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In mathematics, value may refer to several, strongly related notions. Though in general, a mathematical value is a broad term that refers to any definite entity that can be manipulated with operators according to the well-defined rules of its mathematical system.

Certain values can correspond to the real world, although most values in mathematics generally exists purely as abstract objects with no connection to the real world.

Numerical values

[edit]

Numbers (specifically the reals) are values that represent quantities. In that sense, numerical values are values that comprises or are made up of said numbers. In more simpler terms, a numerical value are represented by numbers. Both numbers and numerical values tend to be synonymous and interchangeable with each other.^[1]

The following table shows certain values that are considered numerical values themselves.

Value	Brief description
Digit value	Digit value of a place of a number would simply be its digit or numeral.
Place value	The contribution of a digit to the value of a number is the value of the digit multiplied by a factor of 10 raised to the power of the digit's position.
Ratio	How many times one number contains another.
Rates	The quotient of two quantites.
Percentage	A number or ratio expressed as a fraction of 100
Central tendencies	A typical value for a probability distribution.

Because numerical values can also be a part of composite objects, various terminologies are given. For example, a complex number $z=a+bi$ , has $a$ as considered its real value, likewise $b$ as its complex value.

Variables

[edit]

A variable is a symbol that represents an unspecified object. Homogeneous to numbers, variables themselves are considered as values.

Functions

[edit]

The value of a function, given the value(s) assigned to its argument(s), is the quantity assumed by the function for these argument values.^[2]^[3]

For example, if the function $f$ is defined by $f$ ( $x$ ) = 2 $x$ ² − 3 $x$ + 1, then assigning the value 3 to its argument $x$ yields the function value 10, since $f$ (3) = 2·3² − 3·3 + 1 = 10.

If the variable, expression or function only assumes real values, it is called real-valued. Likewise, a complex-valued variable, expression or function only assumes complex values.

References

[edit]

^ Collins, Joseph Victor (1893). Text-book of Algebra: Through Quadratic Equations. Albert, Scott & Company. p. 64.
^ "Value".
^ Meschkowski, Herbert (1968). Introduction to Modern Mathematics. George G. Harrap & Co. Ltd. p. 32. ISBN 0245591095.

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Value (mathematics)

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In mathematics, a value refers to a specific numerical quantity or result obtained from evaluating an expression, function, or operation, representing the concrete outcome of a mathematical process.^[1] This concept is foundational across various branches of mathematics, where it denotes the worth or magnitude associated with numbers, variables, or symbols after computation.^[2] One of the most common applications is the value of an algebraic expression, which is determined by substituting specific numbers for variables and performing the indicated operations; for instance, evaluating $ f(x) = 2x + 3 $ at $ x = 4 $ yields the value 11.^[3] In number theory and analysis, absolute value (or modulus) measures the distance of a number from zero on the real line, always yielding a non-negative result, such as $ | -5 | = 5 $, and extends to complex numbers and more abstract fields via valuations.^[4] Similarly, place value in positional numeral systems assigns worth to digits based on their position relative to the base, enabling the representation of large numbers; for example, in the decimal system, the digit 7 in 732 has a place value of 700.^[5] Beyond these, the term encompasses specialized notions like eigenvalues in linear algebra, which are scalar values $ \lambda $ satisfying $ Av = \lambda v $ for a matrix $ A $ and nonzero vector $ v $, crucial for understanding transformations and stability in systems. In probability and statistics, expected value quantifies the long-run average outcome of a random variable, calculated as $ E(X) = \sum x_i p_i $ for discrete cases.^[6] These diverse uses highlight how "value" serves as a versatile descriptor for quantifiable aspects in mathematical structures, from elementary arithmetic to advanced theoretical frameworks.

Fundamental Concepts

Definition and Scope

In mathematics, a value refers to a specific numerical quantity or result obtained from evaluating an expression, function, or operation, such as a number or output from computations within a formal mathematical system.^[2] This conception encompasses outputs from calculations or derivations, distinguishing values as definite entities within mathematical discourse.^[7] Values can be categorized as concrete or abstract. Concrete values are specific instances, such as the numerical approximation π ≈ 3.14159, representing a fixed quantity derived from definitions. In contrast, abstract values involve symbolic representations, like the rule-bound outcome of an algebraic expression, emphasizing form.^[7] Historically, the term "value" in mathematics originated in early arithmetic contexts, denoting the magnitude or quantity of an entity. This usage evolved from ancient views of mathematics as the study of continuous magnitudes, transitioning in the 16th–19th centuries through symbolic algebra—pioneered by figures like François Viète and Augustin-Louis Cauchy—to encompass abstract structures in modern fields like algebra, where values are elements satisfying axiomatic relations rather than mere quantities.^[7] The scope of values extends across arithmetic, algebra, analysis, and other branches, unifying applications from numerical computations to theoretical constructs, while excluding non-mathematical interpretations such as economic worth. For instance, numerical values provide concrete exemplars in basic calculations, and function values illustrate outputs from mappings, both fitting within this framework.^[2]

