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System of arithmetic in proof theory
logic, elementary function arithmetic (EFA), also called elementary arithmetic and exponential function arithmetic, is the system of arithmetic with the
Elementary function arithmetic
Elementary_function_arithmetic
Function whose domain is the positive integers
ln(x) or loge(x). In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function whose domain is the set of positive
Arithmetic_function
Type of mathematical function
elementary function is a function of a single variable (real or complex) that is typically encountered by beginners. The basic elementary functions are
Elementary_function
Numbers and the basic operations on them
Elementary arithmetic is a branch of mathematics involving addition, subtraction, multiplication, and division. Due to its low level of abstraction, broad
Elementary_arithmetic
Concept in computability theory
elementary was originally introduced by László Kalmár in the context of computability theory. He defined the class of elementary recursive functions ("Kalmár
Elementary_recursive_function
Function defined on integers in number theory
In number theory, the Lagarias arithmetic derivative or number derivative is a function defined for integers, based on prime factorization, by analogy
Arithmetic_derivative
Type of zeta function
mathematics, the arithmetic zeta function is a zeta function associated with a scheme of finite type over integers. The arithmetic zeta function generalizes
Arithmetic_zeta_function
Topics referred to by the same term
Libraries Elementary abelian group, an abelian group in which every nontrivial element is of prime order Elementary algebra Elementary arithmetic Elementary charge
Elementary
Method for bounding the errors of numerical computations
errors in mathematical computation by computing function bounds. Numerical methods involving interval arithmetic can guarantee relatively reliable and mathematically
Interval_arithmetic
Criterion for integration in terms of elementary functions
fields Elementary function – Type of mathematical function Elementary function arithmetic – System of arithmetic in proof theory Liouvillian function – Elementary
Liouville's theorem (differential algebra)
Liouville's_theorem_(differential_algebra)
outline is provided as an overview of and topical guide to arithmetic: Arithmetic is an elementary branch of mathematics that deals with numerical operations
Outline_of_arithmetic
Mathematical function of two positive real arguments
geometric means. The arithmetic–geometric mean is used in fast algorithms for exponential, trigonometric functions, and other special functions, as well as some
Arithmetic–geometric_mean
Mathematical technique used in proof theory
rudimentary function arithmetic. IΔ0, arithmetic with induction on Δ0-predicates without any axiom asserting that exponentiation is total. EFA, elementary function
Ordinal_analysis
Proof that only uses basic techniques
Theorem is not elementary. However, there are other simple statements about arithmetic such as the existence of iterated exponential functions that cannot
Elementary_proof
Multiplicative function in number theory
the OEIS). In number theory another arithmetic function closely related to the Möbius function is the Mertens function, defined by M ( n ) = ∑ k = 1 n μ
Möbius_function
Arithmetic function related to the divisors of an integer
theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number
Divisor_function
Formalization of the natural numbers
{-}}|x-y|=0} . Elementary recursive arithmetic Finite-valued logic Heyting arithmetic Peano arithmetic Primitive recursive function Robinson arithmetic Second-order
Primitive recursive arithmetic
Primitive_recursive_arithmetic
Digit transferred from one column to another
In elementary arithmetic, a carry is a digit that is transferred from one column of digits to another column of more significant digits. It is part of
Carry_(arithmetic)
Analytic function that does not satisfy a polynomial equation
involving the basic arithmetic operations. This definition can be extended to functions of several variables. The transcendental functions sine and cosine
Transcendental_function
Decidable first-order theory of the natural numbers with addition
The language of Presburger arithmetic contains constants 0 {\displaystyle 0} and 1 {\displaystyle 1} and a binary function + {\displaystyle +} , interpreted
Presburger_arithmetic
Number of partitions of an integer
is credited with discovering that the partition function has nontrivial patterns in modular arithmetic. For instance the number of partitions is divisible
Partition function (number theory)
Partition_function_(number_theory)
17th-century conjecture proved by Andrew Wiles in 1994
Fermat's last theorem) can be proved using only elementary function arithmetic, such a proof need be 'elementary' only in a technical sense and could involve
Fermat's_Last_Theorem
Topics referred to by the same term
military European Fighter Aircraft, now the Eurofighter Typhoon Elementary function arithmetic Essential fatty acid Exploratory factor analysis Egyptian Football
EFA
Branch of mathematical logic
comprehension can be defined. The weak system RCA* 0 consists of elementary function arithmetic EFA (the basic axioms plus Δ0 0 induction in the enriched language
Reverse_mathematics
