Argument of a functionFrom Wikipedia, the free encyclopedia

Contents

1 Argument of a function 11.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Constant (mathematics) 22.1 Constant function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Context-dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Notable mathematical constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 Constants in calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Constant function 53.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Constant term 84.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5 Term (logic) 105.1 Elementary mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.2 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5.2.1 Term structure vs. representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.2.2 Structural equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.2.3 Ground and linear terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.2.4 Building formulas from terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5.3 Operations with terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.4 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.4.1 Sorted terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.4.2 Lambda terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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ii CONTENTS

5.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6 Value (mathematics) 166.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

7 Variable (mathematics) 177.1 Etymology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.2 Genesis and evolution of the concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.3 Specific kinds of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7.3.1 Dependent and independent variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7.4 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.6 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Chapter 1

Argument of a function

In mathematics, an argument of a function is a specific input in the function, also known as an independent variable.When it is clear from the context which argument is meant, the argument is often denoted by the abbreviation arg.[1]

Amathematical function has one or more arguments in the form of independent variables designated in the function’sdefinition, which can also contain parameters. The independent variables are mentioned in the list of arguments thatthe function takes, whereas the parameters are not. For example, in the logarithmic function f(x) = logb(x) , thebase b is considered a parameter.A function that takes a single argument as input (such as f(x) = x2 ) is called a unary function. A function of two ormore variables is considered to have a domain consisting of ordered pairs or tuples of argument values. For example,the binary function f(x, y) = x2 + y2 has two arguments, x and y , in an ordered pair (x, y) . The hypergeometricfunction is an example of a four-argument function. The number of arguments that a function takes is called the arityof the function.The argument of a circular function is an angle. The argument of a hyperbolic function is a hyperbolic angle.

1.1 See also• Value (mathematics)

1.2 References[1] Aleksandrov, A. D.; Kolmogorov, A. N., eds. (1999). Mathematics: Its Content, Methods and Meaning. Courier Dover.

1.3 External links• Weisstein, Eric W., “Argument”, MathWorld.

• argument at PlanetMath.org.

1

Chapter 2

Constant (mathematics)

In mathematics, the adjective constant means non-varying. The noun constant may have two different meanings. Itmay refer to a fixed and well defined number or other mathematical object. The term mathematical constant (and alsophysical constant) is sometimes used to distinguish this meaning from the other one. A constant may also refer to aconstant function or its value (it is a common usage to identify them). Such a constant is commonly represented bya variable which does not depend on the main variable(s) of the studied problem. This is the case, for example, fora constant of integration which is an arbitrary constant function (not depending on the variable of integration) addedto a particular antiderivative to get all the antiderivatives of the given function.For example, a general quadratic function is commonly written as:

ax2 + bx+ c ,

where a, b and c are constants (or parameters), while x is the variable, a placeholder for the argument of the functionbeing studied. A more explicit way to denote this function is

x 7→ ax2 + bx+ c ,

which makes the function-argument status of x clear, and thereby implicitly the constant status of a, b and c. In thisexample a, b and c are coefficients of the polynomial. Since c occurs in a term that does not involve x, it is called theconstant term of the polynomial and can be thought of as the coefficient of x0; any polynomial term or expression ofdegree zero is a constant.[1]:18

2.1 Constant function

Main articles: Constant function and Nullary

A constant may be used to define a constant function that ignores its arguments and always gives the same value. Aconstant function of a single variable, such as f(x) = 5 , has a graph that is a horizontal straight line, parallel tothe x-axis. Such a function always takes the same value (in this case, 5) because its argument does not appear in theexpression defining the function.

2.2 Context-dependence

The context-dependent nature of the concept of “constant” can be seen in this example from elementary calculus:

ddx2

x = limh→02x+h−2x

h = limh→0 2x 2h−1

h

= 2x limh→02h−1

h sincex on depend not does (i.e. constant is h)= 2x · constant, where constant on depending not means x.

2

2.3. NOTABLE MATHEMATICAL CONSTANTS 3

“Constant” means not depending on some variable; not changing as that variable changes. In the first case above, itmeans not depending on h; in the second, it means not depending on x.

2.3 Notable mathematical constants

Main article: Mathematical constant

Some values occur frequently in mathematics and are conventionally denoted by a specific symbol. These standardsymbols and their values are called mathematical constants. Examples include:

• 0 (zero).

• 1 (one), the natural number after zero.

• π (pi), the constant representing the ratio of a circle’s circumference to its diameter, approximately equal to3.141592653589793238462643...[2]

• e, approximately equal to 2.718281828459045235360287...

• i, the imaginary unit such that i2 = −1.

•√2 (square root of 2), the length of the diagonal of a square with unit sides, approximately equal to 1.414213562373095048801688.

• φ (golden ratio), approximately equal to 1.618033988749894848204586, or algebraically, 1+√5

2 .

2.4 Constants in calculus

In calculus, constants are treated in several different ways depending on the operation. For example, the derivativeof a constant function is zero. This is because the derivative measures the rate of change of a function with respectto a variable, and since constants, by definition, do not change, their derivative is therefore zero. Conversely, whenintegrating a constant function, the constant is multiplied by the variable of integration. During the evaluation of alimit, the constant remains the same as it was before and after evaluation.Integration of a function of one variable often involves a constant of integration. This arises because of the integraloperator’s nature as the inverse of the differential operator, meaning the aim of integration is to recover the originalfunction before differentiation. The differential of a constant function is zero, as noted above, and the differentialoperator is a linear operator, so functions that only differ by a constant term have the same derivative. To acknowledgethis, a constant of integration is added to an indefinite integral; this ensures that all possible solutions are included.The constant of integration is generally written as 'c' and represents a constant with a fixed but undefined value.