Numerical Values

Numerical values in mathematics refer to fixed quantities within established number systems, serving as the foundational building blocks for arithmetic and more advanced computations. These include natural numbers (positive integers starting from 1 in many educational contexts, though sometimes including 0 in advanced mathematics), integers (whole numbers including negatives and zero), rational numbers (fractions of integers), real numbers (all points on the number line, encompassing irrationals), and complex numbers (extending reals with imaginary units).^[8]^[9] In positional numeral systems, such as the common base-10 decimal system, numerical values are represented through place value, where the position of each digit determines its contribution to the total. For instance, the number 123 is calculated as $ 1 \times 10^2 + 2 \times 10^1 + 3 \times 10^0 $, emphasizing how digit values scale by powers of the base. This system allows efficient encoding of large quantities using a limited set of symbols (digits 0-9), with the rightmost digit representing units and positions shifting leftward for higher powers.^[10]^[11] Numerical values extend beyond basic integers to include derived forms like ratios, rates, percentages, and measures of central tendency. A ratio, such as 3:4, compares two quantities of the same unit, equivalent to the fraction $ \frac{3}{4} $. Rates involve different units, like 60 miles per hour (60 mph), quantifying change over a dimension such as time. Percentages express proportions per hundred, where 25% equals 0.25 or $ \frac{25}{100} $. In statistics, central tendencies summarize datasets: the mean (arithmetic average), median (middle value in ordered data), and mode (most frequent value) provide representative numerical values for typicality.^[12]^[13]^[14] Complex numbers incorporate numerical values through their real and imaginary parts, expressed as $ z = a + bi $, where $ a $ and $ b $ are real numbers, and $ i $ is the imaginary unit satisfying $ i^2 = -1 $. Here, $ a $ denotes the real part (projection on the real axis), and $ b $ the imaginary part (scaled by $ i $), enabling solutions to equations like $ x^2 + 1 = 0 $. This structure preserves arithmetic properties while extending the real number system.^[15]^[16] Prominent examples of irrational numerical values include $ \pi \approx 3.14159 $, the ratio of a circle's circumference to its diameter, and $ e \approx 2.71828 $, the base of the natural logarithm, arising in exponential growth and calculus. These constants, while approximable, are exact fixed quantities essential for geometric and analytic contexts.

Variables and Expressions

Variables

In mathematics, a variable is a symbol, such as

x

y

, that serves as a placeholder for an unspecified element from a specified domain, most commonly the real numbers, but extensible to other mathematical objects like complex numbers, vectors, sets, or functions.^[17]^[18] This notation facilitates the representation of general relationships without committing to particular values, allowing mathematicians to explore properties that hold across a range of possibilities.^[19] Variables play a crucial role in abstracting and generalizing mathematical statements, particularly in algebra and logic, where they enable the formulation of equations and logical expressions that can be solved or analyzed parametrically. For instance, in the linear equation

ax + b = 0

, the variable

x

represents the unknown quantity to be solved for, while

a

and

b

may act as parameters or constants.^[20] In formal logic and higher mathematics, variables are distinguished as free or bound: free variables are those that remain unbound by quantifiers or operators and can take arbitrary values from their domain, whereas bound variables are temporarily fixed within the scope of a quantifier (e.g.,

\forall x

) or an operation like summation, effectively serving as local placeholders without affecting the overall expression's value.^[21]^[22] Variables acquire specific values through assignment or substitution, a process where a concrete element from the domain replaces the symbol, such as declaring

x = 5

to evaluate an expression containing

x

.^[23] Certain types of variables arise in specific contexts; for example, in functional relations, an independent variable is one whose value can be freely chosen (e.g., input to a function), while a dependent variable's value is determined by the independent one (e.g., output).^[24]^[25] Additionally, dummy variables appear in indefinite integrals or summations as non-referential placeholders that do not persist in the result, as in the integral

\int f(x) \, dx = F(x) + C,

where

x

is dummy and can be replaced by any other symbol without altering the antiderivative

F

.^[26] The concept of variables as symbolic unknowns emerged in the 17th century, with René Descartes pioneering their systematic use in his 1637 treatise La Géométrie, where he employed letters like

x

y

, and

z

to denote line segments of unknown lengths, distinguishing them from known quantities represented by earlier letters.^[27]^[28] This innovation bridged algebra and geometry, laying the foundation for analytic geometry and modern symbolic mathematics. Numerical values often serve as the concrete assignments that instantiate these variables in practical computations.