Mathematical formula involving a given set of operations
variables, and a set of functions considered as basic and connected by arithmetic operations (+, −, ×, /, and integer powers) and function composition. Commonly
Closed-form_expression
Number of integers coprime to and less than n
primitive dth roots of unity. The formula can also be derived from elementary arithmetic. For example, let n = 20 and consider the positive fractions up
Euler's_totient_function
3-volume treatise on mathematics, 1910–1913
(there exists); predicate symbol: "=" (equals); function symbols: "+" (arithmetic addition), "∙" (arithmetic multiplication), " ′ " (successor); individual
Principia_Mathematica
Number taken as representative of a list of numbers
statistics Taylor's law Arithmetic-geometric mean Arithmetic-harmonic mean Cesàro mean Chisini mean Contraharmonic mean Elementary symmetric mean Geometric-harmonic
Average
Branch of pure mathematics
of mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of
Number_theory
Branch of elementary mathematics
Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a
Arithmetic
types of functions Elementary functions are functions built from basic operations (e.g. addition, exponentials, logarithms...) Algebraic functions are functions
List of mathematical functions
List_of_mathematical_functions
Elementary functions and their finitely iterated integrals
Liouvillian functions comprise a set of functions including the elementary functions and their repeated integrals. Liouvillian functions can be recursively
Liouvillian_function
Computer arithmetic standards
Part 1: Integer and floating point arithmetic, second edition published 2012. Part 2: Elementary numerical functions, first edition published 2001. Part
ISO/IEC_10967
Computation modulo a fixed integer
In mathematics, modular arithmetic is a system of arithmetic operations for integers, differing from the usual ones in that numbers "wrap around" when
Modular_arithmetic
Theories in mathematical logic
fragments of Peano arithmetic. The case n = 1 has about the same strength as primitive recursive arithmetic (PRA). Exponential function arithmetic (EFA) is IΣ0
List_of_first-order_theories
Arithmetic operation
denoted with the plus sign +, is one of the four basic operations of arithmetic, the other three being subtraction, multiplication, and division. The
Addition
Class of mathematical expression
the dividend (numerator). The usual definition of the quotient in elementary arithmetic is the number which yields the dividend when multiplied by the divisor
Division_by_zero
arithmetic function is some simpler or better-understood function which takes the same values "on average". Let f {\displaystyle f} be an arithmetic function
Average order of an arithmetic function
Average_order_of_an_arithmetic_function
Integers have unique prime factorizations
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every
Fundamental theorem of arithmetic
Fundamental_theorem_of_arithmetic
Concept in mathematics
replaces each arithmetic operation or elementary function call in the formula by a call to the corresponding AA library routine. For smooth functions, the approximation
Affine_arithmetic
S-shaped curve
A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with the equation f ( x ) = L 1 + e − k ( x − x 0 ) {\displaystyle f(x)={\frac
Logistic_function
Axioms for the natural numbers
define the arithmetical properties of the natural numbers. The naturals are assumed to be closed under a single-valued "successor" function S. For every
Peano_axioms
Association of one output to each input
a combination of arithmetic operations and previously defined functions; such a formula allows computing the value of the function from the value of
Function_(mathematics)
Basic concepts of algebra
(fractional power). The operations of elementary algebra may be performed on numbers, in which case they are often called arithmetic operations. They may also be
Elementary_algebra
Number divisible only by 1 and itself
there are arbitrarily long finite arithmetic progressions consisting only of primes. Euler noted that the function n2−n+41 yields prime numbers for 1≤n≤40
Prime_number
IEEE standard for floating-point arithmetic
numbers during arithmetic and conversions operations: arithmetic and other operations (such as trigonometric functions) on arithmetic formats exception
IEEE_754
Computer approximation for real numbers
In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a significand (a signed sequence of a fixed number of
Floating-point_arithmetic
value property). Also subharmonic function and superharmonic function. Elementary function: composition of arithmetic operations, exponentials, logarithms
List_of_types_of_functions
Mathematical function, denoted exp(x) or e^x
In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted
Exponential_function
Mathematical system
In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative
Second-order_arithmetic
Theorem on the number of primes in arithmetic sequences
rigorous proof.) Selberg, Atle (1949). "An Elementary Proof of Dirichlet's Theorem About Primes in an Arithmetic Progression". Annals of Mathematics. 50
Dirichlet's theorem on arithmetic progressions
Dirichlet's_theorem_on_arithmetic_progressions
Logarithm to the base of the mathematical constant e