2.4.1 Examples

f(x) = 72 ⇒ f ′(x) = 0f(x) = 72 ⇒

∫72 dx = 72x+ c

f(x) = 72 ⇒ limx→∞ 72 = 72

2.5 See also

• Expression

• Physical constant

• Constant (disambiguation)

4 CHAPTER 2. CONSTANT (MATHEMATICS)

2.6 References[1] Foerster, Paul A. (2006). Algebra and Trigonometry: Functions and Applications, Teacher’s Edition (Classics ed.). Upper

Saddle River, NJ: Prentice Hall. ISBN 0-13-165711-9.

[2] Arndt, Jörg; Haenel, Christoph (2001). Pi - Unleashed. Springer. p. 240. ISBN 978-3540665724.

Chapter 3

Constant function

Not to be confused with function constant.In mathematics, a constant function is a function whose (output) value is the same for every input value.[1][2][3] Forexample, the function y(x) = 4 is a constant function because the value of y(x) is 4 regardless of the input value x(see image).

3.1 Basic properties

As a real-valued function of a real-valued argument, a constant function has the general form y(x) = c or just y = c.

Example: The function y(x) = 2 or just y = 2 is the specific constant function where the output valueis c = 2 . The domain of this function is the set of all real numbers ℝ. The codomain of this functionis just {2}. The independent variable x does not appear on the right side of the function expression andso its value is “vacuously substituted”. Namely y(0)=2, y(−2.7)=2, y(π)=2,.... No matter what value ofx is input, the output is “2”.

Real-world example: A store where every item is sold for the price of 1 euro.

The graph of the constant function y = c is a horizontal line in the plane that passes through the point (0, c) .[4]

In the context of a polynomial in one variable x, the non-zero constant function is a polynomial of degree 0 andits general form is f(x) = c , c ̸= 0 . This function has no intersection point with the x-axis, that is, it has no root(zero). On the other hand, the polynomial f(x) = 0 is the identically zero function. It is the (trivial) constantfunction and every x is a root. Its graph is the x-axis in the plane.[5]

A constant function is an even function, i.e. the graph of a constant function is symmetric with respect to the y-axis.In the context where it is defined, the derivative of a function is a measure of the rate of change of function valueswith respect to change in input values. Because a constant function does not change, its derivative is 0.[6] This isoften written: (c)′ = 0 . The converse is also true. Namely, if y'(x)=0 for all real numbers x, then y(x) is a constantfunction.[7]

Example: Given the constant function y(x) = −√2 . The derivative of y is the identically zero function

y′(x) = (−√2)′ = 0 .

3.2 Other properties

For functions between preordered sets, constant functions are both order-preserving and order-reversing; conversely,if f is both order-preserving and order-reversing, and if the domain of f is a lattice, then f must be constant.

• Every constant function whose domain and codomain are the same is idempotent.

5

6 CHAPTER 3. CONSTANT FUNCTION

Constant function y=4

• Every constant function between topological spaces is continuous.

• A constant function factors through the one-point set, the terminal object in the category of sets. This obser-vation is instrumental for F. William Lawvere's axiomatization of set theory, the Elementary Theory of theCategory of Sets (ETCS).[8]

• Every set X is isomorphic to the set of constant functions into it. For each element x and any set Y, there is aunique function x̃ : Y → X such that x̃(y) = x for all y ∈ Y . Conversely, if a function f : X → Y satisfiesf(y) = f(y′) for all y, y′ ∈ Y , f is by definition a constant function.

• As a corollary, the one-point set is a generator in the category of sets.• Every set X is canonically isomorphic to the function set X1 , or hom set hom(1, X) in the categoryof sets, where 1 is the one-point set. Because of this, and the adjunction between cartesian productsand hom in the category of sets (so there is a canonical isomorphism between functions of two vari-ables and functions of one variable valued in functions of another (single) variable, hom(X × Y, Z) ∼=

3.3. REFERENCES 7

hom(X(hom(Y,Z)) ) the category of sets is a closed monoidal category with the cartesian product ofsets as tensor product and the one-point set as tensor unit. In the isomorphisms λ : 1 × X ∼= X ∼=X × 1 : ρ natural in X, the left and right unitors are the projections p1 and p2 the ordered pairs (∗, x)and (x, ∗) respectively to the element x , where ∗ is the unique point in the one-point set.

A function on a connected set is locally constant if and only if it is constant.

3.3 References[1] Tanton, James (2005). Encyclopedia of Mathematics. Facts on File, New York. p. 94. ISBN 0-8160-5124-0.

[2] C.Clapham, J.Nicholson (2009). “Oxford Concise Dictionary of Mathematics, Constant Function” (PDF). Addison-Wesley. p. 175. Retrieved January 2014.

[3] Weisstein, Eric (1999). CRC Concise Encyclopedia of Mathematics. CRC Press, London. p. 313. ISBN 0-8493-9640-9.

[4] Dawkins, Paul (2007). “College Algebra”. Lamar University. p. 224. Retrieved January 2014.

