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72-605: Simplification , Simplify , or Simplified may refer to: Simplification is the process of replacing a mathematical expression by an equivalent one that is simpler (usually shorter), according to a well-founded ordering. Examples include: Expression (mathematics) In mathematics , an expression is a written arrangement of symbols following the context-dependent, syntactic conventions of mathematical notation . Symbols can denote numbers , variables , operations , and functions . Other symbols include punctuation marks and brackets , used for grouping where there

144-445: A i k {\textstyle \sum _{i<k}a_{ik}} , depending on the context, the variable i {\textstyle i} can be free and k {\textstyle k} bound, or vice-versa, but they cannot both be free. Determining which value is assumed to be free depends on context and semantics . An expression is often used to define a function , or denote compositions of funtions, by taking

216-532: A n − 1 x n − 1 {\displaystyle a_{n-1}x^{n-1}} and so on for a total of n ( n + 1 ) 2 {\textstyle {\frac {n(n+1)}{2}}} multiplications and n {\displaystyle n} additions. Using better methods, such as Horner's rule , this can be reduced to n {\displaystyle n} multiplications and n {\displaystyle n} additions. If some preprocessing

288-467: A n − 1 x n − 1 + ⋯ + a 0 , {\textstyle a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{0},} the most naive method would use n {\displaystyle n} multiplications to compute a n x n {\displaystyle a_{n}x^{n}} , use n − 1 {\textstyle n-1} multiplications to compute

360-524: A constant function . 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. The equivalence between two expressions is called an identity and is sometimes denoted with ≡ . {\displaystyle \equiv .} For example, in the expression ∑ n = 1 3 ( 2 n x ) , {\textstyle \sum _{n=1}^{3}(2nx),}

432-402: A group under multiplication. The properties of quadratic residues are widely used in number theory . More generally, in rings, the square function may have different properties that are sometimes used to classify rings. Zero may be the square of some non-zero elements. A commutative ring such that the square of a non zero element is never zero is called a reduced ring . More generally, in

504-415: A rational number ). For example, 3 x − 2 xy + c is an algebraic expression. 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 Square function The square of an integer may also be called a square number or a perfect square . In algebra ,

576-584: A commutative ring, a radical ideal is an ideal  I such that x 2 ∈ I {\displaystyle x^{2}\in I} implies x ∈ I {\displaystyle x\in I} . Both notions are important in algebraic geometry , because of Hilbert's Nullstellensatz . An element of a ring that is equal to its own square is called an idempotent . In any ring, 0 and 1 are idempotents. There are no other idempotents in fields and more generally in integral domains . However,

648-463: A finite number of nodes. Formal languages allow formalizing the concept of well-formed expressions. In the 1930s, a new type of expressions, called lambda expressions , were introduced by Alonzo Church and Stephen Kleene for formalizing functions and their evaluation. They form the basis for lambda calculus , a formal system used in mathematical logic and the theory of programming languages . The equivalence of two lambda expressions

720-492: A function call f(a,b) may first evaluate the arguments a and b , store the results in references or memory locations ref_a and ref_b , then evaluate the function's body with those references passed in. 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

792-811: A kind of formal language , and a well-formed expression can be defined recursively as follows: The alphabet consists of: With this alphabet, the recursive rules for forming a well-formed expression (WFE) are as follows: A well-formed expression can be thought as a syntax tree . The leaf nodes are always atomic expressions. Operations + {\displaystyle +} and ∪ {\displaystyle \cup } have exactly two child nodes, while operations x {\textstyle {\sqrt {x}}} , ln ( x ) {\textstyle {\text{ln}}(x)} and d d x {\textstyle {\frac {d}{dx}}} have exactly one. There are countably infinitely many WFE's, however, each WFE has

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864-413: A root. The real closed fields cannot be distinguished from the field of real numbers by their algebraic properties: every property of the real numbers, which may be expressed in first-order logic (that is expressed by a formula in which the variables that are quantified by ∀ or ∃ represent elements, not sets), is true for every real closed field, and conversely every property of the first-order logic, which