ln(x) is undefined at 0, the function itself does not have a Maclaurin series, unlike many other elementary functions. Instead, one looks for Taylor
Natural_logarithm
Replacing a number with a simpler value
when dividing two numbers in integer or fixed-point arithmetic; when computing mathematical functions such as square roots, logarithms, and sines; or when
Rounding
Numerical calculations carrying along derivatives
executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.) and elementary functions (exp, log, sin, cos
Automatic_differentiation
Axiomatic set theories based on the principles of mathematical constructivism
of arithmetics include elementary function arithmetic E F A {\displaystyle {\mathsf {EFA}}} , which includes induction for just bounded arithmetical formulas
Constructive_set_theory
Generalization of the Riemann zeta function for algebraic number fields
functional equation. Values of Dedekind zeta functions encode important arithmetic data of K. The Dedekind zeta function is named for Richard Dedekind, who introduced
Dedekind_zeta_function
Mathematical function that can be computed by a program
first-order Peano arithmetic). A function that can be proven to be computable is called provably total. The set of provably total functions is recursively
Computable_function
Limitative results in mathematical logic
which contains elementary arithmetic. Then F⊬Cons(F)." Franzén 2005, p. 106. Shelah 1974. S. G. Simpson, Subsystems of Second-Order Arithmetic (2009). Perspectives
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Mathematical function characterizing set membership
generalization of the inverse of the indicator function in elementary number theory, the Möbius function. (See paragraph below about the use of the inverse in
Indicator_function
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
Axiomatic set theory devised by W.V.O. Quine
exponentiation is total. 𝖤𝖥𝖠 see elementary function arithmetic. 𝖯𝖠 see Peano arithmetic. 𝖹2 see Second-order arithmetic. Alternative set theory Axiomatic
New_Foundations
Arithmetic operation
Division is one of the four basic operations of arithmetic. The other operations are addition, subtraction, and multiplication. What is being divided is
Division_(mathematics)
Set of all true first-order statements about the arithmetic of natural numbers
multiplication. The signature of Peano arithmetic includes the addition, multiplication, and successor function symbols, the equality and less-than relation
True_arithmetic
Branch of algebraic geometry
Rational points can be directly characterized by height functions which measure their arithmetic complexity. The structure of algebraic varieties defined
Arithmetic_geometry
Gödel's β function is a function used to permit quantification over finite sequences of natural numbers in formal theories of arithmetic. The β function is used
Gödel's_β_function
Overview of and topical guide to discrete mathematics
of arithmetic – Integers have unique prime factorizations Modular arithmetic – Computation modulo a fixed integer Successor function – Elementary operation
Outline of discrete mathematics
Outline_of_discrete_mathematics
Mathematical problem
12 modulo isomorphism. Elementary function – Type of mathematical function Elementary function arithmetic – System of arithmetic in proof theory Liouville's
Tarski's high school algebra problem
Tarski's_high_school_algebra_problem
Algorithmic runtime requirements for common math procedures
in Borwein & Borwein. The elementary functions are constructed by composing arithmetic operations, the exponential function ( exp {\displaystyle \exp
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
Symbolic description of a mathematical object
define the same function; such an equality is called a "semantic equality", that is, both expressions "mean the same thing." In elementary algebra, a variable
Expression_(mathematics)
Number raised to the third power
In arithmetic and algebra, the cube of a number n is its third power, that is, the result of multiplying three instances of n together. The cube of a number
Cube_(algebra)
Theorem that arithmetical truth cannot be defined in arithmetic
formal semantics. Informally, the theorem states that "arithmetical truth cannot be defined in arithmetic". The theorem applies more generally to any sufficiently
Tarski's undefinability theorem
Tarski's_undefinability_theorem
Number which when multiplied by x equals 1
reciprocal function, the function f(x) that maps x to 1 x , {\displaystyle {\tfrac {1}{x}},} is one of the simplest examples of a function which is its
Multiplicative_inverse
Characterization of how many integers are prime
PNT. One possible definition of an "elementary" proof is "one that can be carried out in first-order Peano arithmetic." There are number-theoretic statements
Prime_number_theorem
Function returning one of only two values
function", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Janković, Dragan; Stanković, Radomir S.; Moraga, Claudio (November 2003). "Arithmetic expressions
Boolean_function
Theorem about natural numbers
theorem is unprovable in Peano arithmetic (but it can be proven in stronger systems, such as second-order arithmetic or Zermelo–Fraenkel set theory)
Goodstein's_theorem
Non-contradiction of a theory
falsity, there is no contradiction in general. In theories of arithmetic, such as Peano arithmetic, there is an intricate relationship between the consistency