[5] Carter, John A.; Cuevas, Gilbert J.; Holliday, Berchie; Marks, Daniel; McClure, Melissa S.publisher=Glencoe/McGraw-Hill School Pub Co (2005). “1”. Advanced Mathematical Concepts - Pre-calculus with Applications, Student Edition (1 ed.).p. 22. ISBN 978-0078682278.

[6] Dawkins, Paul (2007). “Derivative Proofs”. Lamar University. Retrieved January 2014.

[7] “Zero Derivative implies Constant Function”. Retrieved January 2014.

[8] Leinster, Tom (27 Jun 2011). “An informal introduction to topos theory”. Retrieved 11 November 2014.

• Herrlich, Horst and Strecker, George E., Category Theory, Heldermann Verlag (2007).

3.4 External links• Weisstein, Eric W., “Constant Function”, MathWorld.

• Constant function at PlanetMath.org.

Chapter 4

Constant term

In mathematics, a constant term is a term in an algebraic expression that has a value that is constant or cannot change,because it does not contain any modifiable variables. For example, in the quadratic polynomial

x2 + 2x+ 3,

the 3 is a constant term.After like terms are combined, an algebraic expression will have at most one constant term. Thus, it is common tospeak of the quadratic polynomial

ax2 + bx+ c,

where x is the variable, and has a constant term of c. If c = 0, then the constant term will not actually appear whenthe quadratic is written.It is notable that a term that is constant, with a constant as a multiplicative coefficient added to it (although thisexpression could be more simply written as their product), still constitutes a constant term as a variable is still notpresent in the new term. Although the expression is modified, the term (and coefficient) itself classifies as constant.However, should this introduced coefficient contain a variable, while the original number has a constant meaning, thishas no bearing if the new term stays constant as the introduced coefficient will always override the constant expression- for example, in (x + 1)(x − 2) when x is multiplied by 2, the result, 2x, is not constant; while 1 * −2 is −2 andstill a constant.Any polynomial written in standard form has a unique constant term, which can be considered a coefficient of x0. Inparticular, the constant term will always be the lowest degree term of the polynomial. This also applies to multivariatepolynomials. For example, the polynomial

x2 + 2xy + y2 − 2x+ 2y − 4

has a constant term of −4, which can be considered to be the coefficient of x0y0, where the variables are becomeeliminated by exponentiated to 0 (any number exponentiated to 0 becomes 1). For any polynomial, the constantterm can be obtained by substituting in 0 instead of each variable; thus, eliminating each variable. The concept ofexponentiation to 0 can be extended to power series and other types of series, for example in this power series:

a0 + a1x+ a2x2 + a3x

3 + · · · ,

a0 is the constant term. In general a constant term is one that does not involve any variables at all. However inexpressions that involve terms with other types of factors than constants and powers of variables, the notion of constantterm cannot be used in this sense, since that would lead to calling “4” the constant term of (x − 3)2 + 4 , whereassubstituting 0 for x in this polynomial makes it evaluate to 13.

8

4.1. SEE ALSO 9

4.1 See also• Constant (mathematics)

Chapter 5

Term (logic)

In analogy to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact, inmathematical logic, a term denotes a mathematical object and a formula denotes a mathematical fact. In particular,terms appear as components of a formula.A first-order term is recursively constructed from constant symbols, variables and function symbols. An expressionformed by applying a predicate symbol to an appropriate number of terms is called an atomic formula, which evaluatesto true or false in bivalent logics, given an interpretation. For example, (x+1)*(x+1) is a term built from the constant1, the variable x, and the binary function symbols + and *; it is part of the atomic formula (x+1)*(x+1) ≥ 0 whichevaluates to true for each real-numbered value of x.Besides in logic, terms play important roles in universal algebra, and rewriting systems.

5.1 Elementary mathematics

In the context of polynomials, sometimes term is used for a monomial with a coefficient: to 'collect like terms' in apolynomial is the operation of making it a linear combination of distinct monomials. Terms, in this sense, are thingsthat are added or subtracted. A series is often represented as the sum of a sequence of terms. Individual factors in anexpression representing a product are multiplicative terms. For example, in 6 + 3x − 2, 6, 3x, and −2 are all terms.In elementary mathematics,[1]

• each argument term of the addition operator + is called an addend,

• the first and second argument term of the subtraction operator - is called aminuend and subtrahend, respectively,

• each argument term of the multiplication operator ⋅ is called a factor, the first and second argument term isalso called multiplicand and multiplier, respectively,

• the first and second argument term of the division operator / is called dividend and divisor, respectively,

• if the division operator is written as fraction bar, the top and bottom terms are called numerator and denomi-nator, respectively.

5.2 Formal definition

Given a set V of variable symbols, a set C of constant symbols and sets Fn of n-ary function symbols, also calledoperator symbols, for each natural number n ≥ 1, the set of (unsorted first-order) terms T is recursively defined to bethe smallest set with the following properties:[2]

• every variable symbol is a term: V ⊆ T,

• every constant symbol is a term: C ⊆ T,

10

5.2. FORMAL DEFINITION 11

Tree structure of terms (n (n+1))/2 and n ((n+1)/2)

• from every n terms t1,...,tn, and every n-ary function symbol f ∈ Fn, a larger term f(t1, ..., tn) can be built.