936-710: A similar system evolved, with numbers written in a base-60 ( sexagesimal ) format on clay tablets written in Cuneiform , a technique originating with the Sumerians around 3000 BC. This base-60 system persists today in measuring time and angles . 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 ), with little interest in algebraic symbols, until

1008-405: A specification of the sequence can be omitted. The subtraction operation is non-associative; despite that, there is a convention that a − b − c {\displaystyle a-b-c} is shorthand for ( a − b ) − c {\displaystyle (a-b)-c} , thus it is considered "well-defined". On the other hand, Division

1080-425: A value simultaneously express a condition that is assumed to hold, for instance those involving the operator ⊕ {\displaystyle \oplus } to designate an internal direct sum . In algebra , an expression may be used to designate a value, which might depend on values assigned to variables occurring in the expression. The determination of this value depends on the semantics attached to

1152-507: Is n  times greater. This holds for areas in three dimensions as well as in the plane: for instance, the surface area of a sphere is proportional to the square of its radius, a fact that is manifested physically by the inverse-square law describing how the strength of physical forces such as gravity varies according to distance. The square function is related to distance through the Pythagorean theorem and its generalization,

1224-427: Is undecidable . 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 ). An algebraic expression is an expression built up from algebraic constants , variables , and the algebraic operations ( addition , subtraction , multiplication , division and exponentiation by

1296-458: Is a constant or the product of a constant and one or more variables. Some examples include 7 , 5 x , 13 x 2 y , 4 b {\displaystyle 7,\;5x,\;13x^{2}y,\;4b} The constant of the product is called the coefficient . Terms that are either constants or have the same variables raised to the same powers are called like terms . If there are like terms in an expression, you can simplify

1368-536: Is a relation specifying a rewrite for each object or term, compatible with a given reduction relation. A rewriting strategy specifies, out of all the reducible subterms ( redexes ), which one should be reduced ( contracted ) within a term. One of the most common systems involves lambda calculus . The language of mathematics exhibits a kind of grammar (called formal grammar ) about how expressions may be written. There are two considerations for well-definedness of mathematical expressions, syntax and semantics . Syntax

1440-505: Is about attaching meaning to expressions. An expression that defines a unique value or meaning is said to be well-defined . Otherwise, the expression is said to be ill defined or ambiguous. In general the meaning of expressions is not limited to designating values; for instance, an expression might designate a condition, or an equation that is to be solved, or it can be viewed as an object in its own right that can be manipulated according to certain rules. Certain expressions that designate

1512-409: Is allowed, even more savings are possible. A computation is any type of arithmetic or non-arithmetic calculation that is "well-defined". The notion that mathematical statements should be 'well-defined' had been argued by mathematicians since at least the 1600s , but agreement on a suitable definition proved elusive. A candidate definition was proposed independently by several mathematicians in

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1584-644: Is an expression, while the inequality 8 x − 5 ≥ 3 {\displaystyle 8x-5\geq 3} is a formula. To evaluate an expression means to find a numerical value equivalent to the expression. Expressions can be evaluated or simplified by replacing operations that appear in them with their result. For example, the expression 8 × 2 − 5 {\displaystyle 8\times 2-5} simplifies to 16 − 5 {\displaystyle 16-5} , and evaluates to 11. {\displaystyle 11.} An expression

1656-464: 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. Expressions that are syntactically correct are called well-formed . Semantics is concerned with the meaning of these well-formed expressions. Expressions that are semantically correct are called well-defined . The syntax of mathematical expressions can be described somewhat informally as follows:

1728-409: Is non-associative, and in the case of a / b / c {\displaystyle a/b/c} , parenthesization conventions are not well established; therefore, this expression is often considered ill-defined. Unlike with functions, notational ambiguities can be overcome by means of additional definitions (e.g., rules of precedence , associativity of the operator). For example, in

