P\; \backslash leftrightarrow\; (Q\; \backslash leftrightarrow\; R))\; \backslash leftrightarrow\; ((P\; \backslash leftrightarrow\; Q)\; \backslash leftrightarrow\; R)$
Joint denial is an example of a truth functional connective that is ''not'' associative.
Non-associative operation

A binary operation $*$ on a set ''S'' that does not satisfy the associative law is called non-associative. Symbolically, :$(x*y)*z\backslash ne\; x*(y*z)\backslash qquad\backslash mboxx,y,z\backslash in\; S.$ For such an operation the order of evaluation ''does'' matter. For example: *Subtraction
Subtraction is an arithmetic operation
Arithmetic (from the Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located ...

:$(5-3)-2\; \backslash ,\; \backslash ne\; \backslash ,\; 5-(3-2)$
* Division
Division or divider may refer to:
Mathematics
*Division (mathematics), the inverse of multiplication
*Division algorithm, a method for computing the result of mathematical division
Military
*Division (military), a formation typically consisting o ...

:$(4/2)/2\; \backslash ,\; \backslash ne\; \backslash ,\; 4/(2/2)$
* Exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " raised to the power of ". When is a positive integer
An integer (from the Latin wikt: ...

:$2^\; \backslash ,\; \backslash ne\; \backslash ,\; (2^1)^2$
Also note that infinite sums are not generally associative, for example:
:$(1+-1)+(1+-1)+(1+-1)+(1+-1)+(1+-1)+(1+-1)+\backslash dots\; \backslash ,\; =\; \backslash ,\; 0$
whereas
:$1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)+\backslash dots\; \backslash ,\; =\; \backslash ,\; 1$
The study of non-associative structures arises from reasons somewhat different from the mainstream of classical algebra. One area within non-associative algebra
A non-associative algebra (or distributive algebra) is an algebra over a field
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebr ...

that has grown very large is that of Lie algebra
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s. There the associative law is replaced by the Jacobi identity
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

. Lie algebras abstract the essential nature of infinitesimal transformation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s, and have become ubiquitous in mathematics.
There are other specific types of non-associative structures that have been studied in depth; these tend to come from some specific applications or areas such as combinatorial mathematics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite set, finite Mathematical structure, structures. It is closely related to many other area ...

. Other examples are quasigroup
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

, quasifieldIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

, non-associative ring, non-associative algebra
A non-associative algebra (or distributive algebra) is an algebra over a field
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebr ...

and commutative non-associative magmas.
Nonassociativity of floating point calculation

In mathematics, addition and multiplication of real numbers is associative. By contrast, in computer science, the addition and multiplication offloating point
In computing
Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and soft ...

numbers is ''not'' associative, as rounding errors are introduced when dissimilar-sized values are joined together.
To illustrate this, consider a floating point representation with a 4-bit mantissa
Mantissa () may refer to:
* Mantissa (logarithm), the fractional part of the common (base-10) logarithm
* Mantissa (floating point number)
The significand (also mantissa or coefficient, sometimes also argument, or ambiguously fraction or characte ...

:
(1.000

1.000

Kahan summation algorithmIn numerical analysis, the Kahan summation algorithm, also known as compensated summation, significantly reduces the numerical error in the total obtained by adding a sequence of finite-decimal precision, precision floating-point numbers, compared to ...

are ways to minimise the errors. It can be especially problematic in parallel computing.)

Notation for non-associative operations

In general, parentheses must be used to indicate the order of operations, order of evaluation if a non-associative operation appears more than once in an expression (unless the notation specifies the order in another way, like $\backslash dfrac$). However,mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry) ...

s agree on a particular order of evaluation for several common non-associative operations. This is simply a notational convention to avoid parentheses.
A left-associative operation is a non-associative operation that is conventionally evaluated from left to right, i.e.,
:$\backslash left.\; \backslash begin\; x*y*z=(x*y)*z\backslash qquad\backslash qquad\backslash quad\backslash ,\; \backslash \backslash \; w*x*y*z=((w*x)*y)*z\backslash quad\; \backslash \backslash \; \backslash mbox\backslash qquad\backslash qquad\backslash qquad\backslash qquad\backslash qquad\backslash qquad\backslash \; \backslash \; \backslash ,\; \backslash end\; \backslash right\backslash \}\; \backslash mboxw,x,y,z\backslash in\; S$
while a right-associative operation is conventionally evaluated from right to left:
:$\backslash left.\; \backslash begin\; x*y*z=x*(y*z)\backslash qquad\backslash qquad\backslash quad\backslash ,\; \backslash \backslash \; w*x*y*z=w*(x*(y*z))\backslash quad\; \backslash \backslash \; \backslash mbox\backslash qquad\backslash qquad\backslash qquad\backslash qquad\backslash qquad\backslash qquad\backslash \; \backslash \; \backslash ,\; \backslash end\; \backslash right\backslash \}\; \backslash mboxw,x,y,z\backslash in\; S$
Both left-associative and right-associative operations occur. Left-associative operations include the following:
* Subtraction and division of real numbers:Virginia Department of EducationUsing Order of Operations and Exploring Properties