Using an intuitive, pseudo-grammatical notation, this is sometimes written as: t ::= x | c | f(t1, ..., tn). Usually, onlythe first few function symbol sets Fn are inhabited. Well-known examples are the unary function symbols sin, cos ∈F1, and the binary function symbols +, −, ⋅, / ∈ F2, while ternary operations are less known, let alone higher-arityfunctions. Many authors consider constant symbols as 0-ary function symbols F0, thus needing no special syntacticclass for them.A term denotes a mathematical object from the domain of discourse. A constant c denotes a named object from thatdomain, a variable x ranges over the objects in that domain, and an n-ary function f maps n-tuples of objects to objects.For example, if n ∈ V is a variable symbol, 1 ∈ C is a constant symbol, and add ∈ F2 is a binary function symbol,then n ∈ T, 1 ∈ T, and (hence) add(n, 1) ∈ T by the first, second, and third term building rule, respectively. The latterterm is usually written as n+1, using infix notation and the more common operator symbol + for convenience.

5.2.1 Term structure vs. representation

Originally, logicians defined a term to be a character string adhering to certain building rules.[3] However, since theconcept of tree became popular in computer science, it turned out to be more convenient to think of a term as a tree.For example, several distinct character strings, like "(n⋅(n+1))/2”, "((n⋅(n+1)))/2”, and " n(n+1)

2 ", denote the sameterm and correspond to the same tree, viz. the left tree in the above picture. Separating the tree structure of a termfrom its graphical representation on paper, it is also easy to account for parentheses (being only representation, notstructure) and invisible multiplication operators (existing only in structure, not in representation).

5.2.2 Structural equality

Two terms are said to be structurally, literally, or syntactically equal if they correspond to the same tree. Forexample, the left and the right tree in the above picture are structurally unequal terms, although they might beconsidered "semantically equal" as they always evaluate to the same value in rational arithmetic. While structuralequality can be checked without any knowledge about the meaning of the symbols, semantic equality cannot. If thefunction / is e.g. interpreted not as rational but as truncating integer division, then at n=2 the left and right termevaluates to 3 and 2, respectively. Structural equal terms need to agree in their variable names.In contrast, a term t is called a renaming, or a variant, of a term u if the latter resulted from consistently renamingall variables of the former, i.e. if u = tσ for some renaming substitution σ. In that case, u is a renaming of t, too,since a renaming substitution σ has an inverse σ−1, and t = uσ−1. Both terms are then also said to be equal modulorenaming. In many contexts, the particular variable names in a term don't matter, e.g. the commutativity axiom foraddition can be stated as x+y=y+x or as a+b=b+a; in such cases the whole term may be replaced by a renamed term,while an arbitrary subterm usually may not, e.g. x+y=b+a is not a valid version of the commutativity axiom. [note 1][note 2]

12 CHAPTER 5. TERM (LOGIC)

5.2.3 Ground and linear terms

The set of variables of a term t is denoted by vars(t). A term that doesn't contain any variables is called a groundterm; a term that doesn't contain multiple occurrences of a variable is called a linear term. For example, 2+2 is aground term and hence also a linear term, x⋅(n+1) is a linear term, n⋅(n+1) is a non-linear term. These properties areimportant e.g. in term rewriting.Given a signature for the function symbols, the set of all terms forms the free term algebra. The set of all groundterms forms the initial term algebra.Abbreviating the number of constants as f0, and the number of i-ary function symbols as fi, the number θh ofdistinct ground terms of a height up to h can be computed by the following recursion formula:

• θ0 = f0, since a ground term of height 0 can only be a constant,

• θh+1 =∑∞

i=0 fi · θih , since a ground term of height up to h+1 can be obtained by composing any i groundterms of height up to h, using an i-ary root function symbol. The sum has a finite value if fi = 0 for all i beyonda maximal arity, which is usually the case.

5.2.4 Building formulas from terms

Given a set Rn of n-ary relation symbols for each natural number n ≥ 1, an (unsorted first-order) atomic formula isobtained by applying an n-ary relation symbol to n terms. As for function symbols, a relation symbol set Rn is usuallynon-empty only for small n. In mathematical logic, more complex formulas are built from atomic formulas usinglogical connectives and quantifiers. For example, letting ℝ denote the set of real numbers, ∀x: x ∈ ℝ ⇒ (x+1)⋅(x+1)≥ 0 is a mathematical formula evaluating to true in the algebra of complex numbers. An atomic formula is calledground if it is build entirely from ground terms; all ground atomic formulas composable from a given set of functionand predicate symbols make up the Herbrand universe for these symbol sets.

5.3 Operations with terms

Tree structure of black example term a∗((a+1)∗(a+2))1∗(2∗3) , with blue redex x*(y*z)

• Since a term has the structure of a tree hierarchy, to each of its nodes a position, or path, can be assigned,that is, a string of decimal numbers indicating the node’s place in the hierarchy. The empty string, commonly

5.4. RELATED CONCEPTS 13

denoted by ε, is assigned to the root node. Position strings within the black term are indicated in red in thepicture.

• At each position p of a term t, a unique subterm starts, which is commonly denoted by t|p. For example, atposition 122 of the black term in the picture, the subterm a+2 has its root. The relation “is a subterm of” is apartial order on the set of terms; it is reflexive since each term is trivially a subterm of itself.