1800-420: Is not a well-defined order of operations . Expressions are commonly distinguished from formulas : expressions are a kind of mathematical object , whereas formulas are statements about mathematical objects. This is analogous to natural language , where a noun phrase refers to an object, and a whole sentence refers to a fact . For example, 8 x − 5 {\displaystyle 8x-5}

1872-416: Is often used to define a function , by taking the variables to be arguments , or inputs, of the function, and assigning the output to be the evaluation of the resulting expression. For example, x ↦ x 2 + 1 {\displaystyle x\mapsto x^{2}+1} and f ( x ) = x 2 + 1 {\displaystyle f(x)=x^{2}+1} define

1944-403: Is part of the language of mathematics , that is to say, it is not defined within mathematics, but taken as a primitive part of the language. To attempt to define the term would not be doing mathematics, but rather, one would be engaging in a kind of metamathematics (the metalanguage of mathematics), usually mathematical logic . Within mathematical logic, mathematics is usually described as

2016-426: Is strictly negative. Zero is the square of only one number, itself. For this reason, it is possible to define the square root function, which associates with a non-negative real number the non-negative number whose square is the original number. No square root can be taken of a negative number within the system of real numbers , because squares of all real numbers are non-negative . The lack of real square roots for

2088-648: 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 . Ancient Egypt developed a symbolic system using hieroglyphics , assigning symbols for powers of ten and using addition and subtraction symbols resembling legs in motion. This system, recorded in texts like the Rhind Mathematical Papyrus (c. 2000–1800 BC), influenced other Mediterranean cultures . In Mesopotamia ,

2160-418: Is the set of nonnegative real numbers. The square function preserves the order of positive numbers: larger numbers have larger squares. In other words, the square is a monotonic function on the interval [0, +∞) . On the negative numbers, numbers with greater absolute value have greater squares, so the square is a monotonically decreasing function on (−∞,0] . Hence, zero is the (global) minimum of

2232-419: Is true for a specific real closed field is also true for the real numbers. There are several major uses of the square function in geometry. The name of the square function shows its importance in the definition of the area : it comes from the fact that the area of a square with sides of length   l is equal to l . The area depends quadratically on the size: the area of a shape n  times larger

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2304-408: Is unambiguous because ( a × b ) × c = a × ( b × c ) {\displaystyle (a\times b)\times c=a\times (b\times c)} ; hence the notation is said to be well defined . This property, also known as associativity of multiplication, guarantees the result does not depend on the sequence of multiplications; therefore,

2376-406: Is zero. It is easier to compute than the absolute value (no square root), and is a smooth real-valued function . Because of these two properties, the absolute square is often preferred to the absolute value for explicit computations and when methods of mathematical analysis are involved (for example optimization or integration ). For complex vectors , the dot product can be defined involving

2448-405: The absolute value of a complex number is called its absolute square , squared modulus , or squared magnitude . It is the product of the complex number with its complex conjugate , and equals the sum of the squares of the real and imaginary parts of the complex number. The absolute square of a complex number is always a nonnegative real number, that is zero if and only if the complex number

2520-462: The complex number field with quadratic form x + y , and then doubling again to obtain quaternions. The doubling procedure is called the Cayley–Dickson construction , and has been generalized to form algebras of dimension 2 over a field F with involution. The square function z is the "norm" of the composition algebra C {\displaystyle \mathbb {C} } , where

2592-406: The conjugate transpose , leading to the squared norm . Squares are ubiquitous in algebra, more generally, in almost every branch of mathematics, and also in physics where many units are defined using squares and inverse squares: see below . Least squares is the standard method used with overdetermined systems . Squaring is used in statistics and probability theory in determining

2664-464: The execution of computer algorithms . A calculation is a deliberate mathematical process that transforms one or more inputs into one or more outputs or results . For example, multiplying 7 by 6 is a simple algorithmic calculation. Extracting the square root or the cube root of a number using mathematical models is a more complex algorithmic calculation. Expressions can be computed by means of an evaluation strategy . To illustrate, executing