Expression Evaluation

Expression evaluation in mathematics refers to the process of determining the numerical value of an algebraic or arithmetic expression by substituting specific values for variables and then applying the appropriate operations to simplify it to a single output value.^[29] This process transforms a symbolic expression into a concrete number, enabling practical computations in various mathematical contexts.^[30] A key aspect of expression evaluation is adherence to the order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division—from left to right—and Addition and Subtraction—from left to right) in the United States, or BODMAS (Brackets, Orders/Of, Division and Multiplication—from left to right—and Addition and Subtraction—from left to right) in other regions.^[31]^[32] These conventions ensure consistent results by prioritizing operations: first resolving any enclosed groupings like parentheses, then handling exponentiation, followed by multiplicative and divisive operations in sequence, and finally additive and subtractive ones.^[33] To illustrate, consider the expression

2x + 3

where the variable

x

is assigned the value 4. Substituting yields

2(4) + 3

, which evaluates to

8 + 3 = 11

by first performing the multiplication and then the addition, in line with the order of operations.^[34] Variables serve as placeholders that provide the necessary inputs for this substitution step.^[35] Expressions can be arithmetic, yielding purely numerical results upon full evaluation, or algebraic, where symbolic simplification may precede substitution to simplify the form.^[36] For instance, the algebraic identity

(x + y)(x - y) = x^2 - y^2

allows simplification before assigning values to

x

and

y

, reducing computational complexity.^[37] Such simplifications often involve combining like terms or applying known identities to produce an equivalent, more manageable expression.^[38] However, not all expressions can be fully evaluated to a finite numerical value; certain forms remain unevaluated due to inherent indeterminacy or infinite processes. For example, limits of functions as a variable approaches a specific value may result in indeterminate forms like $ \frac{0}{0} $ or $ \infty - \infty $, requiring advanced techniques rather than direct substitution.^[39] Similarly, indefinite integrals represent families of functions and cannot be reduced to a single numerical value without specified bounds, while some definite integrals, such as improper ones extending to infinity, may diverge and lack a finite evaluation.^[40] These limitations highlight that complete numerical evaluation is not always possible or meaningful in higher mathematics.^[41] The resulting numerical values from successful evaluations form the basis for further quantitative analysis in applied contexts.^[29]

Functions

Function Values

The value of a function

f

, denoted

f(x)

, is the unique output in the codomain assigned to an input

x

from the domain under the function's mapping, ensuring each input corresponds to exactly one output.^[42] This concept underpins the relational structure of functions, distinguishing them from general relations by the single-valued property.^[43] Function notation typically expresses this as

f(x) =

a defining formula, facilitating computation by substitution. For instance, given

f(x) = 2x^2 - 3x + 1

, the value

f(3)

is obtained by replacing

x

with 3:

f(3) = 2(3)^2 - 3(3) + 1 = 10

.^[44] Evaluation involves direct substitution for simple algebraic forms, but for piecewise-defined functions—which apply distinct formulas across subintervals of the domain—the appropriate expression is selected based on the input's location.^[43] For example, a piecewise function might define

f(x) = x^2

x < 0

and

f(x) = x

x \geq 0

, yielding

f(-1) = 1

and

f(1) = 1

.^[45] Functions are categorized by their output types: real-valued functions map to real numbers, suitable for modeling physical quantities, while complex-valued functions map to complex numbers, essential in fields like signal processing.^[46]^[47] Constant functions represent a special case, producing the same fixed output for all domain inputs, independent of the input variation.^[48] In applications, function values quantify real-world phenomena; for example, in population modeling, an exponential growth function

f(t) = P_0 e^{kt}

provides the population size

f(t)

at time

t

, where

P_0

is the initial population and

k > 0

is the growth rate.^[49]