• The term obtained by replacing in a term t the subterm at a position p by a new term u is commonly denotedby t[u]p. That term t[u]p can also be viewed as resulting from a generalized concatenation of the term u witha term-like object t[.]; the latter is called a context, or a term with a hole (indicated by "."; its position beingp), in which u is said to be embedded. For example, if t is the black term in the picture, then t[b+1]12 resultsin the term a∗(b+1)

1∗(2∗3) . The latter term also results from embedding the term b+1 into the context a∗( . )1∗(2∗3) . In

an informal sense, the operations of instantiating and embedding are converse to each other: while the formerappends function symbols at the bottom of the term, the latter appends them at the top. The encompassmentordering relates a term and any result of appends at either sides.

• To each node of a term, its depth (called height by some authors) can be assigned, i.e. its distance (numberof edges) from the root. In this setting, the depth of a node always equals the length of its position string. Inthe picture, depth levels in the black term are indicated in green.

• The size of a term commonly refers to the number of its nodes, or, equivalently, to the length of the term’swritten representation, counting symbols without parentheses. The black and the blue term in the picture hasthe size 15 and 5, respectively.

• A term u matches a term t, if an instance of u structurally equals a subterm of t, or formally, if uσ = t|p forsome position p in t and some substitution σ. In this case, u, t, and σ is called the pattern term, the subjectterm, and the matching substitution, respectively. In the picture, the blue pattern term x*(y*z) matches theblack subject term at position 1, with the matching substitution { x ↦ a, y ↦ a+1, z ↦ a+2 } indicated byblue variables immediately left to their black substitutes. Intuitively, the pattern, except for its variables, mustbe contained in the subject; if a variable occurs multiply in the pattern, equal subterms are required at therespective positions of the subject.

• unifying terms

• term rewriting

5.4 Related concepts

5.4.1 Sorted terms

Main article: Many-sorted logic

When the domain of discourse contains elements of basically different kinds, it is useful to split the set of all termsaccordingly. To this end, a sort (sometimes also called type) is assigned to each variable and each constant symbol,and a declaration [note 3] of domain sorts and range sort to each function symbol. A sorted term f(t1,...,tn) may becomposed from sorted subterms t1,...,tn only if the ith subterm’s sort matches the declared ith domain sort of f. Sucha term is also called well-sorted; any other term (i.e. obeying the unsorted rules only) is called ill-sorted.For example, a vector space comes with an associated field of scalar numbers. LetW and N denote the sort of vectorsand numbers, respectively, let VW and VN be the set of vector and number variables, respectively, and CW and CNthe set of vector and number constants, respectively. Then e.g. 0⃗ ∈ CW and 0 ∈ CN, and the vector addition, thescalar multiplication, and the inner product is declared as +:W×W→W, *:W×N→W, and ⟨.,.⟩:W×W→N, respectively.Assuming variable symbols v⃗, w⃗ ∈ VW and a,b ∈ VN, the term ⟨(v⃗+ 0⃗) ∗ a, w⃗ ∗ b⟩ is well-sorted, while v⃗+ a is not(since + doesn't accept a term of sort N as 2nd argument). In order to make a ∗ v⃗ a well-sorted term, an additionaldeclaration *:N×W→W is required. Function symbols having several declarations are called overloaded.See many-sorted logic for more information, including extensions of themany-sorted framework described here.

14 CHAPTER 5. TERM (LOGIC)

5.4.2 Lambda terms

Main article: Lambda term

Motivation

Mathematical notations as shown in the table do not fit into the scheme of a first-order term as defined above, as theyall introduce an own local, or bound, variable that may not appear outside the notation’s scope, e.g. t ·

∫ b

asin(k · t) dt

doesn't make sense. In contrast, the other variables, referred to as free, behave like ordinary first-order term variables,e.g. k ·

∫ b

asin(k · t) dt does make sense.

All these operators can be viewed as taking a function rather than a value term as one of their arguments. For example,the lim operator is applied to a sequence, i.e. to a mapping from positive integer to e.g. real numbers. As anotherexample, a C function to implement the second example from the table, ∑, would have a function pointer argument(see box below).Lambda terms can be used to denote anonymous functions to be supplied as arguments to lim, ∑, ∫, etc.For example, the function square from the C program below can be written anonymously as a lambda term λi. i2. Thegeneral sum operator ∑ can then be considered as a ternary function symbol taking a lower bound value, an upperbound value and a function to be summed-up. Due to its latter argument, the ∑ operator is called a second-orderfunction symbol. As another example, the lambda term λn. x/n denotes a function that maps 1, 2, 3, ... to x/1, x/2,x/3, ..., respectively, that is, it denotes the sequence (x/1, x/2, x/3, ...). The lim operator takes such a sequence andreturns its limit (if defined).The rightmost column of the table indicates how each mathematical notation example can be represented by a lambdaterm, also converting common infix operators into prefix form.int sum(int lwb, int upb, int fct(int)) { // implements general sum operator int res = 0; for (int i=lwb; i<=upb; ++i)res += fct(i); return res; } int square(int i) { return i*i; } // implements anonymous function (lambda i. i*i); however,C requires a name for it #include <stdio.h> int main(void) { int n; scanf(" %d”,&n); printf("%d\n”, sum(1,n,square)); // applies sum operator to sum up squares return 0; }

5.5 See also

• Equation

• Expression (mathematics)

5.6 Notes[1] Strictly speaking, x+y=y+x is an atomic formula, not a term, since = is a predicate, not a function symbol. However, since

atomic formulas can be viewed as trees, too, and renaming is essentially a concept on trees, atomic (and, more generally,quantifier-free) formulas can be renamed in a similar way as terms. In fact, some authors consider a quantifier-free formulaas a term (of type bool rather than e.g. int, cf. #Sorted terms below).