2736-406: The parallelogram law . Euclidean distance is not a smooth function : the three-dimensional graph of distance from a fixed point forms a cone , with a non-smooth point at the tip of the cone. However, the square of the distance (denoted d or r ), which has a paraboloid as its graph, is a smooth and analytic function . The dot product of a Euclidean vector with itself is equal to

2808-406: The standard deviation of a set of values, or a random variable . The deviation of each value  x i from the mean   x ¯ {\displaystyle {\overline {x}}} of the set is defined as the difference x i − x ¯ {\displaystyle x_{i}-{\overline {x}}} . These deviations are squared, then

2880-476: The 1930s. The best-known variant was formalised by the mathematician Alan Turing , who defined a well-defined statement or calculation as any statement that could be expressed in terms of the initialisation parameters of a Turing machine . Turing's definition apportioned "well-definedness" to a very large class of mathematical statements, including all well-formed algebraic statements , and all statements written in modern computer programming languages. Despite

2952-716: The 7th century, Brahmagupta used different colours to represent the unknowns in algebraic equations in the Brāhmasphuṭasiddhānta . Greek and other ancient mathematical advances, were often trapped in cycles of bursts of creativity, followed by long periods of stagnation, but this began to change as knowledge spread in the early modern period . 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 . The plus sign (+) appeared around 1351 with Nicole Oresme , likely derived from

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3024-555: The Latin et (meaning "and"), while the minus sign (−) was first used in 1489 by Johannes Widmann . 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. François Viète ’s New Algebra (1591) formalized modern symbolic manipulation. The multiplication sign (×)

3096-408: The addition operation. In a totally ordered ring , x ≥ 0 for any x . Moreover, x = 0  if and only if  x = 0 . In a supercommutative algebra where 2 is invertible, the square of any odd element equals zero. If A is a commutative semigroup , then one has In the language of quadratic forms , this equality says that the square function is a "form permitting composition". In fact,

3168-473: 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. Strings of symbols that conform to the rules of syntax are called well-formed , and those that are not well-formed are called, ill-formed , and are do not constitute mathematical expressions. For example, in arithmetic ,

3240-481: The arrival of Diophantus of Alexandria , who pioneered a form of syncopated algebra in his Arithmetica , which introduced symbolic manipulation of expressions. His notation represented unknowns and powers symbolically, but without modern symbols for relations (such as equality or inequality ) or exponents . An unknown number was called ζ {\displaystyle \zeta } . The square of ζ {\displaystyle \zeta }

3312-636: The expression 4 x 2 + 8 {\displaystyle 4x^{2}+8} ; it can be evaluated at x = 3 in the following steps: 4 ( 3 ) 2 + 3 {\textstyle 4(3)^{2}+3} , (replace x with 3) 4 ⋅ ( 3 ⋅ 3 ) + 8 {\displaystyle 4\cdot (3\cdot 3)+8} (use definition of exponent ) 4 ⋅ 9 + 8 {\displaystyle 4\cdot 9+8} (simplify) 36 + 8 {\displaystyle 36+8} 44 {\displaystyle 44} A term

3384-411: The expression 1 + 2 × 3 is well-formed, but is not. However, being well-formed is not enough to be considered well-defined. For example in arithmetic, the expression 1 0 {\textstyle {\frac {1}{0}}} is well-formed, but it is not well-defined. (See Division by zero ). Such expressions are called undefined . Semantics is the study of meaning. Formal semantics

3456-429: The expression by combining the like terms. We add the coefficients and keep the same variable. 4 x + 7 x + 2 x = 15 x {\displaystyle 4x+7x+2x=15x} Any variable can be classified as being either a free variable or a bound variable . For a given combination of values for the free variables, an expression may be evaluated, although for some combinations of values of

