Algebraic expression, which is expressed by numbers andletterAfter finite addition, subtraction, multiplication, divisionPowerAnd prescription, etcAlgebraic operationAcquiredFormula, or mathematical expressions containing letters are called algebraic expressions.For example:ax+2b，－2/3，b^2/26，√a+√ 2 etc.
Algebraic expression is a common analytic expression. Analytic expressions limited to finite algebraic operations (addition, subtraction, multiplication, division, power, square root) for variable number denominators are called algebraic expressions, such as
All are algebraic expressions. A single number or letter is also called algebraic expression.
2. Can haveabsolute value。For example:|x|, | -2.25 | etc.
matters needing attention
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The following two points should be noted about the classification of algebraic expressions:
1. To be classified according to the initial form given by the algebraic expression, for example
Although it can be reduced to
, but it is still fractional;Another example
Although it can be reduced to xtwo, but it is still irrational.
2. The classification shall be based on the operation implemented on the specified variable letter.For example, for variable number x, the formula
Is a rational formula
It is irrational.
development
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The formation and development of the concept of algebraic expression has gone through a long historical process. In the 13th century,Fibonacci(Fibonacci, L.) began to use letters to represent operands, but has not yet used symbols,Weida(Viete, F.) introduced the system of mathematical symbols between 1584 and 1589, making algebra the theory of equations, so people generally believe that he is the founder of algebraic expressions,Descartes(Descartes, R.) improved Veda's letter usage, using the letters a, b, c,... in the front of the Latin alphabet to represent known numbers, and some letters x, y, z,... at the end to represent unknown numbers,Leibniz(Leibniz, G, W.) carried out systematic research on various symbolic notation, developed and improved the representation of algebraic expressions.[1]
Rational expressions includeIntegral form（DivisorNone inletterRational expression of) andfraction(There are letters in the divisor and the divisor is not 0).In this algebraic expression, letters can only be added, subtracted, multiplied, divided and summed for a limited number of timesintegersecondPowerThese operations.
Integers include monomials (the product of numbers or letters, or a single number or letter) and polynomials (the sum of several monomials).
The algebraic sum of several monomials is called polynomial;Each single term in a polynomial is called a polynomial term.ExcludingletterThe item of is calledConstant term。
Degree of polynomial: the degree of the highest degree item in the polynomial is the degree of the polynomial.Homogeneous polynomial: Polynomials with the same degree are called homogeneous polynomials.
Irreducible polynomial: Rational with degree greater than zerocoefficientThe polynomial of is called irreducible polynomial in the range of rational number when it cannot be decomposed into the product of two rational number coefficient polynomials whose degree is greater than zero.real numberIrreducible polynomials in the range are first order or some quadratic polynomials,complexThe irreducible polynomial within the norm is a polynomial of degree one.
Symmetric polynomial: onMultivariate polynomialIn, if the result of any two elements exchanged with each other is the same as the original formula, then this polynomial is said to be a symmetric polynomial about these elements.
Cognate term: A term with the same letter and the same exponent of the same letter in a polynomial is called a homogeneous term.
Irrational formula
We put theRadical, non letterintegersecondPowerOr the formula with non algebraic operations is called irrational.Irrational form includes radical form and transcendental form.We call an algebraic expression that can be reduced to a rational form and whose root index does not contain letters a radical form.
Figure 1
We call rational form and radical form algebraic form, and irrational form other than radical formTranscendental。
Writing format
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(1) When multiplying two letters, numbers and letters, letters and parentheses, and parentheses and parentheses, the multiplication sign can be omitted. For example, "the product of x and y" can be written as "xy";"The product of a and 2" should be written as "2a", and "2 times the sum of m and n" should be written as "2 (m+n)".
(2) When multiplying letters by numbers or numbers by brackets, the multiplication sign can be omitted, but numbers must be written in front. For example, "x × 2" should be written as "2x", not "x2";"The perimeter of a rectangle whose length and width are a and b respectively" should be written as "2 (a+b)", not "(a+b) 2".