[2] Renaming of the commutativity axiom can be viewed as alpha-conversion on the universal closure of the axiom: "x+y=y+x"actually means "∀x,y: x+y=y+x", which is synonymous to "∀a,b: a+b=b+a"; see also #Lambda terms below.

[3] called “symbol type” in the Signature (logic)#Many-sorted signatures article

5.7 References

• Franz Baader; Tobias Nipkow (1999). Term Rewriting and All That. Cambridge University Press. pp. 1–2 and34–35. ISBN 978-0-521-77920-3.

5.7. REFERENCES 15

[1] Schwartzman, Steven (1994). The words of mathematics: An etymological dictionary of mathematical terms used in English.The Mathematical Association of America. p. 219. ISBN 0-88385-511-9.

[2] C.C. Chang; H. Jerome Keisler (1977). Model Theory. Studies in Logic and the Foundation of Mathematics 73. NorthHolland.; here: Sect.1.3

[3] Hermes, Hans (1973). Introduction to Mathematical Logic. Springer London. ISBN 3540058192. ISSN 1431-4657.; here:Sect.II.1.3

Chapter 6

Value (mathematics)

In mathematics, value may refer to several, strongly related notions:

• The value of a variable or a constant is any number or other mathematical object assigned to it.

• The value of a mathematical expression is the result of the computation described by this expression when thevariables and constants in it are replaced by some numbers.

• The value of a function is the number implied by the function as a result of a particular number being assignedto its argument (also called the variable of the function).[1][2]

For example, if the function f is defined by f(x) = 2x2 − 3x + 1 , then, given the value 3 to the variable x yieldsthe function value 10 (since indeed 2 · 32 – 3 · 3 + 1 = 10). This is denoted f(3) = 10.

6.1 See also• Value function

• Value (computer science)

• Absolute value

• Truth value

6.2 References[1] http://mathworld.wolfram.com/Value.html

[2] Meschkowski, Herbert (1968). Introduction to Modern Mathematics. George G. Harrap & Co. Ltd. p. 32. ISBN0245591095.

16

Chapter 7

Variable (mathematics)

For variables in computer science, see Variable (computer science). For other uses, see Variable (disambiguation).

In elementary mathematics, a variable is an alphabetic character representing a number, called the value of thevariable, which is either arbitrary or not fully specified or unknown. Making algebraic computations with variables asif they were explicit numbers allows one to solve a range of problems in a single computation. A typical example isthe quadratic formula, which allows one to solve every quadratic equation by simply substituting the numeric valuesof the coefficients of the given equation to the variables that represent them.The concept of variable is also fundamental in calculus. Typically, a function y = f(x) involves two variables, y andx, representing respectively the value and the argument of the function. The term “variable” comes from the fact that,when the argument (also called the “variable of the function”) varies, then the value varies accordingly.[1]

In more advanced mathematics, a variable is a symbol that denotes a mathematical object, which could be a number,a vector, a matrix, or even a function. In this case, the original property of “variability” of a variable is not kept(except, sometimes, for informal explanations).Similarly, in computer science, a variable is a name (commonly an alphabetic character or a word) representingsome value represented in computer memory. In mathematical logic, a variable is either a symbol representing anunspecified term of the theory, or a basic object of the theory, which is manipulated without referring to its possibleintuitive interpretation.

7.1 Etymology

“Variable” comes from a Latin word, variābilis, with "vari(us)"' meaning “various” and "-ābilis"' meaning "-able”,meaning “capable of changing”.[2]

7.2 Genesis and evolution of the concept

François Viète introduced at the end of 16th century the idea of representing known and unknown numbers by letters,nowadays called variables, and of computing with them as if they were numbers, in order to obtain, at the end, theresult by a simple replacement. François Viète's convention was to use consonants for known values and vowels forunknowns.[3]

In 1637, René Descartes “invented the convention of representing unknowns in equations by x, y, and z, and knownsby a, b, and c".[4] Contrarily to Viète’s convention, Descartes’ one is still commonly in use.Starting in the 1660s, Isaac Newton and Gottfried Wilhelm Leibniz independently developed the infinitesimal calcu-lus, which essentially consists of studying how an infinitesimal variation of a variable quantity induces a correspondingvariation of another quantity which is a function of the first variable (quantity). Almost a century later Leonhard Eu-ler fixed the terminology of infinitesimal calculus and introduced the notation y = f(x) for a function f, its variablex and its value y. Until the end of the 19th century, the word variable referred almost exclusively to the argumentsand the values of functions.

17

18 CHAPTER 7. VARIABLE (MATHEMATICS)

In the second half of the 19th century, it appeared that the foundation of infinitesimal calculus was not formalizedenough to deal with apparent paradoxes such as a continuous function which is nowhere differentiable. To solve thisproblem, Karl Weierstrass introduced a new formalism consisting of replacing the intuitive notion of limit by a formaldefinition. The older notion of limit was “when the variable x varies and tends toward a, then f(x) tends toward L",without any accurate definition of “tends”. Weierstrass replaced this sentence by the formula

(∀ϵ > 0)(∃η > 0)(∀x) |x− a| < η ⇒ |L− f(x)| < ϵ,

in which none of the five variables is considered as varying.This static formulation led to the modern notion of variable which is simply a symbol representing a mathematicalobject which either is unknown or may be replaced by any element of a given set; for example, the set of real numbers.