Domain and Codomain

In mathematics, the domain of a function

f: A \to B

is the set

A

consisting of all possible input values for which the function is defined and produces a valid output.^[50] This set specifies the allowable elements that can be mapped by the function, ensuring that the rule defining

f

applies without encountering undefined operations, such as division by zero or taking the square root of a negative number.^[51] The codomain of the function is the set

B

, which represents the target collection of potential output values, though not all elements in

B

need to be achieved. Unlike the domain, the codomain is part of the function's specification and may be larger than the actual outputs produced; it provides a broader context for where function values are expected to lie.^[52] The range, also known as the image of the function, is the subset of the codomain that consists precisely of the output values attained by applying

f

to elements of the domain.^[53] For instance, if the codomain is the set of real numbers

\mathbb{R}

, the range is the specific collection of reals that

f

outputs, which must be contained within

\mathbb{R}

.^[54] Domains often arise with natural restrictions based on the function's formula to avoid invalid expressions. For the square root function

f(x) = \sqrt{x}

, the natural domain is all non-negative real numbers

\{x \in \mathbb{R} \mid x \geq 0\}

, as the square root is undefined for negative inputs in the real numbers.^[55] Similarly, for the rational function

f(x) = 1/x

, the domain excludes zero to prevent division by zero, yielding

\{x \in \mathbb{R} \mid x \neq 0\}

.^[51] These restrictions define the maximal set where the function operates without issue, though a function may be explicitly defined on a smaller subset of this natural domain.^[50] Examples illustrate these concepts clearly. Consider a constant function

f(x) = c

where

c

is a fixed real number; its domain can be all real numbers

\mathbb{R}

, while the codomain might be

\mathbb{R}

or the singleton set

\{c\}

, with the range being exactly

\{c\}

.^[56] For the squaring function

f(x) = x^2

with domain

\mathbb{R}

and codomain

\mathbb{R}

, the range is the non-negative reals

[0, \infty)

, as outputs are always non-negative.^[54] The identity function

id: \mathbb{R} \to \mathbb{R}

defined by

id(x) = x

has domain and codomain both equal to

\mathbb{R}

, with range also

\mathbb{R}

, mapping each input directly to itself.^[57]

Specialized Types

Absolute Value

The absolute value of a real number

x

, denoted

|x|

, is defined as the non-negative distance from

x

to 0 on the real number line. Formally, it is given by the piecewise function

|x| = \begin{cases} x & \text{if } x \geq 0, \\ -x & \text{if } x < 0. \end{cases}

This ensures

|x| \geq 0

for all real

x

, with equality holding if and only if

x =

.^[58] Geometrically, the absolute value measures the magnitude of

x

without regard to direction or sign, making it a fundamental tool for quantifying distances in one dimension./01%3A_Functions/1.06%3A_Absolute_Value_Functions) Key properties of the absolute value for real numbers

x

and

y

include multiplicativity,

|xy| = |x||y|

, which preserves the product of magnitudes; the triangle inequality,

|x + y| \leq |x| + |y|

, stating that the magnitude of a sum is at most the sum of magnitudes; and non-negativity,

|x| \geq 0

. These properties establish the absolute value as a norm on the real numbers, enabling proofs of inequalities and bounds in analysis. Equality in the triangle inequality holds when

x

and

y

have the same sign (or one is zero), reflecting alignment in direction.^[59] The absolute value extends naturally to complex numbers, where for

z = a + bi

with real parts

a

and

b

, the absolute value—also termed the modulus—is

|z| = \sqrt{a^2 + b^2}

. This formula computes the Euclidean distance from

z

to the origin in the complex plane, yielding a non-negative real number.^[60] The modulus inherits analogous properties, such as

|zw| = |z||w|

and the triangle inequality

|z + w| \leq |z| + |w|

for complex

z

and

w

.^[61] In complex analysis, the notation

|z|

specifically denotes the modulus, distinguishing it from the real case while sharing the same interpretive role as magnitude. Applications of absolute value span multiple fields: it defines norms in vector spaces, where the modulus generalizes to the Euclidean norm

\|\mathbf{v}\| = \sqrt{\sum v_i^2}

for vectors

\mathbf{v}

; in numerical analysis, error bounds like

|\tilde{x} - x| < \epsilon

quantify approximation accuracy, ensuring computed values lie within

\epsilon

of the true value

x

.^[62]^[63] Historically, the concept of absolute value emerged in the 17th century amid advancements in algebra, particularly for denoting the positive magnitude in solutions to equations like

x^2 = a

, where

|x| = \sqrt{a}

. The modern vertical bar notation

| \cdot |

was introduced by Karl Weierstrass in 1841 to simplify expressions involving magnitudes, especially in complex analysis.^[64]