section 9Bronstein: :de:Taschenbuch der Mathematik, pages 115-120, chapter: 2.4.1.1, ::$x-y-z=(x-y)-z$ ::$x/y/z=(x/y)/z$ * Function application: ::$(f\; \backslash ,\; x\; \backslash ,\; y)\; =\; ((f\; \backslash ,\; x)\; \backslash ,\; y)$ :This notation can be motivated by the

currying
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

isomorphism.
Right-associative operations include the following:
* Exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " raised to the power of ". When is a positive integer
An integer (from the Latin wikt: ...

of real numbers in superscript notation:
::$x^=x^$
:Exponentiation is commonly used with brackets or right-associatively because a repeated left-associative exponentiation operation is of little use. Repeated powers would mostly be rewritten with multiplication:
::$(x^y)^z=x^$
:Formatted correctly, the superscript inherently behaves as a set of parentheses; e.g. in the expression $2^$ the addition is performed the exponentiation despite there being no explicit parentheses $2^$ wrapped around it. Thus given an expression such as $x^$, the full exponent $y^z$ of the base $x$ is evaluated first. However, in some contexts, especially in handwriting, the difference between $^z=(x^y)^z$, $x^=x^$ and $x^=x^$ can be hard to see. In such a case, right-associativity is usually implied.
* Function definition
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriented ...

::$\backslash mathbb\; \backslash rarr\; \backslash mathbb\; \backslash rarr\; \backslash mathbb\; =\; \backslash mathbb\; \backslash rarr\; (\backslash mathbb\; \backslash rarr\; \backslash mathbb)$
::$x\; \backslash mapsto\; y\; \backslash mapsto\; x\; -\; y\; =\; x\; \backslash mapsto\; (y\; \backslash mapsto\; x\; -\; y)$
:Using right-associative notation for these operations can be motivated by the Curry–Howard correspondence
In programming language theory and proof theory
Proof may refer to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Formal sciences
* Formal proof, a c ...

and by the currying
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

isomorphism.
Non-associative operations for which no conventional evaluation order is defined include the following.
* Exponentiation of real numbers in infix notation:Exponentiation Associativity and Standard Math NotationCodeplea. 23 August 2016. Retrieved 20 September 2016. ::$(x^\backslash wedge\; y)^\backslash wedge\; z\backslash ne\; x^\backslash wedge(y^\backslash wedge\; z)$ * Knuth's up-arrow operators: ::$a\; \backslash uparrow\; \backslash uparrow\; (b\; \backslash uparrow\; \backslash uparrow\; c)\; \backslash ne\; (a\; \backslash uparrow\; \backslash uparrow\; b)\; \backslash uparrow\; \backslash uparrow\; c$ ::$a\; \backslash uparrow\; \backslash uparrow\; \backslash uparrow\; (b\; \backslash uparrow\; \backslash uparrow\; \backslash uparrow\; c)\; \backslash ne\; (a\; \backslash uparrow\; \backslash uparrow\; \backslash uparrow\; b)\; \backslash uparrow\; \backslash uparrow\; \backslash uparrow\; c$ * Taking the

cross product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of three vectors:
::$\backslash vec\; a\; \backslash times\; (\backslash vec\; b\; \backslash times\; \backslash vec\; c)\; \backslash neq\; (\backslash vec\; a\; \backslash times\; \backslash vec\; b\; )\; \backslash times\; \backslash vec\; c\; \backslash qquad\; \backslash mbox\; \backslash vec\; a,\backslash vec\; b,\backslash vec\; c\; \backslash in\; \backslash mathbb^3$
* Taking the pairwise average
In colloquial, ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2 ...

of real numbers:
::$\backslash ne\; \backslash qquad\; \backslash mboxx,y,z\backslash in\backslash mathbb\; \backslash mboxx\backslash ne\; z.$
* Taking the relative complement
In set theory, the complement of a Set (mathematics), set , often denoted by (or ), are the Element (mathematics), elements not in .
When all sets under consideration are considered to be subsets of a given set , the absolute complement of is t ...

of sets $(A\backslash backslash\; B)\backslash backslash\; C$ is not the same as $A\backslash backslash\; (B\backslash backslash\; C)$. (Compare material nonimplication
Material nonimplication or abjunction (Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through ...

in logic.)
See also

* Light's associativity test * Telescoping series, the use of addition associativity for cancelling terms in an infinite series (mathematics), series * A semigroup is a set with an associative binary operation. * Commutativity and distributivity are two other frequently discussed properties of binary operations. * Power associativity, alternativity, flexible algebra, flexibility and N-ary associativity are weak forms of associativity. * Moufang loop, Moufang identities also provide a weak form of associativity.References

{{reflist Properties of binary operations Elementary algebra Functional analysis Rules of inference