7.3 Specific kinds of variables

It is common that many variables appear in the same mathematical formula, which play different roles. Some namesor qualifiers have been introduced to distinguish them. For example, in the general cubic equation

ax3 + bx2 + cx+ d = 0,

there are five variables. Four of them, a, b, c, d represent given numbers, and the last one, x, represents the unknownnumber, which is a solution of the equation. To distinguish them, the variable x is called an unknown, and the othervariables are called parameters or coefficients, or sometimes constants, although this last terminology is incorrect foran equation and should be reserved for the function defined by the left-hand side of this equation.In the context of functions, the term variable refers commonly to the arguments of the functions. This is typicallythe case in sentences like "function of a real variable", "x is the variable of the function f: x↦ f(x)", "f is a functionof the variable x" (meaning that the argument of the function is referred to by the variable x).In the same context, the variables that are independent of x define constant functions and are therefore called constant.For example, a constant of integration is an arbitrary constant function that is added to a particular antiderivativeto obtain the other antiderivatives. Because the strong relationship between polynomials and polynomial function,the term “constant” is often used to denote the coefficients of a polynomial, which are constant functions of theindeterminates.This use of “constant” as an abbreviation of “constant function” must be distinguished from the normal meaning ofthe word in mathematics. A constant, or mathematical constant is a well and unambiguously defined number orother mathematical object, as, for example, the numbers 0, 1, π and the identity element of a group.Here are other specific names for variables.

• A unknown is a variable in which an equation has to be solved for.

• An indeterminate is a symbol, commonly called variable, that appears in a polynomial or a formal powerseries. Formally speaking, an indeterminate is not a variable, but a constant in the polynomial ring of the ringof formal power series. However, because of the strong relationship between polynomials or power series andthe functions that they define, many authors consider indeterminates as a special kind of variables.

• A parameter is a quantity (usually a number) which is a part of the input of a problem, and remains constantduring the whole solution of this problem. For example, in mechanics the mass and the size of a solid bodyare parameters for the study of its movement. It should be noted that in computer science, parameter has adifferent meaning and denotes an argument of a function.

• Free variables and bound variables

• A random variable is a kind of variable that is used in probability theory and its applications.

It should be emphasized that all these denominations of variables are of semantic nature and that the way of computingwith them (syntax) is the same for all.

7.4. NOTATION 19

7.3.1 Dependent and independent variables

Main article: Dependent and independent variables

In calculus and its application to physics and other sciences, it is rather common to consider a variable, say y, whosepossible values depend of the value of another variable, say x. In mathematical terms, the dependent variable yrepresents the value of a function of x. To simplify formulas, it is often useful to use the same symbol for thedependent variable y and the function mapping x onto y. For example, the state of a physical system depends onmeasurable quantities such as the pressure, the temperature, the spatial position, ..., and all these quantities varieswhen the system evolves, that is, they are function of the time. In the formulas describing the system, these quantitiesare represented by variables which are dependent on the time, and thus considered implicitly as functions of the time.Therefore, in a formula, a dependent variable is a variable that is implicitly a function of another (or several other)variables. An independent variable is a variable that is not dependent.[5]

The property of a variable to be dependent or independent depends often of the point of view and is not intrinsic. Forexample, in the notation f(x, y, z), the three variables may be all independent and the notation represents a functionof three variables. On the other hand, if y and z depend on x (are dependent variables) then the notation represent afunction of the single independent variable x.[6]

7.3.2 Examples

If one defines a function f from the real numbers to the real numbers by

f(x) = x2 + sin(x+ 4)

then x is a variable standing for the argument of the function being defined, which can be any real number. In theidentity

n∑i=1

i =n2 + n

2

the variable i is a summation variable which designates in turn each of the integers 1, 2, ..., n (it is also called indexbecause its variation is over a discrete set of values) while n is a parameter (it does not vary within the formula).In the theory of polynomials, a polynomial of degree 2 is generally denoted as ax2 + bx + c, where a, b and c are calledcoefficients (they are assumed to be fixed, i.e., parameters of the problem considered) while x is called a variable.When studying this polynomial for its polynomial function this x stands for the function argument. When studyingthe polynomial as an object in itself, x is taken to be an indeterminate, and would often be written with a capital letterinstead to indicate this status.