3528-407: The former case, polynomials are evaluated using floating-point arithmetic , which is not exact. Thus different schemes for the evaluation will, in general, give slightly different answers. In the latter case, the polynomials are usually evaluated in a finite field , in which case the answers are always exact. For evaluating the univariate polynomial a n x n +

3600-470: The free variables, the value of the expression may be undefined . Thus an expression represents an operation over constants and free variables and whose output is the resulting value of the expression. For a non-formalized language, that is, in most mathematical texts outside of mathematical logic , for an individual expression it is not always possible to identify which variables are free and bound. For example, in ∑ i < k

3672-404: The function that associates to each number its square plus one. An expression with no variables would define a constant function . Usually, two expressions are considered equal or equivalent if they define the same function. Such an equality is called a " semantic equality", that is, both expressions "mean the same thing." A formal expression is a kind of string of symbols , created by

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3744-469: The identity function forms a trivial involution to begin the Cayley–Dickson constructions leading to bicomplex, biquaternion, and bioctonion composition algebras. On complex numbers , the square function z → z 2 {\displaystyle z\to z^{2}} is a twofold cover in the sense that each non-zero complex number has exactly two square roots. The square of

3816-491: The late 17th century, with Leibniz's notation becoming the standard. In elementary algebra , a variable in an expression is a letter that represents a number whose value may change. To evaluate an expression with a variable means to find the value of the expression when the variable is assigned a given number. Expressions can be evaluated or simplified by replacing operations that appear in them with their result, or by combining like-terms . For example, take

3888-420: The negative numbers can be used to expand the real number system to the complex numbers , by postulating the imaginary unit i , which is one of the square roots of −1. The property "every non-negative real number is a square" has been generalized to the notion of a real closed field , which is an ordered field such that every non-negative element is a square and every polynomial of odd degree has

3960-418: The numbers modulo an odd prime number p . A non-zero element of this field is called a quadratic residue if it is a square in Z / p Z , and otherwise, it is called a quadratic non-residue. Zero, while a square, is not considered to be a quadratic residue. Every finite field of this type has exactly ( p − 1)/2 quadratic residues and exactly ( p − 1)/2 quadratic non-residues. The quadratic residues form

4032-414: The operation of squaring is often generalized to polynomials , other expressions , or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial x + 1 is the quadratic polynomial ( x + 1) = x + 2 x + 1 . One of the important properties of squaring, for numbers as well as in many other mathematical systems, is that (for all numbers x ),

4104-433: The operations of addition , subtraction , multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms. The problem of polynomial evaluation arises frequently in practice. In computational geometry , polynomials are used to compute function approximations using Taylor polynomials . In cryptography and hash tables , polynomials are used to compute k -independent hashing . In

4176-403: The programming language C , the operator - for subtraction is left-to-right-associative , which means that a-b-c is defined as (a-b)-c , and the operator = for assignment is right-to-left-associative , which means that a=b=c is defined as a=(b=c) . In the programming language APL there is only one rule: from right to left – but parentheses first. The term 'expression'

4248-440: The references. This is the call-by-reference evaluation strategy. Evaluation strategy is part of the semantics of the programming language definition. Some languages, such as PureScript , have variants with different evaluation strategies. Some declarative languages , such as Datalog , support multiple evaluation strategies. Some languages define a calling convention . In rewriting , a reduction strategy or rewriting strategy

4320-408: The ring of the integers modulo   n has 2 idempotents, where k is the number of distinct prime factors of  n . A commutative ring in which every element is equal to its square (every element is idempotent) is called a Boolean ring ; an example from computer science is the ring whose elements are binary numbers , with bitwise AND as the multiplication operation and bitwise XOR as

4392-502: The same production rules as standard expressions, however, they are used without regard to the meaning of the expression. In this way, two formal expressions are considered equal only if they are syntactically equal, that is, if they are the exact same expression. For instance, the formal expressions "2" and "1+1" are not equal. The earliest written mathematics likely began with tally marks , where each mark represented one unit, carved into wood or stone. An example of early counting