Consistency
Function computable with bounded loops
ISBN 978-1-107-04348-0 Thoralf Skolem (1923) "The foundations of elementary arithmetic" in Jean van Heijenoort, translator and ed. (1967) From Frege to
Primitive_recursive_function
Conjecture on zeros of the zeta function
every arithmetic scheme or a scheme of finite type over integers. The arithmetic zeta function of a regular connected equidimensional arithmetic scheme
Riemann_hypothesis
Concept in model theory
non-standard models of Peano arithmetic, which contain other objects than just the numbers 0, 1, 2, etc., and yet are elementarily equivalent to the standard
Elementary_equivalence
Function that preserves distinctness
In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct
Injective_function
Mathematical-logic system
for arithmetic. (In Church's original lambda calculus, the formal parameter of a lambda expression was required to occur at least once in the function body
Lambda_calculus
Number used for counting
principles of arithmetic presented by a new method (Latin: Arithmetices principia, nova methodo exposita). This approach is now called Peano arithmetic. It is
Natural_number
Algebra theorem about convex functions
the majorization inequality, is a theorem in elementary algebra for convex and concave real-valued functions, defined on an interval of the real line. It
Karamata's_inequality
Axiomatic logical system
In mathematics, Robinson arithmetic is a finitely axiomatized fragment of first-order Peano arithmetic (PA), first set out by Raphael M. Robinson in 1950
Robinson_arithmetic
Function in mathematical logic
expressed as a function on the natural numbers. If f is the Gödel mapping and r is an inference rule, then there should be some arithmetical function gr of natural
Gödel_numbering
Field of mathematics
or rational function. A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures. Global arithmetic dynamics is
Arithmetic_dynamics
Number that, when added to the original number, yields the additive identity
opposite number, or the negative of a number. The unary operation of arithmetic negation is closely related to subtraction and is important in solving
Additive_inverse
Possible axiom for set theory in mathematics
Observations Concerning Elementary Extensions of ω-models. II (1973, p.227). Accessed 2021 November 3. W. Marek, ω-models of second-order arithmetic and admissible
Axiom_of_constructibility
Differentiating positive and negative zero
Signed zero is zero with an associated sign. In ordinary arithmetic, the number 0 does not have a sign, and −0, +0 and 0 are three ways of writing the
Signed_zero
Impossible task in computing
real or rational arithmetic can be decided using the simplex algorithm, formulas in linear integer arithmetic (Presburger arithmetic) can be decided using
Entscheidungsproblem
Mathematical table
traditionally taught as an essential part of elementary arithmetic around the world, as it lays the foundation for arithmetic operations with base-ten numbers. Many
Multiplication_table
Algebraic manipulation of "true" and "false"
denoted as ∨, and negation (not) denoted as ¬. Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction
Boolean_algebra
Mathematical concept
prime-counting function π0(x) which is related to the prime-counting function π(x) by π0(x)=12limh→0[π(x+h)+π(x−h)], which takes the arithmetic mean of the
Explicit formulae for L-functions
Explicit_formulae_for_L-functions
Subfield of mathematics
provably total function in intuitionistic arithmetic is computable; this is not true in classical theories of arithmetic such as Peano arithmetic. Algebraic
Mathematical_logic
One of the four basic arithmetic operations
Subtraction (which is signified by the minus sign, −) is one of the four arithmetic operations along with addition, multiplication and division. Subtraction
Subtraction
Exponential function of an exponential function
A double exponential function is a constant raised to the power of an exponential function. The general formula is f ( x ) = a b x = a ( b x ) {\displaystyle
Double_exponential_function
Type of logical system
topic, such as set theory, a theory for groups, or a formal theory of arithmetic, is usually a first-order logic together with a specified domain of discourse
First-order_logic
Inverse functions of sin, cos, tan, etc.
(3rd ed.). Berlin: J. Springer. Translated as Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis. Translated by Hedrick, E
Inverse trigonometric functions
Inverse_trigonometric_functions
Mathematical logic concept
Gerhard Gentzen in 1936. It shows that the Peano axioms of first-order arithmetic do not contain a contradiction (i.e. are "consistent"), as long as a certain
Gentzen's_consistency_proof
French computer scientist (born 1961)
rounding of elementary functions, in particular on the Table Maker's Dilemma, and for having authored two reference textbooks: Elementary Functions: algorithms
Jean-Michel_Muller
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ELEMENTARY FUNCTION-ARITHMETIC
ELEMENTARY FUNCTION-ARITHMETIC
ELEMENTARY FUNCTION-ARITHMETIC
ELEMENTARY FUNCTION-ARITHMETIC
ELEMENTARY FUNCTION-ARITHMETIC
ELEMENTARY FUNCTION-ARITHMETIC
ELEMENTARY FUNCTION-ARITHMETIC
ELEMENTARY FUNCTION-ARITHMETIC
ELEMENTARY FUNCTION-ARITHMETIC
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