(3) The division sign cannot appear in the algebraic expression, and the division relationship should be written in the form of fraction
(4) When multiplying numbers, the multiplication sign (can also be written as ·) should still be reserved and cannot be omitted, or the result can be calculated directly. For example, "3 × 7xy" cannot be written as "37xy", but should be written as "21xy".
Operation of number expression
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Merge Similar Items:polynomialThe consolidation of the same category items into one item is called consolidation of the same category items.The rule for merging similar items is:coefficientAdd them together, and the result is taken as a coefficient. Letters and letter indices remain unchanged.
Bracket removal rule: The parenthesis is preceded by the "+" sign. Remove the parenthesis and the "+" sign in front of it, and all items in the parenthesis remain unchanged;The "-" sign is in front of the bracket. Remove the bracket and the "-" sign in front of it, and all items in the bracket will change their symbols.
Parenthesis rule: After the bracket is added, the "+" sign is in front of the bracket, and all items enclosed in the bracket remain unchanged;After adding parentheses, the sign "-" is in front of the parentheses,
Everything enclosed in brackets changes the symbol.
There is no doubt that algebra evolved from arithmetic.As for when it came into beingAlgebraIt is difficult to explain this subject clearly.For example, if you think "algebra" refers to the skill of solving algebraic equations represented by symbols such as bx+k=0.This "algebra" was developed only in the sixteenth century.
If we do not require algebraic symbols to be as concise as they are now, then algebra can be traced back to an earlier era.Westerners regard Diao Fan, an ancient Greek mathematician in the third century BC, as the ancestor of algebra.In China, algebraic problems expressed in words appeared earlier.
"Algebra", as a proper noun of mathematics and representing a branch of mathematics, was officially used in China as early as 1859.In that year, among mathematicians in the Qing DynastyLi ShanlanAnd the BritishVeoliaLi co translated a book written by an Englishman, Dimegan, whose name is Algebra.Of course, the contents and methods of algebra have long been produced in ancient China, such as《Chapter Nine Arithmetic》InequationQuestion.
elementary algebra The central content ofalgebraic equationTherefore, for a long timeAlgebraAs the science of algebraic equations, mathematicians also focus on the study of algebraic equations.itsresearch methodIt is highly computational.
To discuss algebraic equations, one of the first problems encountered is how to list the algebraic expressions with unknowns in the actual quantitative relations, and thenEquimetric relationList algebraic equations.So an important content of elementary algebra is algebraic expression.Due to the different quantitative relations in things, elementary algebra has formedIntegral form、fractionandRadicalThese three kinds of algebraic expressions.Algebraic expressions are the incarnation of numbers, so in algebra, they can beFour arithmetic operations, obeyBasic operationLaw, and can also be used toPowerAnd square root.These six operations are usually called algebraic operations to distinguish them from onlycontainOf four operationsArithmetic operation。
stayelementary algebra In the process of generation and development of, through the study of algebraic equations, it also promoted the further development of the concept of numberarithmeticDiscussed inintegerThe concept of sum fraction is extended to the range of rational numbers, so that numbers include positive and negative integers, positive and negative fractions, and zero.This is another important content of elementary algebra, which is the extension of the concept of number.
With rational numbers, the problems that elementary algebra can solve are greatly expanded.But somealgebraic equationThere is still no solution in the rational range.Thus, the concept of number was expanded toreal number, and further expanded tocomplex。
Is there still an algebraic equation that has not been solved in the complex number range, and the complex number must be expanded again?Mathematicians said: No, no.This is a famous theorem in algebra——Fundamental theorem of algebra。This theorem is simplynSub equation hasnRoot (s).December 15, 1742 Swiss mathematicianEulerHe made a clear statement in a letter, and then another mathematician, Gauss of Germany, gave a strict proof in 1799.