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4464-430: The square function is the foundation upon which other quadratic forms are constructed which also permit composition. The procedure was introduced by L. E. Dickson to produce the octonions out of quaternions by doubling. The doubling method was formalized by A. A. Albert who started with the real number field R {\displaystyle \mathbb {R} } and the square function, doubling it to obtain

4536-404: The square function. The square x of a number x is less than x (that is x < x ) if and only if 0 < x < 1 , that is, if x belongs to the open interval (0,1) . This implies that the square of an integer is never less than the original number x . Every positive real number is the square of exactly two numbers, one of which is strictly positive and the other of which

4608-410: The square of x is the same as the square of its additive inverse − x . That is, the square function satisfies the identity x = (− x ) . This can also be expressed by saying that the square function is an even function . The squaring operation defines a real function called the square function or the squaring function . Its domain is the whole real line , and its image

4680-403: The square of its length: v ⋅ v = v . This is further generalised to quadratic forms in linear spaces via the inner product . The inertia tensor in mechanics is an example of a quadratic form. It demonstrates a quadratic relation of the moment of inertia to the size ( length ). There are infinitely many Pythagorean triples , sets of three positive integers such that the sum of

4752-409: The squares of the first two equals the square of the third. Each of these triples gives the integer sides of a right triangle. The square function is defined in any field or ring . An element in the image of this function is called a square , and the inverse images of a square are called square roots . The notion of squaring is particularly important in the finite fields Z / p Z formed by

4824-423: The symbols of the expression. The choice of semantics depends on the context of the expression. The same syntactic expression 1 + 2 × 3 can have different values (mathematically 7, but also 9), depending on the order of operations implied by the context (See also Operations § Calculators ). For real numbers , the product a × b × c {\displaystyle a\times b\times c}

4896-627: The variable n is bound, and the variable x is free. This expression is equivalent to the simpler expression 12 x ; that is ∑ n = 1 3 ( 2 n x ) ≡ 12 x . {\displaystyle \sum _{n=1}^{3}(2nx)\equiv 12x.} The value for x = 3 is 36, which can be denoted ∑ n = 1 3 ( 2 n x ) | x = 3 = 36. {\displaystyle \sum _{n=1}^{3}(2nx){\Big |}_{x=3}=36.} A polynomial consists of variables and coefficients , that involve only

4968-480: The variables to be arguments , or inputs, of the function, and assigning the output to be the evaluation of the resulting expression. For example, x ↦ x 2 + 1 {\displaystyle x\mapsto x^{2}+1} and f ( x ) = x 2 + 1 {\displaystyle f(x)=x^{2}+1} define the function that associates to each number its square plus one. An expression with no variables would define

5040-562: The widespread uptake of this definition, there are some mathematical concepts that have no well-defined characterisation under this definition. This includes the halting problem and the busy beaver game . 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. All statements characterised in modern programming languages are well-defined, including C++ , Python , and Java . Common examples of computation are basic arithmetic and

5112-718: Was Δ v {\displaystyle \Delta ^{v}} ; the cube was K v {\displaystyle K^{v}} ; the fourth power was Δ v Δ {\displaystyle \Delta ^{v}\Delta } ; and the fifth power was Δ K v {\displaystyle \Delta K^{v}} . So for example, what would be written in modern notation as: x 3 − 2 x 2 + 10 x − 1 , {\displaystyle x^{3}-2x^{2}+10x-1,} Would be written in Diophantus's syncopated notation as: In

5184-545: Was first used by William Oughtred and the division sign (÷) by Johann Rahn . René Descartes further advanced algebraic symbolism in La Géométrie (1637), where he introduced the use of letters at at the end of the alphabet (x, y, z) for variables , along with the Cartesian coordinate system , which bridged algebra and geometry. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus in

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