Expression (mathematics)

[1] Other symbols include punctuation marks and brackets, used for grouping where there is not a well-defined order of operations.

An example of early counting is the Ishango bone, found near the Nile and dating back over 20,000 years ago, which is thought to show a six-month lunar calendar.

[7][8] This system, recorded in texts like the Rhind Mathematical Papyrus (c. 2000–1800 BC), influenced other Mediterranean cultures.

The "syncopated" stage of mathematics introduced symbolic abbreviations for commonly used operations and quantities, marking a shift from purely geometric reasoning.

Ancient Greek mathematics, largely geometric in nature, drew on Egyptian numerical systems (especially Attic numerals),[9] with little interest in algebraic symbols, until the arrival of Diophantus of Alexandria,[10] who pioneered a form of syncopated algebra in his Arithmetica, which introduced symbolic manipulation of expressions.

Would be written in Diophantus's syncopated notation as: In the 7th century, Brahmagupta used different colours to represent the unknowns in algebraic equations in the Brāhmasphuṭasiddhānta.

The transition to fully symbolic algebra began with Ibn al-Banna' al-Marrakushi (1256–1321) and Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī, (1412–1482) who introduced symbols for operations using Arabic characters.

[19] Luca Pacioli included these symbols in his works, though much was based on earlier contributions by Piero della Francesca.

The radical symbol (√) for square root was introduced by Christoph Rudolff in the 1500s, and parentheses for precedence by Niccolò Tartaglia in 1556.

René Descartes further advanced algebraic symbolism in La Géométrie (1637), where he introduced the use of letters at the end of the alphabet (x, y, z) for variables, along with the Cartesian coordinate system, which bridged algebra and geometry.

In elementary algebra, a variable in an expression is a letter that represents a number whose value may change.

Expressions can be evaluated or simplified by replacing operations that appear in them with their result, or by combining like-terms.

In this way, two expressions are said to be equivalent if, for each combination of values for the free variables, they have the same output, i.e., they represent the same function.

A polynomial consists of variables and coefficients, that involve only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms.

[27] The notion that mathematical statements should be 'well-defined' had been argued by mathematicians since at least the 1600s,[28] but agreement on a suitable definition proved elusive.

It remains an open question as to whether there exists a more powerful definition of 'well-defined' that is able to capture both computable and 'non-computable' statements.

[a][33] All statements characterised in modern programming languages are well-defined, including C++, Python, and Java.

A calculation is a deliberate mathematical process that transforms one or more inputs into one or more outputs or results.

This gives the function the ability to look up the original argument values passed in through dereferencing the parameters (some languages use specific operators to perform this), to modify them via assignment as if they were local variables, and to return values via the references.

Syntax is concerned with the rules used for constructing, or transforming the symbols of an expression without regard to any interpretation or meaning given to them.

The syntax of mathematical expressions can be described somewhat informally as follows: the allowed operators must have the correct number of inputs in the correct places (usually written with infix notation), the sub-expressions that make up these inputs must be well-formed themselves, have a clear order of operations, etc.

Unlike with functions, notational ambiguities can be overcome by means of additional definitions (e.g., rules of precedence, associativity of the operator).

This is also the case for the expressions representing real numbers, which are built from the integers by using the arithmetical operations, the logarithm and the exponential (Richardson's theorem).

Since taking the square root is the same as raising to the power ⁠1/2⁠, the following is also an algebraic expression: See also: Algebraic equation and Algebraic closure A polynomial expression is an expression built with scalars (numbers of elements of some field), indeterminates, and the operators of addition, multiplication, and exponentiation to nonnegative integer powers; for example

[43] It is a combination of one or more constants, variables, functions, and operators that the programming language interprets (according to its particular rules of precedence and of association) and computes to produce ("to return", in a stateful environment) another value.

In simple settings, the resulting value is usually one of various primitive types, such as string, Boolean, or numerical (such as integer, floating-point, or complex).

Except for numbers and variables, every mathematical expression may be viewed as the symbol of an operator followed by a sequence of operands.

This representation is very flexible, and many things that seem not to be mathematical expressions at first glance, may be represented and manipulated as such.

An expression formed by applying a predicate symbol to an appropriate number of terms is called an atomic formula, which evaluates to true or false in bivalent logics, given an interpretation.

A discourse on the method of correctly conducting one's reason and seeking truth in the sciences.