7.4 Notation

In mathematics, the variables are generally denoted by a single letter. However, this letter is frequently followed bya subscript, as in x2, and this subscript may be a number, another variable (xi), a word or the abbreviation of a word(xᵢ and xₒᵤ ), and even a mathematical expression. Under the influence of computer science, one may encounter inpure mathematics some variable names consisting in several letters and digits.Following the 17th century French philosopher and mathematician, René Descartes, letters at the beginning of thealphabet, e.g. a, b, c are commonly used for known values and parameters, and letters at the end of the alphabet, e.g.x, y, z, and t are commonly used for unknowns and variables of functions.[7] In printed mathematics, the norm is toset variables and constants in an italic typeface.[8]

For example, a general quadratic function is conventionally written as:

ax2 + bx+ c ,

20 CHAPTER 7. VARIABLE (MATHEMATICS)

where a, b and c are parameters (also called constants, because they are constant functions), while x is the variable ofthe function. A more explicit way to denote this function is

x 7→ ax2 + bx+ c ,

which makes the function-argument status of x clear, and thereby implicitly the constant status of a, b and c. Since coccurs in a term that is a constant function of x, it is called the constant term.[9]:18

Specific branches and applications of mathematics usually have specific naming conventions for variables. Variableswith similar roles or meanings are often assigned consecutive letters. For example, the three axes in 3D coordinatespace are conventionally called x, y, and z. In physics, the names of variables are largely determined by the physicalquantity they describe, but various naming conventions exist. A convention often followed in probability and statisticsis to use X, Y, Z for the names of random variables, keeping x, y, z for variables representing corresponding actualvalues.There are many other notational usages. Usually, variables that play a similar role are represented by consecutiveletters or by the same letter with different subscript. Below are some of the most common usages.

• a, b, c, and d (sometimes extended to e and f) often represent parameters or coefficients.

• a0, a1, a2, ... play a similar role, when otherwise too many different letters would be needed.

• ai or ui is often used to denote the i-th term of a sequence or the i-th coefficient of a series.

• f and g (sometimes h) commonly denote functions.

• i, j, and k (sometimes l or h) are often used to denote varying integers or indices in an indexed family.

• l and w are often used to represent the length and width of a figure.

• l is also used to denote a line. In number theory, l often denotes a prime number not equal to p.

• n usually denotes a fixed integer, such as a count of objects or the degree of an equation.

• When two integers are needed, for example for the dimensions of a matrix, one uses commonly m and n.

• p often denotes a prime numbers or a probability.

• q often denotes a prime power or a quotient

• r often denotes a remainder.

• t often denotes time.

• x, y and z usually denote the three Cartesian coordinates of a point in Euclidean geometry. By extension, theyare used to name the corresponding axes.

• z typically denotes a complex number, or, in statistics, a normal random variable.

• α, β, γ, θ and φ commonly denote angle measures.

• ε usually represents an arbitrarily small positive number.

• ε and δ commonly denote two small positives.

• λ is used for eigenvalues.

• σ often denotes a sum, or, in statistics, the standard deviation.

7.5. SEE ALSO 21

7.5 See also• Free variables and bound variables (Bound variables are also known as dummy variables)

• Variable (programming)

• Mathematical expression

• Physical constant

• Coefficient

• Constant of integration

• Constant term of a polynomial

• Indeterminate (variable)

• Lambda calculus

7.6 Bibliography• J. Edwards (1892). Differential Calculus. London: MacMillan and Co. pp. 1 ff.

• Karl Menger, “On Variables in Mathematics and in Natural Science”, The British Journal for the Philosophy ofScience 5:18:134-142 (August 1954) JSTOR 685170

• Jaroslav Peregrin, “Variables in Natural Language: Where do they come from?", in M. Boettner, W. Thümmel,eds., Variable-Free Semantics, 2000, p. 46-65.

• W. V. Quine, “Variables Explained Away”, Proceedings of the American Philosophical Society 104:343-347(1960).

7.7 References[1] Syracuse University. “Appendix One Review of Constants and Variables”. cstl.syr.edu.

[2] ""Variable” Origin”. dictionary.com. Retrieved 18 May 2015.

[3] Fraleigh, John B. (1989). A First Course in Abstract Algebra (4 ed.). United States: Addison-Wesley. p. 276. ISBN0-201-52821-5.

[4] Tom Sorell, Descartes: A Very Short Introduction, (2000). New York: Oxford University Press. p. 19.

[5] Edwards Art. 5

[6] Edwards Art. 6

[7] Edwards Art. 4

[8] William L. Hosch (editor), The Britannica Guide to Algebra and Trigonometry, Britannica Educational Publishing, TheRosen Publishing Group, 2010, ISBN 1615302190, 9781615302192, page 71

[9] Foerster, Paul A. (2006). Algebra and Trigonometry: Functions and Applications, Teacher’s Edition (Classics ed.). UpperSaddle River, NJ: Prentice Hall. ISBN 0-13-165711-9.

22 CHAPTER 7. VARIABLE (MATHEMATICS)

7.8 Text and image sources, contributors, and licenses

7.8.1 Text• Argument of a function Source: https://en.wikipedia.org/wiki/Argument_of_a_function?oldid=666766979 Contributors: Gandalf61,

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• File:Question_book-new.svg Source: https://upload.wikimedia.org/wikipedia/en/9/99/Question_book-new.svg License: Cc-by-sa-3.0Contributors:Created from scratch in Adobe Illustrator. Based on Image:Question book.png created by User:Equazcion Original artist:Tkgd2007

• File:Tree_structure_of_mathematical_first-order_terms_svg.svg Source: https://upload.wikimedia.org/wikipedia/commons/a/ac/Tree_structure_of_mathematical_first-order_terms_svg.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: JochenBurghardt

7.8. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 23

• File:Wiki_constant_function_175_200.png Source: https://upload.wikimedia.org/wikipedia/commons/d/d3/Wiki_constant_function_175_200.png License: CC BY-SA 3.0 Contributors: Own work Original artist: Lfahlberg

7.8.3 Content license• Creative Commons Attribution-Share Alike 3.0