Eigenvalue

In linear algebra, an eigenvalue of a square matrix

A

is a scalar

\lambda

such that there exists a non-zero vector

v

, called an eigenvector, satisfying the equation

A \mathbf{v} = \lambda \mathbf{v}

. This represents a non-trivial solution to the homogeneous system

(A - \lambda I)\mathbf{v} = \mathbf{0}

, where

I

is the identity matrix.^[65]^[66] Eigenvalues are determined by solving the characteristic equation

\det(A - \lambda I) = 0

, which yields a polynomial equation of degree

n

for an

n \times n

matrix

A

, producing

n

eigenvalues (counting multiplicities). The roots of this characteristic polynomial, also known as the characteristic roots, fully characterize the scaling factors associated with the linear transformation represented by

A

.^[65]^[66] Key properties of eigenvalues include the fact that their sum equals the trace of

A

(the sum of its diagonal entries), and their product equals the determinant of

A

. The algebraic multiplicity of an eigenvalue

\lambda

is the number of times it appears as a root of the characteristic polynomial, while the geometric multiplicity is the dimension of the corresponding eigenspace, defined as the nullspace of

A - \lambda I

. For diagonalizability, the geometric multiplicity must equal the algebraic multiplicity for each eigenvalue.^[65]^[66] Eigenvalues play a central role in applications such as stability analysis in linear dynamical systems, where eigenvalues with positive real parts indicate exponential growth or instability, while those with negative real parts indicate decay toward equilibrium. They also enable diagonalization: if

A

has a full set of linearly independent eigenvectors, it can be expressed as

A = P D P^{-1}

, where

D

is a diagonal matrix with eigenvalues on the diagonal and

P

contains the eigenvectors, simplifying computations like matrix powers.^[65]^[67] For example, consider the matrix

A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}.

The characteristic equation is

\det\begin{pmatrix} 2 - \lambda & 1 \\ 0 & 3 - \lambda \end{pmatrix} = (2 - \lambda)(3 - \lambda) = 0

, yielding eigenvalues

\lambda = 2

and

\lambda = 3

.^[65]

Expected Value

In probability theory, the expected value of a random variable represents the long-run average value of repetitions of the experiment it represents. For a discrete random variable

X

taking values

x_i

with probabilities

p_i

, the expected value is defined as

E[X] = \sum_i x_i p_i

.^[68] For a continuous random variable

X

with probability density function

f(x)

, it is given by

E[X] = \int_{-\infty}^{\infty} x f(x) \, dx

. This measure provides a central tendency under uncertainty, weighting possible outcomes by their likelihoods.^[69] Expected value exhibits key properties that facilitate computation and analysis. Linearity holds for any random variables

X

and

Y

, and constants

a

and

b

, such that

E[aX + bY] = aE[X] + bE[Y]

.^[70] Additionally, if

X \geq 0

almost surely, then

E[X] \geq 0

, reflecting monotonicity.^[69] Simple examples illustrate these concepts. For a fair six-sided die, where

X

is the outcome and each face has probability

1/6

, the expected value is

E[X] = (1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5

. For a Bernoulli random variable

X

indicating success with probability

p

E[X] = p

.^[71] In applications, expected value informs risk assessment by quantifying average outcomes, such as in insurance where premiums are set near the expected loss to ensure profitability.^[72] The law of large numbers states that the sample mean of independent repetitions converges to the expected value as the number of trials increases, justifying its use in long-run predictions.^[73] A variant is conditional expectation, which refines the average given partial information. For random variables

X

and

Y

E[X \mid Y = y]

is the expected value of

X

under the conditional distribution given

Y = y

.^[74]

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References

Value (mathematics)

Fundamental Concepts

Definition and Scope

Numerical Values

Variables and Expressions

Variables

Expression Evaluation

Functions

Function Values

Domain and Codomain

Specialized Types

Absolute Value

Eigenvalue

Expected Value

References

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Value (mathematics)

Value (mathematics)

Value (mathematics)

Numerical values

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See also

References

Value (mathematics)

Fundamental Concepts

Definition and Scope

Numerical Values

Variables and Expressions

Variables

Expression Evaluation

Functions

Function Values

Domain and Codomain

Specialized Types

Absolute Value

Eigenvalue

Expected Value

References